



Internet Research Task Force (IRTF)                          D. Bourdrez
Request for Comments: 9807                                              
Category: Informational                                      H. Krawczyk
ISSN: 2070-1721                                                      AWS
                                                                 K. Lewi
                                                                    Meta
                                                              C. A. Wood
                                                        Cloudflare, Inc.
                                                               July 2025


    The OPAQUE Augmented Password-Authenticated Key Exchange (aPAKE)
                                Protocol

Abstract

   This document describes the OPAQUE protocol, an Augmented (or
   Asymmetric) Password-Authenticated Key Exchange (aPAKE) protocol that
   supports mutual authentication in a client-server setting without
   reliance on PKI and with security against pre-computation attacks
   upon server compromise.  In addition, the protocol provides forward
   secrecy and the ability to hide the password from the server, even
   during password registration.  This document specifies the core
   OPAQUE protocol and one instantiation based on 3DH.  This document is
   a product of the Crypto Forum Research Group (CFRG) in the IRTF.

Status of This Memo

   This document is not an Internet Standards Track specification; it is
   published for informational purposes.

   This document is a product of the Internet Research Task Force
   (IRTF).  The IRTF publishes the results of Internet-related research
   and development activities.  These results might not be suitable for
   deployment.  This RFC represents the consensus of the Crypto Forum
   Research Group of the Internet Research Task Force (IRTF).  Documents
   approved for publication by the IRSG are not candidates for any level
   of Internet Standard; see Section 2 of RFC 7841.

   Information about the current status of this document, any errata,
   and how to provide feedback on it may be obtained at
   https://www.rfc-editor.org/info/rfc9807.

Copyright Notice

   Copyright (c) 2025 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents
   (https://trustee.ietf.org/license-info) in effect on the date of
   publication of this document.  Please review these documents
   carefully, as they describe your rights and restrictions with respect
   to this document.

Table of Contents

   1.  Introduction
     1.1.  Requirements Notation
     1.2.  Notation
   2.  Cryptographic Dependencies
     2.1.  Oblivious Pseudorandom Function
     2.2.  Key Derivation Function and Message Authentication Code
     2.3.  Hash Functions
   3.  Protocol Overview
     3.1.  Setup
     3.2.  Registration
     3.3.  Online Authenticated Key Exchange
   4.  Client Credential Storage and Key Recovery
     4.1.  Key Recovery
       4.1.1.  Envelope Structure
       4.1.2.  Envelope Creation
       4.1.3.  Envelope Recovery
   5.  Registration
     5.1.  Registration Messages
     5.2.  Registration Functions
       5.2.1.  CreateRegistrationRequest
       5.2.2.  CreateRegistrationResponse
       5.2.3.  FinalizeRegistrationRequest
   6.  Online Authenticated Key Exchange
     6.1.  AKE Messages
     6.2.  AKE Functions
       6.2.1.  GenerateKE1
       6.2.2.  GenerateKE2
       6.2.3.  GenerateKE3
       6.2.4.  ServerFinish
     6.3.  Credential Retrieval
       6.3.1.  Credential Retrieval Messages
       6.3.2.  Credential Retrieval Functions
     6.4.  3DH Protocol
       6.4.1.  3DH Key Exchange Functions
       6.4.2.  Key Schedule Functions
       6.4.3.  3DH Client Functions
       6.4.4.  3DH Server Functions
   7.  Configurations
   8.  Application Considerations
   9.  Implementation Considerations
     9.1.  Implementation Safeguards
     9.2.  Handling Online Guessing Attacks
     9.3.  Error Considerations
   10. Security Considerations
     10.1.  Notable Design Differences
     10.2.  Security Analysis
     10.3.  Identities
     10.4.  Export Key Usage
     10.5.  Static Diffie-Hellman Oracles
     10.6.  Random-Key Robust MACs
     10.7.  Input Validation
     10.8.  OPRF Key Stretching
     10.9.  Client Enumeration
     10.10. Protecting the Registration Masking Key
     10.11. Password Salt and Storage Implications
     10.12. AKE Private Key Storage
     10.13. Client Authentication Using Credentials
   11. IANA Considerations
   12. References
     12.1.  Normative References
     12.2.  Informative References
   Appendix A.  Alternate Key Recovery Mechanisms
   Appendix B.  Alternate AKE Instantiations
     B.1.  HMQV Instantiation Sketch
     B.2.  SIGMA-I Instantiation Sketch
   Appendix C.  Test Vectors
     C.1.  Real Test Vectors
       C.1.1.  OPAQUE-3DH Real Test Vector 1
       C.1.2.  OPAQUE-3DH Real Test Vector 2
       C.1.3.  OPAQUE-3DH Real Test Vector 3
       C.1.4.  OPAQUE-3DH Real Test Vector 4
       C.1.5.  OPAQUE-3DH Real Test Vector 5
       C.1.6.  OPAQUE-3DH Real Test Vector 6
     C.2.  Fake Test Vectors
       C.2.1.  OPAQUE-3DH Fake Test Vector 1
       C.2.2.  OPAQUE-3DH Fake Test Vector 2
       C.2.3.  OPAQUE-3DH Fake Test Vector 3
   Acknowledgments
   Authors' Addresses

1.  Introduction

   Password authentication is ubiquitous in many applications.  In a
   common implementation, a client authenticates to a server by sending
   its client ID and password to the server over a secure connection.
   This makes the password vulnerable to server mishandling, including
   accidentally logging the password or storing it in plaintext in a
   database.  Server compromise resulting in access to these plaintext
   passwords is not an uncommon security incident, even among security-
   conscious organizations.  Moreover, plaintext password authentication
   over secure channels such as TLS is also vulnerable in cases where
   TLS may fail, including PKI attacks, certificate mishandling,
   termination outside the security perimeter, visibility to TLS-
   terminating intermediaries, and more.

   Augmented (or Asymmetric) Password Authenticated Key Exchange (aPAKE)
   protocols are designed to provide password authentication and
   mutually authenticated key exchange in a client-server setting
   without relying on PKI (except during client registration) and
   without disclosing passwords to servers or other entities other than
   the client machine.  A secure aPAKE should provide the best possible
   security for a password protocol.  Indeed, some attacks are
   inevitable, such as online impersonation attempts with guessed client
   passwords and offline dictionary attacks upon the compromise of a
   server and leakage of its credential file.  In the latter case, the
   attacker learns a mapping of a client's password under a one-way
   function and uses such a mapping to validate potential guesses for
   the password.  It is crucially important for the password protocol to
   use an unpredictable one-way mapping.  Otherwise, the attacker can
   pre-compute a deterministic list of mapped passwords leading to
   almost instantaneous leakage of passwords upon server compromise.

   This document describes OPAQUE, an aPAKE protocol that is secure
   against pre-computation attacks (as defined in [JKX18]).  OPAQUE
   provides forward secrecy with respect to password leakage while also
   hiding the password from the server, even during password
   registration.  OPAQUE allows applications to increase the difficulty
   of offline dictionary attacks via iterated hashing or other key-
   stretching schemes.  OPAQUE is also extensible, allowing clients to
   safely store and retrieve arbitrary application data on servers using
   only their password.

   OPAQUE is defined and proven as the composition of three
   functionalities: an Oblivious Pseudorandom Function (OPRF), a key
   recovery mechanism, and an authenticated key exchange (AKE) protocol.
   It can be seen as a "compiler" for transforming any suitable AKE
   protocol into a secure aPAKE protocol.  (See Section 10 for
   requirements of the OPRF and AKE protocols.)  This document specifies
   one OPAQUE instantiation based on [TripleDH].  Other instantiations
   are possible, as discussed in Appendix B, but their details are out
   of scope for this document.  In general, the modularity of OPAQUE's
   design makes it easy to integrate with additional AKE protocols,
   e.g., TLS or HMQV (Hashed Menezes-Qu-Vanstone), and with future AKE
   protocols such as those based on post-quantum techniques.

   OPAQUE consists of two stages: registration and authenticated key
   exchange.  In the first stage, a client registers its password with
   the server and stores information used to recover authentication
   credentials on the server.  Recovering these credentials can only be
   done with knowledge of the client password.  In the second stage, a
   client uses its password to recover those credentials and
   subsequently uses them as input to an AKE protocol.  This stage has
   additional mechanisms to prevent an active attacker from interacting
   with the server to guess or confirm clients registered via the first
   phase.  Servers can use this mechanism to safeguard registered
   clients against this type of enumeration attack; see Section 10.9 for
   more discussion.

   The name "OPAQUE" is a homonym of O-PAKE, where O is for Oblivious.
   The name "OPAKE" was taken.

   This document complies with the requirements for PAKE protocols set
   forth in [RFC8125].  This document represents the consensus of the
   Crypto Forum Research Group (CFRG).  It is not an IETF product and is
   not a standard.

1.1.  Requirements Notation

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
   "OPTIONAL" in this document are to be interpreted as described in BCP
   14 [RFC2119] [RFC8174] when, and only when, they appear in all
   capitals, as shown here.

1.2.  Notation

   The following functions are used throughout this document:

   I2OSP and OS2IP:  Convert a byte string to and from a non-negative
      integer as described in Section 4 of [RFC8017].  Note that these
      functions operate on byte strings in big-endian byte order.

   concat(x0, ..., xN):  Concatenate byte strings.  For example,
      concat(0x01, 0x0203, 0x040506) = 0x010203040506.

   random(n):  Generates a cryptographically secure pseudorandom byte
      string of length n bytes.

   zeroes(n):  Generate a string of n bytes all equal to 0 (zero).

   xor(a,b):  Apply XOR to byte strings.  For example, xor(0xF0F0,
      0x1234) = 0xE2C4.  It is an error to call this function with
      arguments of unequal length.

   ct_equal(a, b):  Return true if a is equal to b, and false otherwise.
      The implementation of this function must be constant-time in the
      length of a and b, which are assumed to be of equal length,
      irrespective of the values a or b.

   Except if said otherwise, random choices in this specification refer
   to drawing with uniform distribution from a given set (i.e., "random"
   is short for "uniformly random").  Random choices can be replaced
   with fresh outputs from a cryptographically strong pseudorandom
   generator, according to the requirements in [RFC4086], or a
   pseudorandom function.  For convenience, we define nil as a lack of
   value.

   All protocol messages and structures defined in this document use the
   syntax from Section 3 of [RFC8446].

2.  Cryptographic Dependencies

   OPAQUE depends on the following cryptographic protocols and
   primitives:

   *  Oblivious Pseudorandom Function (OPRF); Section 2.1

   *  Key Derivation Function (KDF); Section 2.2

   *  Message Authentication Code (MAC); Section 2.2

   *  Cryptographic Hash Function; Section 2.3

   *  Key Stretching Function (KSF); Section 2.3

   This section describes these protocols and primitives in more detail.
   Unless said otherwise, all random nonces and seeds used in these
   dependencies and the rest of the OPAQUE protocol are of length Nn and
   Nseed bytes, respectively, where Nn = Nseed = 32.

2.1.  Oblivious Pseudorandom Function

   An Oblivious Pseudorandom Function (OPRF) is a two-party protocol
   between client and server for computing a Pseudorandom Function
   (PRF), where the PRF key is held by the server and the input to the
   function is provided by the client.  The client does not learn
   anything about the PRF other than the obtained output, and the server
   learns nothing about the client's input or the function output.  This
   specification depends on the prime-order OPRF construction specified
   as modeOPRF (0x00) from Section 3.1 of [RFC9497].

   The following OPRF client APIs are used:

   Blind(element):  Create and output (blind, blinded_element),
      consisting of a blinded representation of input element, denoted
      blinded_element, along with a value to revert the blinding
      process, denoted blind.  This is equivalent to the Blind function
      described in Section 3.3.1 of [RFC9497].

   Finalize(element, blind, evaluated_element):  Finalize the OPRF
      evaluation using input element, random inverter blind, and
      evaluation output evaluated_element, yielding output oprf_output.
      This is equivalent to the Finalize function described in
      Section 3.3.1 of [RFC9497].

   Moreover, the following OPRF server APIs are used:

   BlindEvaluate(k, blinded_element):  Evaluate blinded input
      blinded_element using input key k, yielding output element
      evaluated_element.  This is equivalent to the BlindEvaluate
      function described in Section 3.3.1 of [RFC9497], where k is the
      private key parameter.

   DeriveKeyPair(seed, info):  Create and output (sk, pk), consisting of
      a private and public key derived deterministically from an input
      seed and input info parameter, as described in Section 3.2 of
      [RFC9497].

   Finally, this specification makes use of the following shared APIs
   and parameters:

   SerializeElement(element):  Map input element to a fixed-length byte
      array.

   DeserializeElement(buf):  Attempt to map input byte array buf to an
      OPRF group element.  This function can raise a DeserializeError
      upon failure; see Section 2.1 of [RFC9497] for more details.

   Noe:  The size of a serialized OPRF group element output from
      SerializeElement.

   Nok:  The size of an OPRF private key as output from DeriveKeyPair.

2.2.  Key Derivation Function and Message Authentication Code

   A Key Derivation Function (KDF) is a function that takes some source
   of initial keying material and uses it to derive one or more
   cryptographically strong keys.  This specification uses a KDF with
   the following API and parameters:

   Extract(salt, ikm):  Extract a pseudorandom key of fixed length Nx
      bytes from input keying material ikm and an optional byte string
      salt.

   Expand(prk, info, L):  Expand a pseudorandom key prk, using the
      string info, into L bytes of output keying material.

   Nx:  The output size of the Extract() function in bytes.

   This specification also makes use of a random-key robust Message
   Authentication Code (MAC).  See Section 10.6 for more details on this
   property.  The API and parameters for the random-key robust MAC are
   as follows:

   MAC(key, msg):  Compute a message authentication code over input msg
      with key key, producing a fixed-length output of Nm bytes.

   Nm:  The output size of the MAC() function in bytes.

2.3.  Hash Functions

   This specification makes use of a collision-resistant hash function
   with the following API and parameters:

   Hash(msg):  Apply a cryptographic hash function to input msg,
      producing a fixed-length digest of size Nh bytes.

   Nh:  The output size of the Hash() function in bytes.

   This specification makes use of a Key Stretching Function (KSF),
   which is a slow and expensive cryptographic hash function with the
   following API:

   Stretch(msg):  Apply a key stretching function to stretch the input
      msg and harden it against offline dictionary attacks.  This
      function also needs to satisfy collision resistance.

3.  Protocol Overview

   OPAQUE consists of two stages: registration and authenticated key
   exchange (AKE).  In the first stage, a client registers its password
   with the server and stores its credential file on the server.  In the
   second stage (also called the "login" or "online" stage), the client
   recovers its authentication material and uses it to perform a
   mutually authenticated key exchange.

3.1.  Setup

   Prior to both stages, the client and server agree on a configuration
   that fully specifies the cryptographic algorithm dependencies
   necessary to run the protocol; see Section 7 for details.  The server
   chooses a pair of keys (server_private_key and server_public_key) for
   the AKE protocol and chooses a seed (oprf_seed) of Nh bytes for the
   OPRF.  The server can use server_private_key and server_public_key
   with multiple clients.  The server can also opt to use a different
   seed for each client (i.e., each client can be assigned a single
   seed), so long as they are maintained across the registration and
   online AKE stages and kept consistent for each client (since an
   inconsistent mapping of clients to seeds could leak information as
   described in Section 10.9).

3.2.  Registration

   Registration is the only stage in OPAQUE that requires a server-
   authenticated channel with confidentiality and integrity: either
   physical, out-of-band, PKI-based, etc.

   The client inputs its credentials, which include its password and
   user identifier, and the server inputs its parameters, which include
   its private key and other information.

   The client output of this stage is a single value export_key that the
   client may use for application-specific purposes, e.g., as a
   symmetric key used to encrypt additional information for storage on
   the server.  The server does not have access to this export_key.

   The server output of this stage is a record corresponding to the
   client's registration that it stores in a credential file alongside
   other clients registrations as needed.

   The registration flow is shown in Figure 1, and the process is
   described in more detail in Section 5:

     credentials                               parameters
         |                                         |
         v                                         v
       Client                                    Server
       ------------------------------------------------
                   registration request
                ------------------------->
                   registration response
                <-------------------------
                         record
                ------------------------->
       ------------------------------------------------
         |                                         |
         v                                         v
     export_key                                 record

                                  Figure 1

   These messages are named RegistrationRequest, RegistrationResponse,
   and RegistrationRecord, respectively.  Their contents and wire format
   are defined in Section 5.1.

3.3.  Online Authenticated Key Exchange

   In this second stage, a client obtains credentials previously
   registered with the server, recovers private key material using the
   password, and subsequently uses them as input to the AKE protocol.
   As in the registration phase, the client inputs its credentials,
   including its password and user identifier, and the server inputs its
   parameters and the credential file record corresponding to the
   client.  The client outputs two values, an export_key (matching that
   from registration) and a session_key, the latter of which is the
   primary AKE protocol output.  The server outputs a single value
   session_key that matches that of the client.  Upon completion,
   clients and servers can use these values as needed.

   The authenticated key exchange flow is shown in Figure 2:

     credentials                          (parameters, record)
         |                                         |
         v                                         v
       Client                                    Server
       ------------------------------------------------
                      AKE message 1
                ------------------------->
                      AKE message 2
                <-------------------------
                      AKE message 3
                ------------------------->
       ------------------------------------------------
         |                                         |
         v                                         v
   (export_key, session_key)                  session_key

                                  Figure 2

   These messages are named KE1, KE2, and KE3, respectively.  They carry
   the messages of the concurrent execution of the key recovery process
   (OPRF) and the authenticated key exchange (AKE).  Their corresponding
   wire formats are specified in Section 6.1.

   The rest of this document describes the specifics of these stages in
   detail.  Section 4 describes how client credential information is
   generated, encoded, and stored on the server during registration and
   recovered during login.  Section 5 describes the first registration
   stage of the protocol, and Section 6 describes the second
   authentication stage of the protocol.  Section 7 describes how to
   instantiate OPAQUE using different cryptographic dependencies and
   parameters.

4.  Client Credential Storage and Key Recovery

   OPAQUE makes use of a structure called Envelope to manage client
   credentials.  The client creates its Envelope on registration and
   sends it to the server for storage.  On every login, the server sends
   this Envelope to the client so it can recover its key material for
   use in the AKE.

   Applications may pin key material to identities if desired.  If no
   identity is given for a party, its value MUST default to its public
   key.  The following types of application credential information are
   considered:

   client_private_key:  The encoded client private key for the AKE
      protocol.

   client_public_key:  The encoded client public key for the AKE
      protocol.

   server_public_key:  The encoded server public key for the AKE
      protocol.

   client_identity:  The client identity.  This is an application-
      specific value, e.g., an email address or an account name.  If not
      specified, it defaults to the client's public key.

   server_identity:  The server identity.  This is typically a domain
      name, e.g., example.com.  If not specified, it defaults to the
      server's public key.  See Section 10.3 for information about this
      identity.

   A subset of these credential values are used in the
   CleartextCredentials structure as follows:

   struct {
     uint8 server_public_key[Npk];
     uint8 server_identity<1..2^16-1>;
     uint8 client_identity<1..2^16-1>;
   } CleartextCredentials;

   The function CreateCleartextCredentials constructs a
   CleartextCredentials structure given application credential
   information.

   CreateCleartextCredentials

   Input:
   - server_public_key, the encoded server public key
     for the AKE protocol.
   - client_public_key, the encoded client public key
     for the AKE protocol.
   - server_identity, the optional encoded server identity.
   - client_identity, the optional encoded client identity.

   Output:
   - cleartext_credentials, a CleartextCredentials structure.

   def CreateCleartextCredentials(server_public_key, client_public_key,
                                  server_identity, client_identity):
     # Set identities as public keys if no
     # application-layer identity is provided
     if server_identity == nil
       server_identity = server_public_key
     if client_identity == nil
       client_identity = client_public_key

     cleartext_credentials = CleartextCredentials {
       server_public_key,
       server_identity,
       client_identity
     }

     return cleartext_credentials

4.1.  Key Recovery

   This specification defines a key recovery mechanism that uses the
   stretched OPRF output as a seed to directly derive the private and
   public keys using the DeriveDiffieHellmanKeyPair() function defined
   in Section 6.4.1.

4.1.1.  Envelope Structure

   The key recovery mechanism defines its Envelope as follows:

   struct {
     uint8 envelope_nonce[Nn];
     uint8 auth_tag[Nm];
   } Envelope;

   envelope_nonce:  A randomly sampled nonce of length Nn used to
      protect this Envelope.

   auth_tag:  An authentication tag protecting the contents of the
      Envelope, covering envelope_nonce and CleartextCredentials.

4.1.2.  Envelope Creation

   Clients create an Envelope at registration with the function Store
   defined below.  Note that DeriveDiffieHellmanKeyPair in this function
   can fail with negligible probability.  If this occurs, servers should
   re-run the function, sampling a new envelope_nonce, to completion.

   Store

   Input:
   - randomized_password, a randomized password.
   - server_public_key, the encoded server public key for
     the AKE protocol.
   - server_identity, the optional encoded server identity.
   - client_identity, the optional encoded client identity.

   Output:
   - envelope, the client's Envelope structure.
   - client_public_key, the client's AKE public key.
   - masking_key, an encryption key used by the server with the
     sole purpose of defending against client enumeration attacks.
   - export_key, an additional client key.

   def Store(randomized_password, server_public_key,
             server_identity, client_identity):
     envelope_nonce = random(Nn)
     masking_key = Expand(randomized_password, "MaskingKey", Nh)
     auth_key =
       Expand(randomized_password, concat(envelope_nonce, "AuthKey"),
              Nh)
     export_key =
       Expand(randomized_password, concat(envelope_nonce, "ExportKey"),
              Nh)
     seed =
       Expand(randomized_password, concat(envelope_nonce, "PrivateKey"),
              Nseed)
     (_, client_public_key) = DeriveDiffieHellmanKeyPair(seed)

     cleartext_credentials =
       CreateCleartextCredentials(server_public_key, client_public_key,
                                  server_identity, client_identity)
     auth_tag =
       MAC(auth_key, concat(
         envelope_nonce,
         server_public_key,
         I2OSP(len(cleartext_credentials.server_identity), 2),
         cleartext_credentials.server_identity,
         I2OSP(len(cleartext_credentials.client_identity), 2),
         cleartext_credentials.client_identity
       ))

     envelope = Envelope {
       envelope_nonce,
       auth_tag
     }

     return (envelope, client_public_key, masking_key, export_key)

4.1.3.  Envelope Recovery

   Clients recover their Envelope during login with the Recover function
   defined below.

   Recover

   Input:
   - randomized_password, a randomized password.
   - server_public_key, the encoded server public key for the
     AKE protocol.
   - envelope, the client's Envelope structure.
   - server_identity, the optional encoded server identity.
   - client_identity, the optional encoded client identity.

   Output:
   - client_private_key, the encoded client private key for the
     AKE protocol.
   - cleartext_credentials, a CleartextCredentials structure.
   - export_key, an additional client key.

   Exceptions:
   - EnvelopeRecoveryError, the Envelope fails to be recovered.

   def Recover(randomized_password, server_public_key, envelope,
               server_identity, client_identity):
     auth_key =
       Expand(randomized_password, concat(envelope.nonce, "AuthKey"),
              Nh)
     export_key =
       Expand(randomized_password, concat(envelope.nonce, "ExportKey"),
              Nh)
     seed =
       Expand(randomized_password, concat(envelope.nonce, "PrivateKey"),
              Nseed)
     (client_private_key, client_public_key) =
       DeriveDiffieHellmanKeyPair(seed)

     cleartext_credentials =
       CreateCleartextCredentials(server_public_key, client_public_key,
                                  server_identity, client_identity)
     expected_tag =
       MAC(auth_key, concat(envelope.nonce, cleartext_credentials))
     If !ct_equal(envelope.auth_tag, expected_tag)
       raise EnvelopeRecoveryError
     return (client_private_key, cleartext_credentials, export_key)

   In the case of EnvelopeRecoveryError being raised, all previously
   computed intermediary values in this function MUST be deleted.

5.  Registration

   The registration process proceeds as follows.  The client inputs the
   following values:

   password:  The client's password.

   creds:  The client credentials as described in Section 4.

   The server inputs the following values:

   server_public_key:  The server public key for the AKE protocol.

   credential_identifier:  A unique identifier for the client's
      credential generated by the server.

   client_identity:  The optional client identity as described in
      Section 4.

   oprf_seed:  A seed used to derive per-client OPRF keys.

   The registration protocol then runs as shown below:

     Client                                         Server
    ------------------------------------------------------
    (request, blind) = CreateRegistrationRequest(password)

                           request
                 ------------------------->

    response = CreateRegistrationResponse(request,
                                          server_public_key,
                                          credential_identifier,
                                          oprf_seed)

                           response
                 <-------------------------

    (record, export_key) = FinalizeRegistrationRequest(password,
                                                       blind,
                                                       response,
                                                       server_identity,
                                                       client_identity)

                           record
                 ------------------------->

   Section 5.1 describes the formats for the above messages, and
   Section 5.2 describes details of the functions and the corresponding
   parameters referenced above.

   At the end of this interaction, the server stores the record object
   as the credential file for each client along with the associated
   credential_identifier and client_identity (if different).  Note that
   the values oprf_seed and server_private_key from the server's setup
   phase must also be persisted.  The oprf_seed value SHOULD be used for
   all clients; see Section 10.9 for the justification behind this,
   along with a description of the exception in which applications may
   choose to avoid the use of a global oprf_seed value across clients
   and instead sample OPRF keys uniquely for each client.  The
   server_private_key may be unique for each client.

   Both client and server MUST validate the other party's public key
   before use.  See Section 10.7 for more details.  Upon completion, the
   server stores the client's credentials for later use.  Moreover, the
   client MAY use the output export_key for further application-specific
   purposes; see Section 10.4.

5.1.  Registration Messages

   This section contains definitions of the RegistrationRequest,
   RegistrationResponse, and RegistrationRecord messages exchanged
   between client and server during registration.

   struct {
     uint8 blinded_message[Noe];
   } RegistrationRequest;

   blinded_message:  A serialized OPRF group element.

   struct {
     uint8 evaluated_message[Noe];
     uint8 server_public_key[Npk];
   } RegistrationResponse;

   evaluated_message:  A serialized OPRF group element.

   server_public_key:  The server's encoded public key that will be used
      for the online AKE stage.

   struct {
     uint8 client_public_key[Npk];
     uint8 masking_key[Nh];
     Envelope envelope;
   } RegistrationRecord;

   client_public_key:  The client's encoded public key corresponding to
      the private key client_private_key.

   masking_key:  An encryption key used by the server with the sole
      purpose of defending against client enumeration attacks.

   envelope:  The client's Envelope structure.

5.2.  Registration Functions

   This section contains definitions of the functions used by client and
   server during registration, including CreateRegistrationRequest,
   CreateRegistrationResponse, and FinalizeRegistrationRequest.

5.2.1.  CreateRegistrationRequest

   To begin the registration flow, the client executes the following
   function.  This function can fail with an InvalidInputError error
   with negligible probability.  A different input password is necessary
   in the event of this error.

   CreateRegistrationRequest

   Input:
   - password, an opaque byte string containing the client's password.

   Output:
   - request, a RegistrationRequest structure.
   - blind, an OPRF scalar value.

   Exceptions:
   - InvalidInputError, when Blind fails

   def CreateRegistrationRequest(password):
     (blind, blinded_element) = Blind(password)
     blinded_message = SerializeElement(blinded_element)
     request = RegistrationRequest {
       blinded_message
     }
     return (request, blind)

5.2.2.  CreateRegistrationResponse

   To process the client's registration request, the server executes the
   following function.  This function can fail with a DeriveKeyPairError
   error with negligible probability.  In this case, applications can
   choose a new credential_identifier for this registration record and
   rerun this function.

   CreateRegistrationResponse

   Input:
   - request, a RegistrationRequest structure.
   - server_public_key, the server's public key.
   - credential_identifier, an identifier that uniquely represents
     the credential.
   - oprf_seed, the seed of Nh bytes used by the server to generate
     an oprf_key.

   Output:
   - response, a RegistrationResponse structure.

   Exceptions:
   - DeserializeError, when OPRF element deserialization fails.
   - DeriveKeyPairError, when OPRF key derivation fails.

   def CreateRegistrationResponse(request, server_public_key,
                                  credential_identifier, oprf_seed):
     seed =
      Expand(oprf_seed, concat(credential_identifier, "OprfKey"), Nok)
     (oprf_key, _) = DeriveKeyPair(seed, "OPAQUE-DeriveKeyPair")

     blinded_element = DeserializeElement(request.blinded_message)
     evaluated_element = BlindEvaluate(oprf_key, blinded_element)
     evaluated_message = SerializeElement(evaluated_element)

     response = RegistrationResponse {
       evaluated_message,
       server_public_key
     }

     return response

5.2.3.  FinalizeRegistrationRequest

   To create the user record used for subsequent authentication and
   complete the registration flow, the client executes the following
   function.

   FinalizeRegistrationRequest

   Input:
   - password, an opaque byte string containing the client's password.
   - blind, an OPRF scalar value.
   - response, a RegistrationResponse structure.
   - server_identity, the optional encoded server identity.
   - client_identity, the optional encoded client identity.

   Output:
   - record, a RegistrationRecord structure.
   - export_key, an additional client key.

   Exceptions:
   - DeserializeError, when OPRF element deserialization fails.

   def FinalizeRegistrationRequest(password, blind, response,
                                   server_identity, client_identity):
     evaluated_element = DeserializeElement(response.evaluated_message)
     oprf_output = Finalize(password, blind, evaluated_element)

     stretched_oprf_output = Stretch(oprf_output)
     randomized_password =
       Extract("", concat(oprf_output, stretched_oprf_output))

     (envelope, client_public_key, masking_key, export_key) =
       Store(randomized_password, response.server_public_key,
             server_identity, client_identity)

     record = RegistrationRecord {
       client_public_key,
       masking_key,
       envelope
     }

     return (record, export_key)

   See Section 6 for details about the output export_key usage.

6.  Online Authenticated Key Exchange

   The generic outline of OPAQUE with a 3-message AKE protocol includes
   three messages: KE1, KE2, and KE3.  KE1 and KE2 include key exchange
   shares (e.g., DH values) sent by the client and server, respectively.
   KE3 provides explicit client authentication and full forward security
   (without it, forward secrecy is only achieved against eavesdroppers,
   which is insufficient for OPAQUE security).

   This section describes the online authenticated key exchange protocol
   flow, message encoding, and helper functions.  This stage is composed
   of a concurrent OPRF and key exchange flow.  The key exchange
   protocol is authenticated using the client and server credentials
   established during registration; see Section 5.  In the end, the
   client proves its knowledge of the password, and both client and
   server agree on (1) a mutually authenticated shared secret key and
   (2) any optional application information exchange during the
   handshake.

   In this stage, the client inputs the following values:

   password:  The client's password.

   client_identity:  The client identity as described in Section 4.

   The server inputs the following values:

   server_private_key:  The server's private key for the AKE protocol.

   server_public_key:  The server's public key for the AKE protocol.

   server_identity:  The server identity as described in Section 4.

   record:  The RegistrationRecord object corresponding to the client's
      registration.

   credential_identifier:  An identifier that uniquely represents the
      credential.

   oprf_seed:  The seed used to derive per-client OPRF keys.

   The client receives two outputs: a session secret and an export key.
   The export key is only available to the client and may be used for
   additional application-specific purposes, as outlined in
   Section 10.4.  Clients MUST NOT use the output export_key before
   authenticating the peer in the authenticated key exchange protocol.
   See Appendix A for more details about this requirement.  The server
   receives a single output: a session secret matching the client's.

   The protocol runs as shown below:

     Client                                         Server
    ------------------------------------------------------
     ke1 = GenerateKE1(password)

                            ke1
                 ------------------------->

     ke2 = GenerateKE2(server_identity, server_private_key,
                       server_public_key, record,
                       credential_identifier, oprf_seed, ke1)

                            ke2
                 <-------------------------

       (ke3,
       session_key,
       export_key) = GenerateKE3(client_identity,
                                  server_identity, ke2)

                            ke3
                 ------------------------->

                          session_key = ServerFinish(ke3)

   Both client and server may use implicit internal state objects to
   keep necessary material for the OPRF and AKE, client_state, and
   server_state, respectively.

   The client state ClientState may have the following fields:

   password:  The client's password.

   blind:  The random blinding inverter returned by Blind().

   client_ake_state:  The ClientAkeState as defined in Section 6.4.

   The server state ServerState may have the following fields:

   server_ake_state:  The ServerAkeState as defined in Section 6.4.

   Both of these states are ephemeral and should be erased after the
   protocol completes.

   The rest of this section describes these authenticated key exchange
   messages and their parameters in more detail.  Section 6.1 defines
   the structure of the messages passed between client and server in the
   above setup.  Section 6.2 describes details of the functions and
   corresponding parameters mentioned above.  Section 6.3 discusses
   internal functions used for retrieving client credentials, and
   Section 6.4 discusses how these functions are used to execute the
   authenticated key exchange protocol.

6.1.  AKE Messages

   In this section, we define the KE1, KE2, and KE3 structs that make up
   the AKE messages used in the protocol.  KE1 is composed of a
   CredentialRequest and AuthRequest, and KE2 is composed of a
   CredentialResponse and AuthResponse.

   struct {
     uint8 client_nonce[Nn];
     uint8 client_public_keyshare[Npk];
   } AuthRequest;

   client_nonce:  A fresh randomly generated nonce of length Nn.

   client_public_keyshare:  A serialized client ephemeral public key of
      fixed size Npk.

   struct {
     CredentialRequest credential_request;
     AuthRequest auth_request;
   } KE1;

   credential_request:  A CredentialRequest structure.

   auth_request:  An AuthRequest structure.

   struct {
     uint8 server_nonce[Nn];
     uint8 server_public_keyshare[Npk];
     uint8 server_mac[Nm];
   } AuthResponse;

   server_nonce:  A fresh randomly generated nonce of length Nn.

   server_public_keyshare:  A server ephemeral public key of fixed size
      Npk, where Npk depends on the corresponding prime order group.

   server_mac:  An authentication tag computed over the handshake
      transcript computed using Km2, which is defined below.

   struct {
     CredentialResponse credential_response;
     AuthResponse auth_response;
   } KE2;

   credential_response:  A CredentialResponse structure.

   auth_response:  An AuthResponse structure.

   struct {
     uint8 client_mac[Nm];
   } KE3;

   client_mac:  An authentication tag computed over the handshake
      transcript of fixed size Nm, computed using Km2, defined below.

6.2.  AKE Functions

   In this section, we define the main functions used to produce the AKE
   messages in the protocol.  Note that this section relies on
   definitions of subroutines defined in later sections:

   *  CreateCredentialRequest, CreateCredentialResponse, and
      RecoverCredentials are defined in Section 6.3.

   *  AuthClientStart, AuthServerRespond, AuthClientFinalize, and
      AuthServerFinalize are defined in Sections 6.4.3 and 6.4.4.

6.2.1.  GenerateKE1

   The GenerateKE1 function begins the AKE protocol and produces the
   client's KE1 output for the server.

   GenerateKE1

   State:
   - state, a ClientState structure.

   Input:
   - password, an opaque byte string containing the client's password.

   Output:
   - ke1, a KE1 message structure.

   def GenerateKE1(password):
     request, blind = CreateCredentialRequest(password)
     state.password = password
     state.blind = blind
     ke1 = AuthClientStart(request)
     return ke1

6.2.2.  GenerateKE2

   The GenerateKE2 function continues the AKE protocol by processing the
   client's KE1 message and producing the server's KE2 output.

   GenerateKE2

   State:
   - state, a ServerState structure.

   Input:
   - server_identity, the optional encoded server identity, which is
     set to server_public_key if not specified.
   - server_private_key, the server's private key.
   - server_public_key, the server's public key.
   - record, the client's RegistrationRecord structure.
   - credential_identifier, an identifier that uniquely represents
     the credential.
   - oprf_seed, the server-side seed of Nh bytes used to generate
     an oprf_key.
   - ke1, a KE1 message structure.
   - client_identity, the optional encoded client identity, which is
     set to client_public_key if not specified.

   Output:
   - ke2, a KE2 structure.

   def GenerateKE2(server_identity, server_private_key,
                  server_public_key, record, credential_identifier,
                  oprf_seed, ke1, client_identity):
     credential_response =
       CreateCredentialResponse(ke1.credential_request,
                                server_public_key, record,
                                credential_identifier, oprf_seed)
     cleartext_credentials =
       CreateCleartextCredentials(server_public_key,
                                  record.client_public_key,
                                  server_identity, client_identity)
     auth_response =
       AuthServerRespond(cleartext_credentials, server_private_key,
                         record.client_public_key, ke1,
                         credential_response)

     ke2 = KE2 {
       credential_response,
       auth_response
     }

     return ke2

6.2.3.  GenerateKE3

   The GenerateKE3 function completes the AKE protocol for the client
   and produces the client's KE3 output for the server, as well as the
   session_key and export_key outputs from the AKE.

   GenerateKE3

   State:
   - state, a ClientState structure.

   Input:
   - client_identity, the optional encoded client identity, which is
     set to client_public_key if not specified.
   - server_identity, the optional encoded server identity, which is
     set to server_public_key if not specified.
   - ke2, a KE2 message structure.

   Output:
   - ke3, a KE3 message structure.
   - session_key, the session's shared secret.
   - export_key, an additional client key.

   def GenerateKE3(client_identity, server_identity, ke2):
     (client_private_key, cleartext_credentials, export_key) =
       RecoverCredentials(state.password, state.blind,
                          ke2.credential_response,
                          server_identity, client_identity)
     (ke3, session_key) =
       AuthClientFinalize(cleartext_credentials,
                          client_private_key, ke2)
     return (ke3, session_key, export_key)

6.2.4.  ServerFinish

   The ServerFinish function completes the AKE protocol for the server,
   yielding the session_key.  Since the OPRF is a two-message protocol,
   KE3 has no element of the OPRF.  Therefore, KE3 invokes the AKE's
   AuthServerFinalize directly.  The AuthServerFinalize function takes
   KE3 as input and MUST verify the client authentication material it
   contains before the session_key value can be used.  This verification
   is necessary to ensure forward secrecy against active attackers.

   ServerFinish

   State:
   - state, a ServerState structure.

   Input:
   - ke3, a KE3 structure.

   Output:
   - session_key, the shared session secret if and only if ke3 is valid.

   def ServerFinish(ke3):
     return AuthServerFinalize(ke3)

   This function MUST NOT return the session_key value if the client
   authentication material is invalid and may instead return an
   appropriate error message, such as ClientAuthenticationError, which
   is invoked from AuthServerFinalize.

6.3.  Credential Retrieval

   This section describes the sub-protocol run during authentication to
   retrieve and recover the client credentials.

6.3.1.  Credential Retrieval Messages

   This section describes the CredentialRequest and CredentialResponse
   messages exchanged between client and server to perform credential
   retrieval.

   struct {
     uint8 blinded_message[Noe];
   } CredentialRequest;

   blinded_message:  A serialized OPRF group element.

   struct {
     uint8 evaluated_message[Noe];
     uint8 masking_nonce[Nn];
     uint8 masked_response[Npk + Nn + Nm];
   } CredentialResponse;

   evaluated_message:  A serialized OPRF group element.

   masking_nonce:  A nonce used for the confidentiality of the
      masked_response field.

   masked_response:  An encrypted form of the server's public key and
      the client's Envelope structure.

6.3.2.  Credential Retrieval Functions

   This section describes the CreateCredentialRequest,
   CreateCredentialResponse, and RecoverCredentials functions used for
   credential retrieval.

6.3.2.1.  CreateCredentialRequest

   The CreateCredentialRequest is used by the client to initiate the
   credential retrieval process, and it produces a CredentialRequest
   message and OPRF state.  Like CreateRegistrationRequest, this
   function can fail with an InvalidInputError error with negligible
   probability.  However, this should not occur since registration (via
   CreateRegistrationRequest) will fail when provided the same password
   input.

   CreateCredentialRequest

   Input:
   - password, an opaque byte string containing the client's password.

   Output:
   - request, a CredentialRequest structure.
   - blind, an OPRF scalar value.

   Exceptions:
   - InvalidInputError, when Blind fails

   def CreateCredentialRequest(password):
     (blind, blinded_element) = Blind(password)
     blinded_message = SerializeElement(blinded_element)
     request = CredentialRequest {
       blinded_message
     }
     return (request, blind)

6.3.2.2.  CreateCredentialResponse

   The CreateCredentialResponse function is used by the server to
   process the client's CredentialRequest message and complete the
   credential retrieval process, producing a CredentialResponse.

   There are two scenarios to handle for the construction of a
   CredentialResponse object: either the record for the client exists
   (corresponding to a properly registered client) or it was never
   created (corresponding to an unregistered client identity, possibly
   the result of an enumeration attack attempt).

   In the case of an existing record with the corresponding identifier
   credential_identifier, the server invokes the following function to
   produce a CredentialResponse:

   CreateCredentialResponse

   Input:
   - request, a CredentialRequest structure.
   - server_public_key, the public key of the server.
   - record, an instance of RegistrationRecord which is the server's
     output from registration.
   - credential_identifier, an identifier that uniquely represents
     the credential.
   - oprf_seed, the server-side seed of Nh bytes used to generate
     an oprf_key.

   Output:
   - response, a CredentialResponse structure.

   Exceptions:
   - DeserializeError, when OPRF element deserialization fails.

   def CreateCredentialResponse(request, server_public_key, record,
                                credential_identifier, oprf_seed):
     seed =
       Expand(oprf_seed, concat(credential_identifier, "OprfKey"), Nok)
     (oprf_key, _) = DeriveKeyPair(seed, "OPAQUE-DeriveKeyPair")

     blinded_element = DeserializeElement(request.blinded_message)
     evaluated_element = BlindEvaluate(oprf_key, blinded_element)
     evaluated_message = SerializeElement(evaluated_element)

     masking_nonce = random(Nn)
     credential_response_pad = Expand(record.masking_key,
                                      concat(masking_nonce,
                                      "CredentialResponsePad"),
                                      Npk + Nn + Nm)
     masked_response = xor(credential_response_pad,
                           concat(server_public_key, record.envelope))

     response = CredentialResponse {
       evaluated_message,
       masking_nonce,
       masked_response
     }

     return response

   In the case of a record that does not exist and if client enumeration
   prevention is desired, the server MUST respond to the credential
   request to fake the existence of the record.  The server SHOULD
   invoke the CreateCredentialResponse function with a fake client
   record argument that is configured so that:

   *  record.client_public_key is set to a randomly generated public key
      of length Npk

   *  record.masking_key is set to a random byte string of length Nh

   *  record.envelope is set to the byte string consisting only of zeros
      of length Nn + Nm

   It is RECOMMENDED that a fake client record is created once (e.g., as
   the first user record of the application) and then stored alongside
   legitimate client records to serve subsequent client requests.  This
   allows servers to retrieve the record in a time comparable to that of
   a legitimate client record.

   Note that the responses output by either scenario are
   indistinguishable to an adversary that is unable to guess the
   registered password for the client corresponding to
   credential_identifier.

6.3.2.3.  RecoverCredentials

   The RecoverCredentials function is used by the client to process the
   server's CredentialResponse message and produce the client's private
   key, server public key, and the export_key.

   RecoverCredentials

   Input:
   - password, an opaque byte string containing the client's password.
   - blind, an OPRF scalar value.
   - response, a CredentialResponse structure.
   - server_identity, The optional encoded server identity.
   - client_identity, The encoded client identity.

   Output:
   - client_private_key, the encoded client private key for
     the AKE protocol.
   - cleartext_credentials, a CleartextCredentials structure.
   - export_key, an additional client key.

   Exceptions:
   - DeserializeError, when OPRF element deserialization fails.

   def RecoverCredentials(password, blind, response,
                          server_identity, client_identity):
     evaluated_element = DeserializeElement(response.evaluated_message)

     oprf_output = Finalize(password, blind, evaluated_element)
     stretched_oprf_output = Stretch(oprf_output)

     randomized_password =
       Extract("", concat(oprf_output, stretched_oprf_output))

     masking_key = Expand(randomized_password, "MaskingKey", Nh)

     credential_response_pad =
       Expand(masking_key,
              concat(response.masking_nonce, "CredentialResponsePad"),
              Npk + Nn + Nm)

     concat(server_public_key, envelope) =
       xor(credential_response_pad, response.masked_response)

     (client_private_key, cleartext_credentials, export_key) =
       Recover(randomized_password, server_public_key, envelope,
               server_identity, client_identity)

     return (client_private_key, cleartext_credentials, export_key)

6.4.  3DH Protocol

   This section describes the authenticated key exchange protocol for
   OPAQUE using 3DH, a 3-message AKE that satisfies the forward secrecy
   and KCI properties discussed in Section 10.

   The client AKE state ClientAkeState mentioned in Section 6 has the
   following fields:

   client_secret:  An opaque byte string of length Nsk.

   ke1:  A value of type KE1.

   The server AKE state ServerAkeState mentioned in Section 6 has the
   following fields:

   expected_client_mac:  An opaque byte string of length Nm.

   session_key:  An opaque byte string of length Nx.

   Sections 6.4.3 and 6.4.4 specify the inner workings of client and
   server functions, respectively.

6.4.1.  3DH Key Exchange Functions

   We assume the following functions exist for all Diffie-Hellman key
   exchange variants:

   DeriveDiffieHellmanKeyPair(seed):  Derive a private and public
      Diffie-Hellman key pair deterministically from the input seed.
      The type of the private key depends on the implementation, whereas
      the type of the public key is a byte string of Npk bytes.

   DiffieHellman(k, B):  A function that performs the Diffie-Hellman
      operation between the private input k and public input B.  The
      output of this function is a unique, fixed-length byte string.

   It is RECOMMENDED to use Elliptic Curve Diffie-Hellman for this key
   exchange protocol.  Implementations for recommended groups in
   Section 7, as well as groups covered by test vectors in Appendix C,
   are described in the following sections.

6.4.1.1.  3DH ristretto255

   This section describes the implementation of the Diffie-Hellman key
   exchange functions based on ristretto255 as defined in [RFC9496].

   DeriveDiffieHellmanKeyPair(seed):  This function is implemented as
      DeriveKeyPair(seed, "OPAQUE-DeriveDiffieHellmanKeyPair"), where
      DeriveKeyPair is as specified in Section 3.2 of [RFC9497].  The
      public value from DeriveKeyPair is encoded using SerializeElement
      from Section 2.1 of [RFC9497].

   DiffieHellman(k, B):  Implemented as scalar multiplication as
      described in [RFC9496] after decoding B from its encoded input
      using the Decode function in Section 4.3.1 of [RFC9496].  The
      output is then encoded using the SerializeElement function of the
      OPRF group described in Section 2.1 of [RFC9497].

6.4.1.2.  3DH P-256

   This section describes the implementation of the Diffie-Hellman key
   exchange functions based on NIST P-256 as defined in [NISTCurves].

   DeriveDiffieHellmanKeyPair(seed):  As defined in Section 6.4.1.1.

   DiffieHellman(k, B):  Implemented as scalar multiplication as
      described in [NISTCurves] after decoding B from its encoded input
      using the compressed Octet-String-to-Elliptic-Curve-Point method
      according to [NISTCurves].  The output is then encoded using the
      SerializeElement function of the OPRF group described in
      Section 2.1 of [RFC9497].

6.4.1.3.  3DH Curve25519

   This section describes the implementation of the Diffie-Hellman key
   exchange functions based on Curve25519 as defined in [RFC7748].

   DeriveDiffieHellmanKeyPair(seed):  This function is implemented by
      returning the private key k based on seed (of length Nseed = 32
      bytes) as described in Section 5 of [RFC7748], as well as the
      result of DiffieHellman(k, B), where B is the base point of
      Curve25519.

   DiffieHellman(k, B):  Implemented using the X25519 function in
      Section 5 of [RFC7748].  The output is then used raw with no
      processing.

6.4.2.  Key Schedule Functions

   This section contains functions used for the AKE key schedule.

6.4.2.1.  Transcript Functions

   The OPAQUE-3DH key derivation procedures make use of the functions
   below that are repurposed from TLS 1.3 [RFC8446].

   Expand-Label(Secret, Label, Context, Length) =
       Expand(Secret, CustomLabel, Length)

   Where CustomLabel is specified and encoded (following Section 3.4 of
   [RFC8446]) as:

   struct {
     uint16 length = Length;
     opaque label<8..255> = "OPAQUE-" + Label;
     uint8 context<0..255> = Context;
   } CustomLabel;

   Derive-Secret(Secret, Label, Transcript-Hash) =
       Expand-Label(Secret, Label, Transcript-Hash, Nx)

   Note that the Label parameter is not a NULL-terminated string.

   OPAQUE-3DH can optionally include application-specific, shared
   context information in the transcript, such as configuration
   parameters or application-specific information, e.g., "appXYZ-
   v1.2.3".

   The OPAQUE-3DH key schedule requires a preamble, which is computed as
   follows.

   Preamble

   Parameters:
   - context, optional shared context information.

   Input:
   - client_identity, the optional encoded client identity, which is set
     to client_public_key if not specified.
   - ke1, a KE1 message structure.
   - server_identity, the optional encoded server identity, which is set
     to server_public_key if not specified.
   - credential_response, the corresponding field on the KE2 structure.
   - server_nonce, the corresponding field on the AuthResponse
     structure.
   - server_public_keyshare, the corresponding field on the AuthResponse
     structure.

   Output:
   - preamble, the protocol transcript with identities and messages.

   def Preamble(client_identity, ke1, server_identity,
               credential_response, server_nonce,
               server_public_keyshare):
     preamble = concat("OPAQUEv1-",
                        I2OSP(len(context), 2), context,
                        I2OSP(len(client_identity), 2), client_identity,
                        ke1,
                        I2OSP(len(server_identity), 2), server_identity,
                        credential_response,
                        server_nonce,
                        server_public_keyshare)
     return preamble

6.4.2.2.  Shared Secret Derivation

   The OPAQUE-3DH shared secret derived during the key exchange protocol
   is computed using the following helper function.

   DeriveKeys

   Input:
   - ikm, input key material.
   - preamble, the protocol transcript with identities and messages.

   Output:
   - Km2, a MAC authentication key.
   - Km3, a MAC authentication key.
   - session_key, the shared session secret.

   def DeriveKeys(ikm, preamble):
     prk = Extract("", ikm)
     handshake_secret =
       Derive-Secret(prk, "HandshakeSecret",Hash(preamble))
     session_key =
       Derive-Secret(prk, "SessionKey", Hash(preamble))
     Km2 = Derive-Secret(handshake_secret, "ServerMAC", "")
     Km3 = Derive-Secret(handshake_secret, "ClientMAC", "")
     return (Km2, Km3, session_key)

6.4.3.  3DH Client Functions

   The AuthClientStart function is used by the client to create a KE1
   structure.

   AuthClientStart

   Parameters:
   - Nn, the nonce length.

   State:
   - state, a ClientAkeState structure.

   Input:
   - credential_request, a CredentialRequest structure.

   Output:
   - ke1, a KE1 structure.

   def AuthClientStart(credential_request):
     client_nonce = random(Nn)
     client_keyshare_seed = random(Nseed)
     (client_secret, client_public_keyshare) =
       DeriveDiffieHellmanKeyPair(client_keyshare_seed)

     auth_request = AuthRequest {
       client_nonce,
       client_public_keyshare
     }

     ke1 = KE1 {
       credential_request,
       auth_request
     }

     state.client_secret = client_secret
     state.ke1 = ke1
     return ke1

   The AuthClientFinalize function is used by the client to create a KE3
   message and output session_key using the server's KE2 message and
   recovered credential information.

   AuthClientFinalize

   State:
   - state, a ClientAkeState structure.

   Input:
   - cleartext_credentials, a CleartextCredentials structure.
   - client_private_key, the client's private key.
   - ke2, a KE2 message structure.

   Output:
   - ke3, a KE3 structure.
   - session_key, the shared session secret.

   Exceptions:
   - ServerAuthenticationError, the handshake fails.

   def AuthClientFinalize(cleartext_credentials,
                          client_private_key, ke2):

     dh1 = DiffieHellman(state.client_secret,
                         ke2.auth_response.server_public_keyshare)
     dh2 = DiffieHellman(state.client_secret,
                         cleartext_credentials.server_public_key)
     dh3 = DiffieHellman(client_private_key,
                         ke2.auth_response.server_public_keyshare)
     ikm = concat(dh1, dh2, dh3)

     preamble = Preamble(cleartext_credentials.client_identity,
                         state.ke1,
                         cleartext_credentials.server_identity,
                         ke2.credential_response,
                         ke2.auth_response.server_nonce,
                         ke2.auth_response.server_public_keyshare)
     Km2, Km3, session_key = DeriveKeys(ikm, preamble)
     expected_server_mac = MAC(Km2, Hash(preamble))
     if !ct_equal(ke2.auth_response.server_mac, expected_server_mac),
       raise ServerAuthenticationError
     client_mac = MAC(Km3, Hash(concat(preamble, expected_server_mac)))
     ke3 = KE3 {
       client_mac
     }
     return (ke3, session_key)

6.4.4.  3DH Server Functions

   The AuthServerRespond function is used by the server to process the
   client's KE1 message and public credential information to create a
   KE2 message.

   AuthServerRespond

   Parameters:
   - Nn, the nonce length.

   State:
   - state, a ServerAkeState structure.

   Input:
   - cleartext_credentials, a CleartextCredentials structure.
   - server_private_key, the server's private key.
   - client_public_key, the client's public key.
   - ke1, a KE1 message structure.

   Output:
   - auth_response, an AuthResponse structure.

   def AuthServerRespond(cleartext_credentials, server_private_key,
                         client_public_key, ke1, credential_response):
     server_nonce = random(Nn)
     server_keyshare_seed = random(Nseed)
     (server_private_keyshare, server_public_keyshare) =
       DeriveDiffieHellmanKeyPair(server_keyshare_seed)
     preamble = Preamble(cleartext_credentials.client_identity,
                         ke1,
                         cleartext_credentials.server_identity,
                         credential_response,
                         server_nonce,
                         server_public_keyshare)

     dh1 = DiffieHellman(server_private_keyshare,
                         ke1.auth_request.client_public_keyshare)
     dh2 = DiffieHellman(server_private_key,
                         ke1.auth_request.client_public_keyshare)
     dh3 = DiffieHellman(server_private_keyshare,
                         client_public_key)
     ikm = concat(dh1, dh2, dh3)

     Km2, Km3, session_key = DeriveKeys(ikm, preamble)
     server_mac = MAC(Km2, Hash(preamble))

     state.expected_client_mac =
       MAC(Km3, Hash(concat(preamble, server_mac)))
     state.session_key = session_key

     auth_response = AuthResponse {
       server_nonce,
       server_public_keyshare,
       server_mac
     }

     return auth_response

   The AuthServerFinalize function is used by the server to process the
   client's KE3 message and output the final session_key.

   AuthServerFinalize

   State:
   - state, a ServerAkeState structure.

   Input:
   - ke3, a KE3 structure.

   Output:
   - session_key, the shared session secret if and only if ke3 is valid.

   Exceptions:
   - ClientAuthenticationError, the handshake fails.

   def AuthServerFinalize(ke3):
     if !ct_equal(ke3.client_mac, state.expected_client_mac):
       raise ClientAuthenticationError
     return state.session_key

7.  Configurations

   An OPAQUE-3DH configuration is a tuple (OPRF, KDF, MAC, Hash, KSF,
   Group, Context) such that the following conditions are met:

   *  The OPRF protocol uses the modeOPRF configuration in Section 3.1
      of [RFC9497] and implements the interface in Section 2.  Examples
      include ristretto255-SHA512 and P256-SHA256.

   *  The KDF, MAC, and Hash functions implement the interfaces in
      Section 2.  Examples include HKDF [RFC5869] for the KDF, HMAC
      [RFC2104] for the MAC, and SHA-256 and SHA-512 for the Hash
      functions.  If an extensible output function such as SHAKE128
      [FIPS202] is used, then the output length Nh MUST be chosen to
      align with the target security level of the OPAQUE configuration.
      For example, if the target security parameter for the
      configuration is 128 bits, then Nh SHOULD be at least 32 bytes.

   *  The KSF is determined by the application and implements the
      interface in Section 2.  As noted, collision resistance is
      required.  Examples for KSF include Argon2id [RFC9106], scrypt
      [RFC7914], and PBKDF2 [RFC8018] with fixed parameter choices.  See
      Section 8 for more information about this choice of function.

   *  The Group mode identifies the group used in the OPAQUE-3DH AKE.
      This SHOULD match that of the OPRF.  For example, if the OPRF is
      ristretto255-SHA512, then Group SHOULD be ristretto255.

   Context is the shared parameter used to construct the preamble in
   Section 6.4.2.1.  This parameter SHOULD include any application-
   specific configuration information or parameters that are needed to
   prevent cross-protocol or downgrade attacks.

   Absent an application-specific profile, the following configurations
   are RECOMMENDED:

   *  ristretto255-SHA512, HKDF-SHA-512, HMAC-SHA-512, SHA-512,
      Argon2id(S = zeroes(16), p = 4, T = Nh, m = 2^21, t = 1, v = 0x13,
      K = nil, X = nil, y = 2), ristretto255

   *  P256-SHA256, HKDF-SHA-256, HMAC-SHA-256, SHA-256, Argon2id(S =
      zeroes(16), p = 4, T = Nh, m = 2^21, t = 1, v = 0x13, K = nil, X =
      nil, y = 2), P-256

   *  P256-SHA256, HKDF-SHA-256, HMAC-SHA-256, SHA-256, scrypt(S =
      zeroes(16), N = 32768, r = 8, p = 1, dkLen = 32), P-256

   The above recommended configurations target 128-bit security.

   Future configurations may specify different combinations of dependent
   algorithms with the following considerations:

   1.  The size of AKE public and private keys -- Npk and Nsk,
       respectively -- must adhere to the output length limitations of
       the KDF Expand function.  If HKDF is used, this means Npk, Nsk <=
       255 * Nx, where Nx is the output size of the underlying hash
       function.  See [RFC5869] for details.

   2.  The output size of the Hash function SHOULD be long enough to
       produce a key for MAC of suitable length.  For example, if MAC is
       HMAC-SHA256, then Nh could be 32 bytes.

8.  Application Considerations

   Beyond choosing an appropriate configuration, there are several
   parameters that applications can use to control OPAQUE:

   *  Credential identifier: As described in Section 5, this is a unique
      handle to the client's credential being stored.  In applications
      where there are alternate client identities that accompany an
      account, such as a username or email address, this identifier can
      be set to those alternate values.  For simplicity, applications
      may choose to set credential_identifier to be equal to
      client_identity.  Applications MUST NOT use the same credential
      identifier for multiple clients.

   *  Context information: As described in Section 7, applications may
      include a shared context string that is authenticated as part of
      the handshake.  This parameter SHOULD include any configuration
      information or parameters that are needed to prevent cross-
      protocol or downgrade attacks.  This context information is not
      sent over the wire in any key exchange messages.  However,
      applications may choose to send it alongside key exchange messages
      if needed for their use case.

   *  Client and server identities: As described in Section 4, clients
      and servers are identified with their public keys by default.
      However, applications may choose alternate identities that are
      pinned to these public keys.  For example, servers may use a
      domain name instead of a public key as their identifier.  Absent
      alternate notions of identity, applications SHOULD set these
      identities to nil and rely solely on public key information.

   *  Configuration and envelope updates: Applications may wish to
      update or change their configuration or other parameters that
      affect the client's RegistrationRecord over time.  Some reasons
      for changing these are to use different cryptographic algorithms,
      e.g., a different KSF with improved parameters, or to update key
      material that is cryptographically bound to the
      RegistrationRecord, such as the server's public key
      (server_public_key).  Any such change will require users to
      reregister to create a new RegistrationRecord.  Supporting these
      types of updates can be helpful for applications that anticipate
      such changes in their deployment setting.

   *  Password hardening parameters: Key stretching is done to help
      prevent password disclosure in the event of server compromise; see
      Section 10.8.  There is no ideal or default set of parameters,
      though relevant specifications for KSFs give some reasonable
      defaults.

   *  Enumeration prevention: If servers receive a credential request
      for a non-existent client, they SHOULD respond with a "fake"
      response to prevent active client enumeration attacks as described
      in Section 6.3.2.2.  Servers that implement this mitigation SHOULD
      use the same configuration information (such as the oprf_seed) for
      all clients; see Section 10.9.  In settings where this attack is
      not a concern, servers may choose to not support this
      functionality.

   *  Handling password changes: In the event of a password change, the
      client and server can run the registration phase using the new
      password as a fresh instance (ensuring to resample all random
      values).  The resulting registration record can then replace the
      previous record corresponding to the client's old password
      registration.

9.  Implementation Considerations

   This section documents considerations for OPAQUE implementations.
   This includes implementation safeguards and error handling
   considerations.

9.1.  Implementation Safeguards

   Certain information created, exchanged, and processed in OPAQUE is
   sensitive.  Specifically, all private key material and intermediate
   values, along with the outputs of the key exchange phase, are all
   secret.  Implementations should not retain these values in memory
   when no longer needed.  Moreover, all operations, particularly the
   cryptographic and group arithmetic operations, should be constant-
   time and independent of the bits of any secrets.  This includes any
   conditional branching during the creation of the credential response
   as needed to mitigate client enumeration attacks.

   As specified in Section 5 and Section 6, OPAQUE only requires the
   client password as input to the OPRF for registration and
   authentication.  However, if client_identity can be bound to the
   client's registration record (i.e., the identity will not change
   during the lifetime of the record), then an implementation SHOULD
   incorporate client_identity alongside the password as input to the
   OPRF.  Furthermore, it is RECOMMENDED to incorporate server_identity
   alongside the password as input to the OPRF.  These additions provide
   domain separation for clients and servers; see Section 10.2.

9.2.  Handling Online Guessing Attacks

   Online guessing attacks (against any aPAKE) can be done from both the
   client side and the server side.  In particular, a malicious server
   can attempt to simulate honest responses to learn the client's
   password.  While this constitutes an exhaustive online attack (as
   expensive as a guessing attack from the client side), it can be
   mitigated when the channel between client and server is
   authenticated, e.g., using server-authenticated TLS.  In such cases,
   these online attacks are limited to clients and the authenticated
   server itself.  Moreover, such a channel provides privacy of user
   information, including identity and envelope values.

   Additionally, note that a client participating in the online login
   stage will learn whether or not authentication is successful after
   receiving the KE2 message.  This means that the server should treat
   any client which fails to send a subsequent KE3 message as an
   authentication failure.  This can be handled in applications that
   wish to track authentication failures by, for example, assuming by
   default that any client authentication attempt is a failure unless a
   KE3 message is received by the server and passes ServerFinish without
   error.

9.3.  Error Considerations

   Some functions included in this specification are fallible.  For
   example, the authenticated key exchange protocol may fail because the
   client's password was incorrect or the authentication check failed,
   yielding an error.  The explicit errors generated throughout this
   specification, along with conditions that lead to each error, are as
   follows:

   *  EnvelopeRecoveryError: The Envelope Recover function failed to
      produce any authentication key material; Section 4.1.3.

   *  ServerAuthenticationError: The client failed to complete the
      authenticated key exchange protocol with the server;
      Section 6.4.3.

   *  ClientAuthenticationError: The server failed to complete the
      authenticated key exchange protocol with the client;
      Section 6.4.4.

   Beyond these explicit errors, OPAQUE implementations can produce
   implicit errors.  For example, if protocol messages sent between
   client and server do not match their expected size, an implementation
   should produce an error.  More generally, if any protocol message
   received from the peer is invalid, perhaps because the message
   contains an invalid public key (indicated by the AKE
   DeserializeElement function failing) or an invalid OPRF element
   (indicated by the OPRF DeserializeElement), then an implementation
   should produce an error.

   The errors in this document are meant as a guide for implementors.
   They are not an exhaustive list of all the errors an implementation
   might emit.  For example, an implementation might run out of memory.

10.  Security Considerations

   OPAQUE is defined as the composition of two functionalities: an OPRF
   and an AKE protocol.  It can be seen as a "compiler" for transforming
   any AKE protocol (with Key Compromise Impersonation (KCI) security
   and forward secrecy; see Section 10.2) into a secure aPAKE protocol.
   In OPAQUE, the client derives a private key during password
   registration and retrieves this key each time it needs to
   authenticate to the server.  The OPRF security properties ensure that
   only the correct password can unlock the private key while at the
   same time avoiding potential offline guessing attacks.  This general
   composability property provides great flexibility and enables a
   variety of OPAQUE instantiations, from optimized performance to
   integration with existing authenticated key exchange protocols such
   as TLS.

10.1.  Notable Design Differences

   The specification as written here differs from the original
   cryptographic design in [JKX18] and the corresponding CFRG document
   [Krawczyk20], both of which were used as input to the CFRG PAKE
   competition.  This section describes these differences, including
   their motivation and explanation as to why they preserve the provable
   security of OPAQUE based on [JKX18].

   The following list enumerates important functional differences that
   were made as part of the protocol specification process to address
   application or implementation considerations.

   *  Clients construct envelope contents without revealing the password
      to the server, as described in Section 5, whereas the servers
      construct envelopes in [JKX18].  This change adds to the security
      of the protocol.  [JKX18] considered the case where the envelope
      was constructed by the server for reasons of compatibility with
      previous Universal Composability (UC) security modeling.  [HJKW23]
      analyzes the registration phase as specified in this document.
      This change was made to support registration flows where the
      client chooses the password and wishes to keep it secret from the
      server, and it is compatible with the variant in [JKX18] that was
      originally analyzed.

   *  Envelopes do not contain encrypted credentials.  Instead,
      envelopes contain information used to derive client private key
      material for the AKE.  This change improves the assumption behind
      the protocol by getting rid of equivocality and random key
      robustness for the encryption function.  The random-key robustness
      property defined in Section 2.2 is only needed for the MAC
      function.  This change was made for two reasons.  First, it
      reduces the number of bytes stored in envelopes, which is a
      helpful improvement for large applications of OPAQUE with many
      registered users.  Second, it removes the need for client
      applications to generate private keys during registration.
      Instead, this responsibility is handled by OPAQUE, thereby
      simplifying the client interface to the protocol.

   *  Envelopes are masked with a per-user masking key as a way of
      preventing client enumeration attacks.  See Section 10.9 for more
      details.  This extension is not needed for the security of OPAQUE
      as an aPAKE protocol, but is only used to provide a defense
      against enumeration attacks.  In the analysis, the masking key can
      be simulated as a (pseudo) random key.  This change was made to
      support real-world use cases where client or user enumeration is a
      security (or privacy) risk.

   *  Per-user OPRF keys are derived from a client identity and cross-
      user PRF seed as a mitigation against client enumeration attacks.
      See Section 10.9 for more details.  The analysis of OPAQUE assumes
      OPRF keys of different users are independently random or
      pseudorandom.  Deriving these keys via a single PRF (i.e., with a
      single cross-user key) applied to users' identities satisfies this
      assumption.  This change was made to support real-world use cases
      where client or user enumeration is a security (or privacy) risk.
      Note that the derivation of the OPRF key via a PRF keyed by
      oprf_seed and applied to the unique credential_identifier ensures
      the critical requirement of the per-user OPRF keys being unique
      per client.

   *  The protocol outputs an export key for the client in addition to a
      shared session key that can be used for application-specific
      purposes.  This key is a pseudorandom value derived from the
      client password (among other values) and has no influence on the
      security analysis (it can be simulated with a random output).
      This change was made to support more application use cases for
      OPAQUE, such as the use of OPAQUE for end-to-end encrypted
      backups; see [WhatsAppE2E].

   *  The AKE protocol describes a 3-message protocol where the third
      message includes client authentication material that the server is
      required to verify.  This change (from the original 2-message
      protocol) was made to provide explicit client authentication and
      full forward security.  The 3-message protocol is analyzed in
      [JKX18].

   *  The protocol admits optional application-layer client and server
      identities.  In the absence of these identities, the client and
      server are authenticated against their public keys.  Binding
      authentication to identities is part of the AKE part of OPAQUE.
      The type of identities and their semantics are application-
      dependent and independent of the protocol analysis.  This change
      was made to simplify client and server interfaces to the protocol
      by removing the need to specify additional identities alongside
      their corresponding public authentication keys when not needed.

   *  The protocol admits application-specific context information
      configured out-of-band in the AKE transcript.  This allows domain
      separation between different application uses of OPAQUE.  This is
      a mechanism for the AKE component and is best practice for domain
      separation between different applications of the protocol.  This
      change was made to allow different applications to use OPAQUE
      without the risk of cross-protocol attacks.

   *  Servers use a separate identifier for computing OPRF evaluations
      and indexing into the registration record storage called the
      credential_identifier.  This allows clients to change their
      application-layer identity (client_identity) without inducing
      server-side changes, e.g., by changing an email address associated
      with a given account.  This mechanism is part of the derivation of
      OPRF keys via a single PRF.  As long as the derivation of
      different OPRF keys from a single PRF has different PRF inputs,
      the protocol is secure.  The choice of such inputs is up to the
      application.

   *  [JKX18] comments on a defense against offline dictionary attacks
      upon server compromise or honest-but-curious servers.  The authors
      suggest implementing the OPRF phase as a threshold OPRF [TOPPSS],
      effectively forcing an attacker to act online or control at least
      t key shares (of the total n), where t is the threshold number of
      shares necessary to recombine the secret OPRF key.  Only then
      would an attacker be able to run an offline dictionary attack.
      This implementation only affects the server and changes nothing
      for the client.  Furthermore, if the threshold OPRF servers
      holding these keys are separate from the authentication server,
      then recovering all n shares would still not suffice to run an
      offline dictionary attack without access to the client record
      database.  However, this mechanism is out of scope for this
      document.

   The following list enumerates notable differences and refinements
   from the original cryptographic design in [JKX18] and the
   corresponding CFRG document [Krawczyk20] that were made to make this
   specification suitable for interoperable implementations.

   *  [JKX18] used a generic prime-order group for the DH-OPRF and HMQV
      operations, and includes necessary prime-order subgroup checks
      when receiving attacker-controlled values over the wire.  This
      specification instantiates the prime-order group used for 3DH
      using prime-order groups based on elliptic curves as described in
      Section 2.1 of [RFC9497].  This specification also delegates OPRF
      group choice and operations to Section 4 of [RFC9497].  As such,
      the prime-order group as used in the OPRF and 3DH as specified in
      this document both adhere to the requirements in [JKX18].

   *  Appendix B of [JKX18] specified DH-OPRF to instantiate the OPRF
      functionality in the protocol.  A critical part of DH-OPRF is the
      hash-to-group operation, which was not instantiated in the
      original analysis.  However, the requirements for this operation
      were included.  This specification instantiates the OPRF
      functionality based on Section 3.3.1 of [RFC9497], which is
      identical to the DH-OPRF functionality in [JKX18] and, concretely,
      uses the hash-to-curve functions in [RFC9380].  All hash-to-curve
      methods in [RFC9380] are compliant with the requirement in
      [JKX18], namely, that the output be a member of the prime-order
      group.

   *  [JKX18] and [Krawczyk20] both used HMQV as the AKE for the
      protocol.  However, this document fully specifies 3DH instead of
      HMQV (though a sketch for how to instantiate OPAQUE using HMQV is
      included in Appendix B.1).  Since 3DH satisfies the essential
      requirements for the AKE protocol as described in [JKX18] and
      [Krawczyk20], as recalled in Section 10.2, this change preserves
      the overall security of the protocol. 3DH was chosen for its
      simplicity and ease of implementation.

   *  The DH-OPRF and HMQV instantiation of OPAQUE as shown in Figure 12
      [JKX18] uses a different transcript than that which is described
      in this specification.  In particular, the key exchange transcript
      specified in Section 6.4 is a superset of the transcript as
      defined in [JKX18].  This was done to align with best practices,
      like what is done for key exchange protocols like TLS 1.3
      [RFC8446].

   *  Neither [JKX18] nor [Krawczyk20] included wire format details for
      the protocol, which is essential for interoperability.  This
      specification fills this gap by including such wire format details
      and corresponding test vectors; see Appendix C.

10.2.  Security Analysis

   Jarecki et al. [JKX18] proved the security of OPAQUE (modulo the
   design differences outlined in Section 10.1) in a strong aPAKE model
   that ensures security against precomputation attacks and is
   formulated in the UC framework [Canetti01] under the random oracle
   model.  This assumes security of the OPRF function and the underlying
   key exchange protocol.

   OPAQUE's design builds on a line of work initiated in the seminal
   paper of Ford and Kaliski [FK00] and is based on the HPAKE protocol
   of Xavier Boyen [Boyen09] and the (1,1)-PPSS protocol from Jarecki et
   al. [JKKX16].  None of these papers considered security against
   precomputation attacks or presented a proof of aPAKE security (not
   even in a weak model).

   The KCI property required from AKE protocols for use with OPAQUE
   states that knowledge of a party's private key does not allow an
   attacker to impersonate others to that party.  This is an important
   security property achieved by most public-key-based AKE protocols,
   including protocols that use signatures or public key encryption for
   authentication.  It is also a property of many implicitly
   authenticated protocols, e.g., HMQV, but not all of them.  We also
   note that key exchange protocols based on shared keys do not satisfy
   the KCI requirement, hence they are not considered in the OPAQUE
   setting.  We note that KCI is needed to ensure a crucial property of
   OPAQUE.  Even upon compromise of the server, the attacker cannot
   impersonate the client to the server without first running an
   exhaustive dictionary attack.  Another essential requirement from AKE
   protocols for use in OPAQUE is to provide forward secrecy (against
   active attackers).

   In [JKX18], security is proven for one instance (i.e., one key) of
   the OPAQUE protocol, and without batching.  There is currently no
   security analysis available for the OPAQUE protocol described in this
   document in a setting with multiple server keys or batching.

   As stated in Section 9.1, incorporating client_identity adds domain
   separation, particularly against servers that choose the same OPRF
   key for multiple clients.  The client_identity as input to the OPRF
   also acts as a key identifier that would be required for a proof of
   the protocol in the multi-key setting; the OPAQUE analysis in [JKX18]
   assumes single server-key instances.  Adding server_identity to the
   OPRF input provides domain separation for clients that reuse the same
   client_identity across different server instantiations.

10.3.  Identities

   AKE protocols generate keys that need to be uniquely and verifiably
   bound to a pair of identities.  In the case of OPAQUE, those
   identities correspond to client_identity and server_identity.  Thus,
   it is essential for the parties to agree on such identities,
   including an agreed bit representation of these identities as needed.

   Note that the method of transmission of client_identity from client
   to server is outside the scope of this protocol and it is up to an
   application to choose how this identity should be delivered (for
   instance, alongside the first OPAQUE message or agreed upon before
   the OPAQUE protocol begins).

   Applications may have different policies about how and when
   identities are determined.  A natural approach is to tie
   client_identity to the identity the server uses to fetch the envelope
   (determined during password registration) and tie server_identity to
   the server identity used by the client to initiate an offline
   password registration or online authenticated key exchange session.
   server_identity and client_identity can also be part of the envelope
   or tied to the parties' public keys.  In principle, identities may
   change across different sessions as long as there is a policy that
   can establish if the identity is acceptable or not to the peer.
   However, we note that the public keys of both the server and the
   client must always be those defined at the time of password
   registration.

   The client identity (client_identity) and server identity
   (server_identity) are optional parameters that are left to the
   application to designate as aliases for the client and server.  If
   the application layer does not supply values for these parameters,
   then they will be omitted from the creation of the envelope during
   the registration stage.  Furthermore, they will be substituted with
   client_identity = client_public_key and server_identity =
   server_public_key during the authenticated key exchange stage.

   The advantage of supplying a custom client_identity and
   server_identity (instead of simply relying on a fallback to
   client_public_key and server_public_key) is that the client can then
   ensure that any mappings between client_identity and
   client_public_key (and server_identity and server_public_key) are
   protected by the authentication from the envelope.  Then, the client
   can verify that the client_identity and server_identity contained in
   its envelope match the client_identity and server_identity supplied
   by the server.

   However, if this extra layer of verification is unnecessary for the
   application, then simply leaving client_identity and server_identity
   unspecified (and using client_public_key and server_public_key
   instead) is acceptable.

10.4.  Export Key Usage

   The export key can be used (separately from the OPAQUE protocol) to
   provide confidentiality and integrity to other data that only the
   client should be able to process.  For instance, if the client wishes
   to store secrets with a third party, then this export key can be used
   by the client to encrypt these secrets so that they remain hidden
   from a passive adversary that does not have access to the server's
   secret keys or the client's password.

10.5.  Static Diffie-Hellman Oracles

   While one can expect the practical security of the OPRF function
   (namely, the hardness of computing the function without knowing the
   key) to be in the order of computing discrete logarithms or solving
   Diffie-Hellman, Brown and Gallant [BG04] and Cheon [Cheon06] show an
   attack that slightly improves on generic attacks.  For typical
   curves, the attack requires an infeasible number of calls to the OPRF
   or results in insignificant security loss; see Section 7.2.3 of
   [RFC9497] for more information.  For OPAQUE, these attacks are
   particularly impractical as they translate into an infeasible number
   of failed authentication attempts directed at individual users.

10.6.  Random-Key Robust MACs

   The random-key robustness property for a MAC states that, given two
   random keys k1 and k2, it is infeasible to find a message m such that
   MAC(k1, m) = MAC(k2, m).  Note that in general, not every MAC
   function is key-robust.  In particular, GMAC (which underlies GCM)
   does not satisfy key-robustness, whereas HMAC with a collision-
   resistant hash function does satisfy key-robustness.

   An application can choose to use a non-key-robust MAC within the AKE
   portion of the protocol described in Section 6.4, but it MUST use a
   key-robust MAC for the creation of the auth_tag parameter in
   Section 4.1.2.

10.7.  Input Validation

   Both client and server MUST validate the other party's public key(s)
   used for the execution of OPAQUE.  This includes the keys shared
   during the registration phase, as well as any keys shared during the
   online key agreement phase.  The validation procedure varies
   depending on the type of key.  For example, for OPAQUE instantiations
   using 3DH with P-256, P-384, or P-521 as the underlying group,
   validation is as specified in Section 5.6.2.3.4 of [keyagreement].
   This includes checking that the coordinates are in the correct range,
   that the point is on the curve, and that the point is not the point
   at infinity.  Additionally, validation MUST ensure the Diffie-Hellman
   shared secret is not the point at infinity.

10.8.  OPRF Key Stretching

   Applying a key stretching function to the output of the OPRF greatly
   increases the cost of an offline attack upon the compromise of the
   credential file on the server.  Applications SHOULD select parameters
   for the KSF that balance cost and complexity across different client
   implementations and deployments.  Note that in OPAQUE, the key
   stretching function is executed by the client as opposed to the
   server in common password hashing scenarios.  This means that
   applications must consider a tradeoff between the performance of the
   protocol on clients (specifically low-end devices) and protection
   against offline attacks after a server compromise.

10.9.  Client Enumeration

   Client enumeration refers to attacks where the attacker tries to
   learn whether a given user identity is registered with a server or
   whether a reregistration or change of password was performed for that
   user.  OPAQUE counters these attacks by requiring servers to act with
   unregistered client identities in a way that is indistinguishable
   from their behavior with existing registered clients.  Servers do
   this by simulating a fake CredentialResponse as specified in
   Section 6.3.2.2 for unregistered users and encrypting
   CredentialResponse using a masking key.  In this way, real and fake
   CredentialResponse messages are indistinguishable from one another.
   Implementations must also take care to avoid side-channel leakage
   (e.g., timing attacks) from helping differentiate these operations
   from a regular server response.  Note that this may introduce
   possible abuse vectors since the server's cost of generating a
   CredentialResponse is less than that of the client's cost of
   generating a CredentialRequest.  Server implementations may choose to
   forego the construction of a simulated credential response message
   for an unregistered client if these client enumeration attacks can be
   mitigated through other application-specific means or are otherwise
   not applicable for their threat model.

   OPAQUE does not prevent this type of attack during the registration
   flow.  Servers necessarily react differently during the registration
   flow between registered and unregistered clients.  This allows an
   attacker to use the server's response during registration as an
   oracle for whether a given client identity is registered.
   Applications should mitigate against this type of attack by rate
   limiting or otherwise restricting the registration flow.

   Finally, applications that do not require protection against client
   enumeration attacks can choose to derive independent OPRF keys for
   different clients.  The advantage to using independently-derived OPRF
   keys is that the server avoids keeping the oprf_seed value across
   different clients, which, if leaked, would compromise the security
   for all clients reliant on oprf_seed as noted in [DL24].

10.10.  Protecting the Registration Masking Key

   The user enumeration prevention method described in this document
   uses a symmetric encryption key, masking_key, generated and sent to
   the server by the client during registration.  This requires a
   confidential channel between client and server during registration,
   e.g., using TLS [RFC8446].  If the channel is only authenticated
   (this is a requirement for correct identification of the parties), a
   confidential channel can be established using public-key encryption,
   e.g., with HPKE [RFC9180].  However, the details of this mechanism
   are out of scope for this document.

10.11.  Password Salt and Storage Implications

   In OPAQUE, the OPRF key acts as the secret salt value that ensures
   the infeasibility of precomputation attacks.  No extra salt value is
   needed.  Also, clients never disclose their passwords to the server,
   even during registration.  Note that a corrupted server can run an
   exhaustive offline dictionary attack to validate guesses for the
   client's password; this is inevitable in any (single-server) aPAKE
   protocol.  It can be avoided in the case of OPAQUE by resorting to a
   multi-server threshold OPRF implementation, e.g., [TOPPSS].
   Furthermore, if the server does not sample the PRF seed with
   sufficiently high entropy, or if it is not kept hidden from an
   adversary, then any derivatives from the client's password may also
   be susceptible to an offline dictionary attack to recover the
   original password.

   Some applications may require learning the client's password to
   enforce password rules.  Doing so invalidates this important security
   property of OPAQUE and is NOT RECOMMENDED unless it is not possible
   for applications to move such checks to the client.  Note that
   limited checks at the server are possible to implement, e.g.,
   detecting repeated passwords upon reregistrations or password change.

   In general, passwords should be selected with sufficient entropy to
   avoid being susceptible to recovery through dictionary attacks, both
   online and offline.

10.12.  AKE Private Key Storage

   Server implementations of OPAQUE do not need access to the raw AKE
   private key.  They only require the ability to compute shared secrets
   as specified in Section 6.4.2.  Thus, applications may store the
   server AKE private key in a Hardware Security Module (HSM) or
   similar.  Upon compromise of oprf_seed and client envelopes, this
   would prevent an attacker from using this data to mount a server
   spoofing attack.  Supporting implementations need to consider
   allowing separate AKE and OPRF algorithms in cases where the HSM is
   incompatible with the OPRF algorithm.

10.13.  Client Authentication Using Credentials

   For scenarios in which the client has access to private state that
   can be persisted across registration and login, the client can back
   up the randomized_password variable (as computed in Section 5.2.3) so
   that upon a future login attempt, the client can authenticate to the
   server using randomized_password instead of the original password.
   This can be achieved by supplying an arbitrary password as input to
   CreateCredentialRequest in the login phase, and then using
   randomized_password from the backup in RecoverCredentials (invoked by
   GenerateKE3) rather than computing it from the password.

   This provides an advantage over the regular authentication flow for
   login in that if randomized_password is compromised, an adversary
   cannot use this value to successfully impersonate the server to the
   client during login.  The drawback is that it is only applicable to
   settings where randomized_password can be treated as a credential
   that can be stored securely after registration and retrieved upon
   login.

11.  IANA Considerations

   This document has no IANA actions.

12.  References

12.1.  Normative References

   [RFC2104]  Krawczyk, H., Bellare, M., and R. Canetti, "HMAC: Keyed-
              Hashing for Message Authentication", RFC 2104,
              DOI 10.17487/RFC2104, February 1997,
              <https://www.rfc-editor.org/info/rfc2104>.

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997,
              <https://www.rfc-editor.org/info/rfc2119>.

   [RFC4086]  Eastlake 3rd, D., Schiller, J., and S. Crocker,
              "Randomness Requirements for Security", BCP 106, RFC 4086,
              DOI 10.17487/RFC4086, June 2005,
              <https://www.rfc-editor.org/info/rfc4086>.

   [RFC8174]  Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
              2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
              May 2017, <https://www.rfc-editor.org/info/rfc8174>.

   [RFC9497]  Davidson, A., Faz-Hernandez, A., Sullivan, N., and C. A.
              Wood, "Oblivious Pseudorandom Functions (OPRFs) Using
              Prime-Order Groups", RFC 9497, DOI 10.17487/RFC9497,
              December 2023, <https://www.rfc-editor.org/info/rfc9497>.

12.2.  Informative References

   [BG04]     Brown, D. and R. Galant, "The Static Diffie-Hellman
              Problem", Cryptology ePrint Archive, Paper 2004/306, 2004,
              <https://eprint.iacr.org/2004/306>.

   [Boyen09]  Boyen, X., "HPAKE: Password Authentication Secure against
              Cross-Site User Impersonation", Cryptology and Network
              Security (CANS 2009), Lecture Notes in Computer Science,
              vol. 5888, pp. 279-298, DOI 10.1007/978-3-642-10433-6_19,
              2009, <https://doi.org/10.1007/978-3-642-10433-6_19>.

   [Canetti01]
              Canetti, R., "Universally composable security: A new
              paradigm for cryptographic protocols", 42nd IEEE Symposium
              on Foundations of Computer Science, pp. 136-145,
              DOI 10.1109/SFCS.2001.959888, 2001,
              <https://doi.org/10.1109/SFCS.2001.959888>.

   [Cheon06]  Cheon, J. H., "Security Analysis of the Strong Diffie-
              Hellman Problem", Advances in Cryptology - EUROCRYPT 2006,
              Lecture Notes in Computer Science, vol. 4004, pp. 1-11,
              DOI 10.1007/11761679_1, 2006,
              <https://doi.org/10.1007/11761679_1>.

   [DL24]     Dayanikli, D. and A. Lehmann, "(Strong) aPAKE Revisited:
              Capturing Multi-User Security and Salting", Cryptology
              ePrint Archive, Paper 2024/756, 2024,
              <https://eprint.iacr.org/2024/756>.

   [FIPS202]  NIST, "SHA-3 Standard: Permutation-Based Hash and
              Extendable-Output Functions", NIST FIPS 202,
              DOI 10.6028/NIST.FIPS.202, August 2015,
              <https://nvlpubs.nist.gov/nistpubs/FIPS/
              NIST.FIPS.202.pdf>.

   [FK00]     Ford, W. and B. S. Kaliski, Jr, "Server-assisted
              generation of a strong secret from a password", IEEE 9th
              International Workshops on Enabling Technologies:
              Infrastructure for Collaborative Enterprises (WET ICE
              2000), pp. 176-180, DOI 10.1109/ENABL.2000.883724, 2000,
              <https://doi.org/10.1109/ENABL.2000.883724>.

   [HJKW23]   Hesse, J., Jarecki, S., Krawczyk, H., and C. Wood,
              "Password-Authenticated TLS via OPAQUE and Post-Handshake
              Authentication", Advances in Cryptology - EUROCRYPT 2023,
              Lecture Notes in Computer Science, vol. 14008, pp. 98-127,
              DOI 10.1007/978-3-031-30589-4_4, 2023,
              <https://doi.org/10.1007/978-3-031-30589-4_4>.

   [HMQV]     Krawczyk, H., "HMQV: A High-Performance Secure Diffie-
              Hellman Protocol", Cryptology ePrint Archive, Paper
              2005/176, 2005, <https://eprint.iacr.org/2005/176>.

   [JKKX16]   Jarecki, S., Kiayias, A., Krawczyk, H., and J. Xu,
              "Highly-Efficient and Composable Password-Protected Secret
              Sharing (Or: How to Protect Your Bitcoin Wallet Online)",
              2016 IEEE European Symposium on Security and Privacy
              (EuroS&P), pp. 276-291, DOI 10.1109/EuroSP.2016.30, 2016,
              <https://doi.org/10.1109/EuroSP.2016.30>.

   [JKX18]    Jarecki, S., Krawczyk, H., and J. Xu, "OPAQUE: An
              Asymmetric PAKE Protocol Secure Against Pre-Computation
              Attacks", Cryptology ePrint Archive, Paper 2018/163, 2018,
              <https://eprint.iacr.org/2018/163>.

   [keyagreement]
              Barker, E., Chen, L., Roginsky, A., Vassilev, A., and R.
              Davis, "Recommendation for Pair-Wise Key-Establishment
              Schemes Using Discrete Logarithm Cryptography",
              DOI 10.6028/nist.sp.800-56ar3, NIST SP 800-56Ar3, April
              2018, <https://doi.org/10.6028/nist.sp.800-56ar3>.

   [Krawczyk20]
              Krawczyk, H., "The OPAQUE Asymmetric PAKE Protocol", Work
              in Progress, Internet-Draft, draft-krawczyk-cfrg-opaque-
              06, 19 June 2020, <https://datatracker.ietf.org/doc/html/
              draft-krawczyk-cfrg-opaque-06>.

   [LGR20]    Len, J., Grubbs, P., and T. Ristenpart, "Partitioning
              Oracle Attacks", Cryptology ePrint Archive, Paper
              2020/1491, 2021, <https://eprint.iacr.org/2020/1491.pdf>.

   [NISTCurves]
              NIST, "Digital Signature Standard (DSS)", NIST FIPS 186-5,
              DOI 10.6028/nist.fips.186-5, 2013,
              <https://doi.org/10.6028/nist.fips.186-5>.

   [RFC5869]  Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand
              Key Derivation Function (HKDF)", RFC 5869,
              DOI 10.17487/RFC5869, May 2010,
              <https://www.rfc-editor.org/info/rfc5869>.

   [RFC7748]  Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves
              for Security", RFC 7748, DOI 10.17487/RFC7748, January
              2016, <https://www.rfc-editor.org/info/rfc7748>.

   [RFC7914]  Percival, C. and S. Josefsson, "The scrypt Password-Based
              Key Derivation Function", RFC 7914, DOI 10.17487/RFC7914,
              August 2016, <https://www.rfc-editor.org/info/rfc7914>.

   [RFC8017]  Moriarty, K., Ed., Kaliski, B., Jonsson, J., and A. Rusch,
              "PKCS #1: RSA Cryptography Specifications Version 2.2",
              RFC 8017, DOI 10.17487/RFC8017, November 2016,
              <https://www.rfc-editor.org/info/rfc8017>.

   [RFC8018]  Moriarty, K., Ed., Kaliski, B., and A. Rusch, "PKCS #5:
              Password-Based Cryptography Specification Version 2.1",
              RFC 8018, DOI 10.17487/RFC8018, January 2017,
              <https://www.rfc-editor.org/info/rfc8018>.

   [RFC8125]  Schmidt, J., "Requirements for Password-Authenticated Key
              Agreement (PAKE) Schemes", RFC 8125, DOI 10.17487/RFC8125,
              April 2017, <https://www.rfc-editor.org/info/rfc8125>.

   [RFC8446]  Rescorla, E., "The Transport Layer Security (TLS) Protocol
              Version 1.3", RFC 8446, DOI 10.17487/RFC8446, August 2018,
              <https://www.rfc-editor.org/info/rfc8446>.

   [RFC9106]  Biryukov, A., Dinu, D., Khovratovich, D., and S.
              Josefsson, "Argon2 Memory-Hard Function for Password
              Hashing and Proof-of-Work Applications", RFC 9106,
              DOI 10.17487/RFC9106, September 2021,
              <https://www.rfc-editor.org/info/rfc9106>.

   [RFC9180]  Barnes, R., Bhargavan, K., Lipp, B., and C. Wood, "Hybrid
              Public Key Encryption", RFC 9180, DOI 10.17487/RFC9180,
              February 2022, <https://www.rfc-editor.org/info/rfc9180>.

   [RFC9380]  Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R. S.,
              and C. A. Wood, "Hashing to Elliptic Curves", RFC 9380,
              DOI 10.17487/RFC9380, August 2023,
              <https://www.rfc-editor.org/info/rfc9380>.

   [RFC9496]  de Valence, H., Grigg, J., Hamburg, M., Lovecruft, I.,
              Tankersley, G., and F. Valsorda, "The ristretto255 and
              decaf448 Groups", RFC 9496, DOI 10.17487/RFC9496, December
              2023, <https://www.rfc-editor.org/info/rfc9496>.

   [SIGMA-I]  Krawczyk, H., "SIGMA: The 'SIGn-and-MAc' Approach to
              Authenticated Diffie-Hellman and its Use in the IKE
              Protocols", 2003,
              <https://www.iacr.org/cryptodb/archive/2003/
              CRYPTO/1495/1495.pdf>.

   [TOPPSS]   Jarecki, S., Kiayias, A., Krawczyk, H., and J. Xu,
              "TOPPSS: Cost-Minimal Password-Protected Secret Sharing
              based on Threshold OPRF", Applied Cryptology and Network
              Security - ACNS 2017, Lecture Notes in Computer Science,
              vol. 10355, pp. 39-58, DOI 10.1007/978-3-319-61204-1_3,
              2017, <https://doi.org/10.1007/978-3-319-61204-1_3>.

   [TripleDH] Marlinspike, M., "Simplifying OTR deniability", Signal
              Blog, 27 July 2013,
              <https://signal.org/blog/simplifying-otr-deniability>.

   [WhatsAppE2E]
              WhatsApp, "Security of End-to-End Encrypted Backups",
              WhatsApp Security Whitepaper, 10 September 2021,
              <https://www.whatsapp.com/security/
              WhatsApp_Security_Encrypted_Backups_Whitepaper.pdf>.

Appendix A.  Alternate Key Recovery Mechanisms

   Client authentication material can be stored and retrieved using
   different key recovery mechanisms.  Any key recovery mechanism that
   encrypts data in the envelope MUST use an authenticated encryption
   scheme with random key-robustness (or key-committing).  Deviating
   from the key-robustness requirement may open the protocol to attacks,
   e.g., [LGR20].  This specification enforces this property by using a
   MAC over the envelope contents.

   We remark that export_key for authentication or encryption requires
   no special properties from the authentication or encryption schemes
   as long as export_key is used only after authentication material is
   successfully recovered, i.e., after the MAC in RecoverCredentials
   passes verification.

Appendix B.  Alternate AKE Instantiations

   It is possible to instantiate OPAQUE with other AKEs, such as HMQV
   [HMQV] and SIGMA-I [SIGMA-I].  HMQV is similar to 3DH but varies in
   its key schedule.  SIGMA-I uses digital signatures rather than static
   DH keys for authentication.  Specification of these instantiations is
   left to future documents.  A sketch of how these instantiations might
   change is included in the next subsection for posterity.

   OPAQUE may also be instantiated with any post-quantum (PQ) AKE
   protocol that has the message flow above and security properties (KCI
   resistance and forward secrecy) outlined in Section 10.  Note that
   such an instantiation is not quantum-safe unless the OPRF is quantum-
   safe.  However, an OPAQUE instantiation where the AKE protocol is
   quantum-safe, but the OPRF is not, would still ensure the
   confidentiality and integrity of application data encrypted under
   session_key (or a key derived from it) with a quantum-safe encryption
   function.  However, the only effect of a break of the OPRF by a
   future quantum attacker would be the ability of this attacker to run
   at that time an exhaustive dictionary attack against the old user's
   password and only for users whose envelopes were harvested while in
   use (in the case of OPAQUE run over a TLS channel with the server,
   harvesting such envelopes requires targeted active attacks).

B.1.  HMQV Instantiation Sketch

   An HMQV instantiation would work similarly to OPAQUE-3DH, differing
   primarily in the key schedule [HMQV].  First, the key schedule
   preamble value would use a different constant prefix -- "HMQV"
   instead of "3DH" -- as shown below.

   preamble = concat("HMQV",
                     I2OSP(len(client_identity), 2), client_identity,
                     KE1,
                     I2OSP(len(server_identity), 2), server_identity,
                     KE2.credential_response,
                     KE2.auth_response.server_nonce,
                     KE2.auth_response.server_public_keyshare)

   Second, the IKM derivation would change.  Assuming HMQV is
   instantiated with a cyclic group of prime order p with bit length L,
   clients would compute IKM as follows:

   u' = (eskU + u \* skU) mod p
   IKM = (epkS \* pkS^s)^u'

   Likewise, servers would compute IKM as follows:

   s' = (eskS + s \* skS) mod p
   IKM = (epkU \* pkU^u)^s'

   In both cases, u would be computed as follows:

   hashInput = concat(I2OSP(len(epkU), 2), epkU,
                      I2OSP(len(info), 2), info,
                      I2OSP(len("client"), 2), "client")
   u = Hash(hashInput) mod L

   Likewise, s would be computed as follows:

   hashInput = concat(I2OSP(len(epkS), 2), epkS,
                      I2OSP(len(info), 2), info,
                      I2OSP(len("server"), 2), "server")
   s = Hash(hashInput) mod L

   Hash is the same hash function used in the main OPAQUE protocol for
   key derivation.  Its output length (in bits) must be at least L.

   Both parties should perform validation (as in Section 10.7) on each
   other's public keys before computing the above parameters.

B.2.  SIGMA-I Instantiation Sketch

   A SIGMA-I [SIGMA-I] instantiation differs more drastically from
   OPAQUE-3DH since authentication uses digital signatures instead of
   Diffie-Hellman.  In particular, both KE2 and KE3 would carry a
   digital signature, computed using the server and client private keys
   established during registration, respectively, as well as a MAC,
   where the MAC is computed as in OPAQUE-3DH but it also covers the
   identity of the sender.

   The key schedule would also change.  Specifically, the key schedule
   preamble value would use a different constant prefix -- "SIGMA-I"
   instead of "3DH" -- and the IKM computation would use only the
   ephemeral public keys exchanged between client and server.

Appendix C.  Test Vectors

   This section contains real and fake test vectors for the OPAQUE-3DH
   specification.  Each real test vector in Appendix C.1 specifies the
   configuration information, protocol inputs, intermediate values
   computed during registration and authentication, and protocol
   outputs.

   Similarly, each fake test vector in Appendix C.2 specifies the
   configuration information, protocol inputs, and protocol outputs
   computed during the authentication of an unknown or unregistered
   user.  Note that masking_key, client_private_key, and
   client_public_key are used as additional inputs as described in
   Section 6.3.2.2. client_public_key is used as the fake record's
   public key, and masking_key is used for the fake record's masking key
   parameter.

   All values are encoded in hexadecimal strings.  The configuration
   information includes the (OPRF, Hash, KSF, KDF, MAC, Group, Context)
   tuple, where the Group matches that which is used in the OPRF.  The
   KSF used for each test vector is the identity function (denoted
   Identity), which returns as output the input message supplied to the
   function without any modification, i.e., msg = Stretch(msg).

C.1.  Real Test Vectors

C.1.1.  OPAQUE-3DH Real Test Vector 1

C.1.1.1.  Configuration

   OPRF: ristretto255-SHA512
   Hash: SHA512
   KSF: Identity
   KDF: HKDF-SHA512
   MAC: HMAC-SHA512
   Group: ristretto255
   Context: 4f50415155452d504f43
   Nh: 64
   Npk: 32
   Nsk: 32
   Nm: 64
   Nx: 64
   Nok: 32

C.1.1.2.  Input Values

   oprf_seed: f433d0227b0b9dd54f7c4422b600e764e47fb503f1f9a0f0a47c6606b0
   54a7fdc65347f1a08f277e22358bbabe26f823fca82c7848e9a75661f4ec5d5c1989e
   f
   credential_identifier: 31323334
   password: 436f7272656374486f72736542617474657279537461706c65
   envelope_nonce: ac13171b2f17bc2c74997f0fce1e1f35bec6b91fe2e12dbd323d2
   3ba7a38dfec
   masking_nonce: 38fe59af0df2c79f57b8780278f5ae47355fe1f817119041951c80
   f612fdfc6d
   server_private_key: 47451a85372f8b3537e249d7b54188091fb18edde78094b43
   e2ba42b5eb89f0d
   server_public_key: b2fe7af9f48cc502d016729d2fe25cdd433f2c4bc904660b2a
   382c9b79df1a78
   server_nonce: 71cd9960ecef2fe0d0f7494986fa3d8b2bb01963537e60efb13981e
   138e3d4a1
   client_nonce: da7e07376d6d6f034cfa9bb537d11b8c6b4238c334333d1f0aebb38
   0cae6a6cc
   client_keyshare_seed: 82850a697b42a505f5b68fcdafce8c31f0af2b581f063cf
   1091933541936304b
   server_keyshare_seed: 05a4f54206eef1ba2f615bc0aa285cb22f26d1153b5b40a
   1e85ff80da12f982f
   blind_registration: 76cfbfe758db884bebb33582331ba9f159720ca8784a2a070
   a265d9c2d6abe01
   blind_login: 6ecc102d2e7a7cf49617aad7bbe188556792d4acd60a1a8a8d2b65d4
   b0790308

C.1.1.3.  Intermediate Values

   client_public_key: 76a845464c68a5d2f7e442436bb1424953b17d3e2e289ccbac
   cafb57ac5c3675
   auth_key: 6cd32316f18d72a9a927a83199fa030663a38ce0c11fbaef82aa9003773
   0494fc555c4d49506284516edd1628c27965b7555a4ebfed2223199f6c67966dde822
   randomized_password: aac48c25ab036e30750839d31d6e73007344cb1155289fb7
   d329beb932e9adeea73d5d5c22a0ce1952f8aba6d66007615cd1698d4ac85ef1fcf15
   0031d1435d9
   envelope: ac13171b2f17bc2c74997f0fce1e1f35bec6b91fe2e12dbd323d23ba7a3
   8dfec634b0f5b96109c198a8027da51854c35bee90d1e1c781806d07d49b76de6a28b
   8d9e9b6c93b9f8b64d16dddd9c5bfb5fea48ee8fd2f75012a8b308605cdd8ba5
   handshake_secret: 81263cb85a0cfa12450f0f388de4e92291ec4c7c7a0878b6245
   50ff528726332f1298fc6cc822a432c89504347c7a2ccd70316ae3da6a15e0399e6db
   3f7c1b12
   server_mac_key: 0d36b26cfe38f51f804f0a9361818f32ee1ce2a4e5578653b5271
   84af058d3b2d8075c296fd84d24677913d1baa109290cd81a13ed383f9091a3804e65
   298dfc
   client_mac_key: 91750adbac54a5e8e53b4c233cc8d369fe83b0de1b6a3cd85575e
   eb0bb01a6a90a086a2cf5fe75fff2a9379c30ba9049510a33b5b0b1444a88800fc3ee
   e2260d
   oprf_key: 5d4c6a8b7c7138182afb4345d1fae6a9f18a1744afbcc3854f8f5a2b4b4
   c6d05

C.1.1.4.  Output Values

   registration_request: 5059ff249eb1551b7ce4991f3336205bde44a105a032e74
   7d21bf382e75f7a71
   registration_response: 7408a268083e03abc7097fc05b587834539065e86fb0c7
   b6342fcf5e01e5b019b2fe7af9f48cc502d016729d2fe25cdd433f2c4bc904660b2a3
   82c9b79df1a78
   registration_upload: 76a845464c68a5d2f7e442436bb1424953b17d3e2e289ccb
   accafb57ac5c36751ac5844383c7708077dea41cbefe2fa15724f449e535dd7dd562e
   66f5ecfb95864eadddec9db5874959905117dad40a4524111849799281fefe3c51fa8
   2785c5ac13171b2f17bc2c74997f0fce1e1f35bec6b91fe2e12dbd323d23ba7a38dfe
   c634b0f5b96109c198a8027da51854c35bee90d1e1c781806d07d49b76de6a28b8d9e
   9b6c93b9f8b64d16dddd9c5bfb5fea48ee8fd2f75012a8b308605cdd8ba5
   KE1: c4dedb0ba6ed5d965d6f250fbe554cd45cba5dfcce3ce836e4aee778aa3cd44d
   da7e07376d6d6f034cfa9bb537d11b8c6b4238c334333d1f0aebb380cae6a6cc6e29b
   ee50701498605b2c085d7b241ca15ba5c32027dd21ba420b94ce60da326
   KE2: 7e308140890bcde30cbcea28b01ea1ecfbd077cff62c4def8efa075aabcbb471
   38fe59af0df2c79f57b8780278f5ae47355fe1f817119041951c80f612fdfc6dd6ec6
   0bcdb26dc455ddf3e718f1020490c192d70dfc7e403981179d8073d1146a4f9aa1ced
   4e4cd984c657eb3b54ced3848326f70331953d91b02535af44d9fedc80188ca46743c
   52786e0382f95ad85c08f6afcd1ccfbff95e2bdeb015b166c6b20b92f832cc6df01e0
   b86a7efd92c1c804ff865781fa93f2f20b446c8371b671cd9960ecef2fe0d0f749498
   6fa3d8b2bb01963537e60efb13981e138e3d4a1c4f62198a9d6fa9170c42c3c71f197
   1b29eb1d5d0bd733e40816c91f7912cc4a660c48dae03e57aaa38f3d0cffcfc21852e
   bc8b405d15bd6744945ba1a93438a162b6111699d98a16bb55b7bdddfe0fc5608b23d
   a246e7bd73b47369169c5c90
   KE3: 4455df4f810ac31a6748835888564b536e6da5d9944dfea9e34defb9575fe5e2
   661ef61d2ae3929bcf57e53d464113d364365eb7d1a57b629707ca48da18e442
   export_key: 1ef15b4fa99e8a852412450ab78713aad30d21fa6966c9b8c9fb3262a
   970dc62950d4dd4ed62598229b1b72794fc0335199d9f7fcc6eaedde92cc04870e63f
   16
   session_key: 42afde6f5aca0cfa5c163763fbad55e73a41db6b41bc87b8e7b62214
   a8eedc6731fa3cb857d657ab9b3764b89a84e91ebcb4785166fbb02cedfcbdfda215b
   96f

C.1.2.  OPAQUE-3DH Real Test Vector 2

C.1.2.1.  Configuration

   OPRF: ristretto255-SHA512
   Hash: SHA512
   KSF: Identity
   KDF: HKDF-SHA512
   MAC: HMAC-SHA512
   Group: ristretto255
   Context: 4f50415155452d504f43
   Nh: 64
   Npk: 32
   Nsk: 32
   Nm: 64
   Nx: 64
   Nok: 32

C.1.2.2.  Input Values

   client_identity: 616c696365
   server_identity: 626f62
   oprf_seed: f433d0227b0b9dd54f7c4422b600e764e47fb503f1f9a0f0a47c6606b0
   54a7fdc65347f1a08f277e22358bbabe26f823fca82c7848e9a75661f4ec5d5c1989e
   f
   credential_identifier: 31323334
   password: 436f7272656374486f72736542617474657279537461706c65
   envelope_nonce: ac13171b2f17bc2c74997f0fce1e1f35bec6b91fe2e12dbd323d2
   3ba7a38dfec
   masking_nonce: 38fe59af0df2c79f57b8780278f5ae47355fe1f817119041951c80
   f612fdfc6d
   server_private_key: 47451a85372f8b3537e249d7b54188091fb18edde78094b43
   e2ba42b5eb89f0d
   server_public_key: b2fe7af9f48cc502d016729d2fe25cdd433f2c4bc904660b2a
   382c9b79df1a78
   server_nonce: 71cd9960ecef2fe0d0f7494986fa3d8b2bb01963537e60efb13981e
   138e3d4a1
   client_nonce: da7e07376d6d6f034cfa9bb537d11b8c6b4238c334333d1f0aebb38
   0cae6a6cc
   client_keyshare_seed: 82850a697b42a505f5b68fcdafce8c31f0af2b581f063cf
   1091933541936304b
   server_keyshare_seed: 05a4f54206eef1ba2f615bc0aa285cb22f26d1153b5b40a
   1e85ff80da12f982f
   blind_registration: 76cfbfe758db884bebb33582331ba9f159720ca8784a2a070
   a265d9c2d6abe01
   blind_login: 6ecc102d2e7a7cf49617aad7bbe188556792d4acd60a1a8a8d2b65d4
   b0790308

C.1.2.3.  Intermediate Values

   client_public_key: 76a845464c68a5d2f7e442436bb1424953b17d3e2e289ccbac
   cafb57ac5c3675
   auth_key: 6cd32316f18d72a9a927a83199fa030663a38ce0c11fbaef82aa9003773
   0494fc555c4d49506284516edd1628c27965b7555a4ebfed2223199f6c67966dde822
   randomized_password: aac48c25ab036e30750839d31d6e73007344cb1155289fb7
   d329beb932e9adeea73d5d5c22a0ce1952f8aba6d66007615cd1698d4ac85ef1fcf15
   0031d1435d9
   envelope: ac13171b2f17bc2c74997f0fce1e1f35bec6b91fe2e12dbd323d23ba7a3
   8dfec1ac902dc5589e9a5f0de56ad685ea8486210ef41449cd4d8712828913c5d2b68
   0b2b3af4a26c765cff329bfb66d38ecf1d6cfa9e7a73c222c6efe0d9520f7d7c
   handshake_secret: 5e723bed1e5276de2503419eba9da61ead573109c4012268323
   98c7e08155b885bfe7bc93451f9d887a0c1d0c19233e40a8e47b347a9ac3907f94032
   a4cff64f
   server_mac_key: dad66bb9251073d17a13f8e5500f36e5998e3cde520ca0738e708
   5af62fd97812eb79a745c94d0bf8a6ac17f980cf435504cf64041eeb6bb237796d2c7
   f81e9a
   client_mac_key: f816fe2914f7c5b29852385615d7c7f31ac122adf202d7ccd4976
   06d7aabd48930323d1d02b1cc9ecd456c4de6f46c7950becb18bffd921dd5876381b5
   486ffe
   oprf_key: 5d4c6a8b7c7138182afb4345d1fae6a9f18a1744afbcc3854f8f5a2b4b4
   c6d05

C.1.2.4.  Output Values

   registration_request: 5059ff249eb1551b7ce4991f3336205bde44a105a032e74
   7d21bf382e75f7a71
   registration_response: 7408a268083e03abc7097fc05b587834539065e86fb0c7
   b6342fcf5e01e5b019b2fe7af9f48cc502d016729d2fe25cdd433f2c4bc904660b2a3
   82c9b79df1a78
   registration_upload: 76a845464c68a5d2f7e442436bb1424953b17d3e2e289ccb
   accafb57ac5c36751ac5844383c7708077dea41cbefe2fa15724f449e535dd7dd562e
   66f5ecfb95864eadddec9db5874959905117dad40a4524111849799281fefe3c51fa8
   2785c5ac13171b2f17bc2c74997f0fce1e1f35bec6b91fe2e12dbd323d23ba7a38dfe
   c1ac902dc5589e9a5f0de56ad685ea8486210ef41449cd4d8712828913c5d2b680b2b
   3af4a26c765cff329bfb66d38ecf1d6cfa9e7a73c222c6efe0d9520f7d7c
   KE1: c4dedb0ba6ed5d965d6f250fbe554cd45cba5dfcce3ce836e4aee778aa3cd44d
   da7e07376d6d6f034cfa9bb537d11b8c6b4238c334333d1f0aebb380cae6a6cc6e29b
   ee50701498605b2c085d7b241ca15ba5c32027dd21ba420b94ce60da326
   KE2: 7e308140890bcde30cbcea28b01ea1ecfbd077cff62c4def8efa075aabcbb471
   38fe59af0df2c79f57b8780278f5ae47355fe1f817119041951c80f612fdfc6dd6ec6
   0bcdb26dc455ddf3e718f1020490c192d70dfc7e403981179d8073d1146a4f9aa1ced
   4e4cd984c657eb3b54ced3848326f70331953d91b02535af44d9fea502150b67fe367
   95dd8914f164e49f81c7688a38928372134b7dccd50e09f8fed9518b7b2f94835b3c4
   fe4c8475e7513f20eb97ff0568a39caee3fd6251876f71cd9960ecef2fe0d0f749498
   6fa3d8b2bb01963537e60efb13981e138e3d4a1c4f62198a9d6fa9170c42c3c71f197
   1b29eb1d5d0bd733e40816c91f7912cc4a292371e7809a9031743e943fb3b56f51de9
   03552fc91fba4e7419029951c3970b2e2f0a9dea218d22e9e4e0000855bb6421aa361
   0d6fc0f4033a6517030d4341
   KE3: 7a026de1d6126905736c3f6d92463a08d209833eb793e46d0f7f15b3e0f62c76
   43763c02bbc6b8d3d15b63250cae98171e9260f1ffa789750f534ac11a0176d5
   export_key: 1ef15b4fa99e8a852412450ab78713aad30d21fa6966c9b8c9fb3262a
   970dc62950d4dd4ed62598229b1b72794fc0335199d9f7fcc6eaedde92cc04870e63f
   16
   session_key: ae7951123ab5befc27e62e63f52cf472d6236cb386c968cc47b7e34f
   866aa4bc7638356a73cfce92becf39d6a7d32a1861f12130e824241fe6cab34fbd471
   a57

C.1.3.  OPAQUE-3DH Real Test Vector 3

C.1.3.1.  Configuration

   OPRF: ristretto255-SHA512
   Hash: SHA512
   KSF: Identity
   KDF: HKDF-SHA512
   MAC: HMAC-SHA512
   Group: curve25519
   Context: 4f50415155452d504f43
   Nh: 64
   Npk: 32
   Nsk: 32
   Nm: 64
   Nx: 64
   Nok: 32

C.1.3.2.  Input Values

   oprf_seed: a78342ab84d3d30f08d5a9630c79bf311c31ed7f85d9d4959bf492ec67
   a0eec8a67dfbf4497248eebd49e878aab173e5e4ff76354288fdd53e949a5f7c9f7f1
   b
   credential_identifier: 31323334
   password: 436f7272656374486f72736542617474657279537461706c65
   envelope_nonce: 40d6b67fdd7da7c49894750754514dbd2070a407166bd2a5237cc
   a9bf44d6e0b
   masking_nonce: 38fe59af0df2c79f57b8780278f5ae47355fe1f817119041951c80
   f612fdfc6d
   server_private_key: c06139381df63bfc91c850db0b9cfbec7a62e86d80040a41a
   a7725bf0e79d564
   server_public_key: a41e28269b4e97a66468cc00c5a57753e192e1527669897706
   88aa90486ef031
   server_nonce: 71cd9960ecef2fe0d0f7494986fa3d8b2bb01963537e60efb13981e
   138e3d4a1
   client_nonce: da7e07376d6d6f034cfa9bb537d11b8c6b4238c334333d1f0aebb38
   0cae6a6cc
   client_keyshare_seed: 82850a697b42a505f5b68fcdafce8c31f0af2b581f063cf
   1091933541936304b
   server_keyshare_seed: 05a4f54206eef1ba2f615bc0aa285cb22f26d1153b5b40a
   1e85ff80da12f982f
   blind_registration: c575731ffe1cb0ca5ba63b42c4699767b8b9ab78ba39316ee
   04baddb2034a70a
   blind_login: 6ecc102d2e7a7cf49617aad7bbe188556792d4acd60a1a8a8d2b65d4
   b0790308

C.1.3.3.  Intermediate Values

   client_public_key: 0936ea94ab030ec332e29050d266c520e916731a052d05ced7
   e0cfe751142b48
   auth_key: 7e880ab484f750e80e6f839d975aff476070ce65066d85ea62523d1d576
   4739d91307fac47186a4ab935e6a5c7f70cb47faa9473311947502c022cc67ae9440c
   randomized_password: 3a602c295a9c323d9362fe286f104567ed6862b25dbe30fa
   da844f19e41cf40047424b7118e15dc2c1a815a70fea5c8de6c30aa61440cd4b4b5e8
   f3963fbb2e1
   envelope: 40d6b67fdd7da7c49894750754514dbd2070a407166bd2a5237cca9bf44
   d6e0b20c1e81fef28e92e897ca8287d49a55075b47c3988ff0fff367d79a3e350ccac
   150b4a3ff48b4770c8e84e437b3d4e68d2b95833f7788f7eb93fa6a8afb85ecb
   handshake_secret: 178c8c15e025252380c3edb1c6ad8ac52573b38d536099e2f86
   5786f5e31c642608550c0c6f281c37ce259667dd72768af31630e0eb36f1096a2e642
   1c2aa163
   server_mac_key: f3c6a8e069c54bb0d8905139f723c9e22f5c662dc08848243a665
   4c8223800019b9823523d84da2ef67ca1c14277630aace464c113be8a0a658c39e181
   a8bb71
   client_mac_key: b1ee7ce52dbd0ab72872924ff11596cb196bbabfc319e74aca78a
   de54a0f74dd15dcf5621f6d2e79161b0c9b701381d494836dedbb86e584a65b34267a
   370e01
   oprf_key: 62ef7f7d9506a14600c34f642aaf6ef8019cc82a6755db4fded5248ea14
   6030a

C.1.3.4.  Output Values

   registration_request: 26f3dbfd76b8e5f85b4da604f42889a7d4b1bc919f65538
   1a67de02c59fd5436
   registration_response: 506e8f1b89c098fb89b5b6210a05f7898cafdaea221761
   e8d5272fc39e0f9f08a41e28269b4e97a66468cc00c5a57753e192e15276698977068
   8aa90486ef031
   registration_upload: 0936ea94ab030ec332e29050d266c520e916731a052d05ce
   d7e0cfe751142b486d23c6ed818882f9bdfdcf91389fcbc0b7a3faf92bd0bd6be4a1e
   7730277b694fc7c6ba327fbe786af18487688e0f7c148bbd54dc2fc80c28e7a976d9e
   f53c3540d6b67fdd7da7c49894750754514dbd2070a407166bd2a5237cca9bf44d6e0
   b20c1e81fef28e92e897ca8287d49a55075b47c3988ff0fff367d79a3e350ccac150b
   4a3ff48b4770c8e84e437b3d4e68d2b95833f7788f7eb93fa6a8afb85ecb
   KE1: c4dedb0ba6ed5d965d6f250fbe554cd45cba5dfcce3ce836e4aee778aa3cd44d
   da7e07376d6d6f034cfa9bb537d11b8c6b4238c334333d1f0aebb380cae6a6cc10a83
   b9117d3798cb2957fbdb0268a0d63dbf9d66bde5c00c78affd80026c911
   KE2: 9a0e5a1514f62e005ea098b0d8cf6750e358c4389e6add1c52aed9500fa19d00
   38fe59af0df2c79f57b8780278f5ae47355fe1f817119041951c80f612fdfc6d22cc3
   1127d6f0096755be3c3d2dd6287795c317aeea10c9485bf4f419a786642c19a8f151c
   eb5e8767d175248c62c017de94057398d28bf0ed00d1b50ee4f812fd9afddf98af8cd
   58067ca43b0633b6cadd0e9d987f89623fed4d3583bdf6910c425600e90dab3c6b351
   3188a465461a67f6bbc47aeba808f7f7e2c6d66f5c3271cd9960ecef2fe0d0f749498
   6fa3d8b2bb01963537e60efb13981e138e3d4a141f55f0bef355cfb34ccd468fdacad
   75865ee7efef95f4cb6c25d477f720502676f06a3b806da262139bf3fa76a1090b94d
   ac78bc3bc6f8747d5b35acf94eff3ec2ebe7d49b8cf16be64120b279fe92664e47be5
   da7e60f08f12e91192652f79
   KE3: 550e923829a544496d8316c490da2b979b78c730dd75be3a17f237a26432c19f
   bba54b6a0467b1c22ecbd6794bc5fa5b04215ba1ef974c6b090baa42c5bb984f
   export_key: 9dec51d6d0f6ce7e4345f10961053713b07310cc2e45872f57bbd2fe5
   070fdf0fb5b77c7ddaa2f3dc5c35132df7417ad7fefe0f690ad266e5a54a21d045c9c
   38
   session_key: fd2fdd07c1bcc88e81c1b1d1de5ad62dfdef1c0b8209ff9d671e1fac
   55ce9c34d381c1fb2703ff53a797f77daccbe33047ccc167b8105171e10ec962eea20
   3aa

C.1.4.  OPAQUE-3DH Real Test Vector 4

C.1.4.1.  Configuration

   OPRF: ristretto255-SHA512
   Hash: SHA512
   KSF: Identity
   KDF: HKDF-SHA512
   MAC: HMAC-SHA512
   Group: curve25519
   Context: 4f50415155452d504f43
   Nh: 64
   Npk: 32
   Nsk: 32
   Nm: 64
   Nx: 64
   Nok: 32

C.1.4.2.  Input Values

   client_identity: 616c696365
   server_identity: 626f62
   oprf_seed: a78342ab84d3d30f08d5a9630c79bf311c31ed7f85d9d4959bf492ec67
   a0eec8a67dfbf4497248eebd49e878aab173e5e4ff76354288fdd53e949a5f7c9f7f1
   b
   credential_identifier: 31323334
   password: 436f7272656374486f72736542617474657279537461706c65
   envelope_nonce: 40d6b67fdd7da7c49894750754514dbd2070a407166bd2a5237cc
   a9bf44d6e0b
   masking_nonce: 38fe59af0df2c79f57b8780278f5ae47355fe1f817119041951c80
   f612fdfc6d
   server_private_key: c06139381df63bfc91c850db0b9cfbec7a62e86d80040a41a
   a7725bf0e79d564
   server_public_key: a41e28269b4e97a66468cc00c5a57753e192e1527669897706
   88aa90486ef031
   server_nonce: 71cd9960ecef2fe0d0f7494986fa3d8b2bb01963537e60efb13981e
   138e3d4a1
   client_nonce: da7e07376d6d6f034cfa9bb537d11b8c6b4238c334333d1f0aebb38
   0cae6a6cc
   client_keyshare_seed: 82850a697b42a505f5b68fcdafce8c31f0af2b581f063cf
   1091933541936304b
   server_keyshare_seed: 05a4f54206eef1ba2f615bc0aa285cb22f26d1153b5b40a
   1e85ff80da12f982f
   blind_registration: c575731ffe1cb0ca5ba63b42c4699767b8b9ab78ba39316ee
   04baddb2034a70a
   blind_login: 6ecc102d2e7a7cf49617aad7bbe188556792d4acd60a1a8a8d2b65d4
   b0790308

C.1.4.3.  Intermediate Values

   client_public_key: 0936ea94ab030ec332e29050d266c520e916731a052d05ced7
   e0cfe751142b48
   auth_key: 7e880ab484f750e80e6f839d975aff476070ce65066d85ea62523d1d576
   4739d91307fac47186a4ab935e6a5c7f70cb47faa9473311947502c022cc67ae9440c
   randomized_password: 3a602c295a9c323d9362fe286f104567ed6862b25dbe30fa
   da844f19e41cf40047424b7118e15dc2c1a815a70fea5c8de6c30aa61440cd4b4b5e8
   f3963fbb2e1
   envelope: 40d6b67fdd7da7c49894750754514dbd2070a407166bd2a5237cca9bf44
   d6e0bb4c0eab6143959a650c5f6b32acf162b1fbe95bb36c5c4f99df53865c4d3537d
   69061d80522d772cd0efdbe91f817f6bf7259a56e20b4eb9cbe9443702f4b759
   handshake_secret: 13e7dc6afa5334b9dfffe26bee3caf744ef4add176caee464cd
   eb3d37303b90de35a8bf095df84471ac77d705f12fe232f1571de1d6a001d3e808998
   73a142dc
   server_mac_key: a58135acfb2bde92d506cf59119729a6404ad94eba294e4b52a63
   baf58cfe03f21bcf735222c7f2c27a60bd958be7f6aed50dc03a78f64e7ae4ac1ff07
   1b95aa
   client_mac_key: 1e1a8ba156aadc4a302f707d2193c9dab477b355f430d450dd407
   ce40dc75613f76ec33dec494f8a6bfdcf951eb060dac33e6572c693954fe92e33730c
   9ab0a2
   oprf_key: 62ef7f7d9506a14600c34f642aaf6ef8019cc82a6755db4fded5248ea14
   6030a

C.1.4.4.  Output Values

   registration_request: 26f3dbfd76b8e5f85b4da604f42889a7d4b1bc919f65538
   1a67de02c59fd5436
   registration_response: 506e8f1b89c098fb89b5b6210a05f7898cafdaea221761
   e8d5272fc39e0f9f08a41e28269b4e97a66468cc00c5a57753e192e15276698977068
   8aa90486ef031
   registration_upload: 0936ea94ab030ec332e29050d266c520e916731a052d05ce
   d7e0cfe751142b486d23c6ed818882f9bdfdcf91389fcbc0b7a3faf92bd0bd6be4a1e
   7730277b694fc7c6ba327fbe786af18487688e0f7c148bbd54dc2fc80c28e7a976d9e
   f53c3540d6b67fdd7da7c49894750754514dbd2070a407166bd2a5237cca9bf44d6e0
   bb4c0eab6143959a650c5f6b32acf162b1fbe95bb36c5c4f99df53865c4d3537d6906
   1d80522d772cd0efdbe91f817f6bf7259a56e20b4eb9cbe9443702f4b759
   KE1: c4dedb0ba6ed5d965d6f250fbe554cd45cba5dfcce3ce836e4aee778aa3cd44d
   da7e07376d6d6f034cfa9bb537d11b8c6b4238c334333d1f0aebb380cae6a6cc10a83
   b9117d3798cb2957fbdb0268a0d63dbf9d66bde5c00c78affd80026c911
   KE2: 9a0e5a1514f62e005ea098b0d8cf6750e358c4389e6add1c52aed9500fa19d00
   38fe59af0df2c79f57b8780278f5ae47355fe1f817119041951c80f612fdfc6d22cc3
   1127d6f0096755be3c3d2dd6287795c317aeea10c9485bf4f419a786642c19a8f151c
   eb5e8767d175248c62c017de94057398d28bf0ed00d1b50ee4f812699bff7663be3c5
   d59de94d8e7e58817c7da005b39c25d25555c929e1c5cf6c1b82837b1367c839aab56
   a422c0d97719426a79a16f9869cf852100597b23b5a071cd9960ecef2fe0d0f749498
   6fa3d8b2bb01963537e60efb13981e138e3d4a141f55f0bef355cfb34ccd468fdacad
   75865ee7efef95f4cb6c25d477f72050267cc22c87edbf3ecaca64cb33bc60dc3bfc5
   51e365f0d46a7fed0e09d96f9afbb48868f5bb3c3e05a86ed8c9476fc22c58306c5a2
   91be34388e09548ba9d70f39
   KE3: d16344e791c3f18594d22ba068984fa18ec1e9bead662b75f66826ffd627932f
   cd1ec40cd01dcf5f63f4055ebe45c7717a57a833aad360256cf1e1c20c0eae1c
   export_key: 9dec51d6d0f6ce7e4345f10961053713b07310cc2e45872f57bbd2fe5
   070fdf0fb5b77c7ddaa2f3dc5c35132df7417ad7fefe0f690ad266e5a54a21d045c9c
   38
   session_key: f6116d3aa0e4089a179713bad4d98ed5cb57e5443cae8d36ef78996f
   a60f3dc6e9fcdd63c001596b06dbc1285d80211035cc0e485506b3f7a650cbf78c5bf
   fc9

C.1.5.  OPAQUE-3DH Real Test Vector 5

C.1.5.1.  Configuration

   OPRF: P256-SHA256
   Hash: SHA256
   KSF: Identity
   KDF: HKDF-SHA256
   MAC: HMAC-SHA256
   Group: P256_XMD:SHA-256_SSWU_RO_
   Context: 4f50415155452d504f43
   Nh: 32
   Npk: 33
   Nsk: 32
   Nm: 32
   Nx: 32
   Nok: 32

C.1.5.2.  Input Values

   oprf_seed: 62f60b286d20ce4fd1d64809b0021dad6ed5d52a2c8cf27ae6582543a0
   a8dce2
   credential_identifier: 31323334
   password: 436f7272656374486f72736542617474657279537461706c65
   envelope_nonce: a921f2a014513bd8a90e477a629794e89fec12d12206dde662ebd
   cf65670e51f
   masking_nonce: 38fe59af0df2c79f57b8780278f5ae47355fe1f817119041951c80
   f612fdfc6d
   server_private_key: c36139381df63bfc91c850db0b9cfbec7a62e86d80040a41a
   a7725bf0e79d5e5
   server_public_key: 035f40ff9cf88aa1f5cd4fe5fd3da9ea65a4923a5594f84fd9
   f2092d6067784874
   server_nonce: 71cd9960ecef2fe0d0f7494986fa3d8b2bb01963537e60efb13981e
   138e3d4a1
   client_nonce: ab3d33bde0e93eda72392346a7a73051110674bbf6b1b7ffab8be4f
   91fdaeeb1
   client_keyshare_seed: 633b875d74d1556d2a2789309972b06db21dfcc4f5ad51d
   7e74d783b7cfab8dc
   server_keyshare_seed: 05a4f54206eef1ba2f615bc0aa285cb22f26d1153b5b40a
   1e85ff80da12f982f
   blind_registration: 411bf1a62d119afe30df682b91a0a33d777972d4f2daa4b34
   ca527d597078153
   blind_login: c497fddf6056d241e6cf9fb7ac37c384f49b357a221eb0a802c989b9
   942256c1

C.1.5.3.  Intermediate Values

   client_public_key: 03b218507d978c3db570ca994aaf36695a731ddb2db272c817
   f79746fc37ae5214
   auth_key: 5bd4be1602516092dc5078f8d699f5721dc1720a49fb80d8e5c16377abd
   0987b
   randomized_password: 06be0a1a51d56557a3adad57ba29c5510565dcd8b5078fa3
   19151b9382258fb0
   envelope: a921f2a014513bd8a90e477a629794e89fec12d12206dde662ebdcf6567
   0e51fad30bbcfc1f8eda0211553ab9aaf26345ad59a128e80188f035fe4924fad67b8
   handshake_secret: 83a932431a8f25bad042f008efa2b07c6cd0faa8285f335b636
   3546a9f9b235f
   server_mac_key: 13e928581febfad28855e3e7f03306d61bd69489686f621535d44
   a1365b73b0d
   client_mac_key: afdc53910c25183b08b930e6953c35b3466276736d9de2e9c5efa
   f150f4082c5
   oprf_key: 2dfb5cb9aa1476093be74ca0d43e5b02862a05f5d6972614d7433acdc66
   f7f31

C.1.5.4.  Output Values

   registration_request: 029e949a29cfa0bf7c1287333d2fb3dc586c41aa652f507
   0d26a5315a1b50229f8
   registration_response: 0350d3694c00978f00a5ce7cd08a00547e4ab5fb5fc2b2
   f6717cdaa6c89136efef035f40ff9cf88aa1f5cd4fe5fd3da9ea65a4923a5594f84fd
   9f2092d6067784874
   registration_upload: 03b218507d978c3db570ca994aaf36695a731ddb2db272c8
   17f79746fc37ae52147f0ed53532d3ae8e505ecc70d42d2b814b6b0e48156def71ea0
   29148b2803aafa921f2a014513bd8a90e477a629794e89fec12d12206dde662ebdcf6
   5670e51fad30bbcfc1f8eda0211553ab9aaf26345ad59a128e80188f035fe4924fad6
   7b8
   KE1: 037342f0bcb3ecea754c1e67576c86aa90c1de3875f390ad599a26686cdfee6e
   07ab3d33bde0e93eda72392346a7a73051110674bbf6b1b7ffab8be4f91fdaeeb1022
   ed3f32f318f81bab80da321fecab3cd9b6eea11a95666dfa6beeaab321280b6
   KE2: 0246da9fe4d41d5ba69faa6c509a1d5bafd49a48615a47a8dd4b0823cc147648
   1138fe59af0df2c79f57b8780278f5ae47355fe1f817119041951c80f612fdfc6d2f0
   c547f70deaeca54d878c14c1aa5e1ab405dec833777132eea905c2fbb12504a67dcbe
   0e66740c76b62c13b04a38a77926e19072953319ec65e41f9bfd2ae26837b6ce688bf
   9af2542f04eec9ab96a1b9328812dc2f5c89182ed47fead61f09f71cd9960ecef2fe0
   d0f7494986fa3d8b2bb01963537e60efb13981e138e3d4a103c1701353219b53acf33
   7bf6456a83cefed8f563f1040b65afbf3b65d3bc9a19b50a73b145bc87a157e8c58c0
   342e2047ee22ae37b63db17e0a82a30fcc4ecf7b
   KE3: e97cab4433aa39d598e76f13e768bba61c682947bdcf9936035e8a3a3ebfb66e
   export_key: c3c9a1b0e33ac84dd83d0b7e8af6794e17e7a3caadff289fbd9dc769a
   853c64b
   session_key: 484ad345715ccce138ca49e4ea362c6183f0949aaaa1125dc3bc3f80
   876e7cd1

C.1.6.  OPAQUE-3DH Real Test Vector 6

C.1.6.1.  Configuration

   OPRF: P256-SHA256
   Hash: SHA256
   KSF: Identity
   KDF: HKDF-SHA256
   MAC: HMAC-SHA256
   Group: P256_XMD:SHA-256_SSWU_RO_
   Context: 4f50415155452d504f43
   Nh: 32
   Npk: 33
   Nsk: 32
   Nm: 32
   Nx: 32
   Nok: 32

C.1.6.2.  Input Values

   client_identity: 616c696365
   server_identity: 626f62
   oprf_seed: 62f60b286d20ce4fd1d64809b0021dad6ed5d52a2c8cf27ae6582543a0
   a8dce2
   credential_identifier: 31323334
   password: 436f7272656374486f72736542617474657279537461706c65
   envelope_nonce: a921f2a014513bd8a90e477a629794e89fec12d12206dde662ebd
   cf65670e51f
   masking_nonce: 38fe59af0df2c79f57b8780278f5ae47355fe1f817119041951c80
   f612fdfc6d
   server_private_key: c36139381df63bfc91c850db0b9cfbec7a62e86d80040a41a
   a7725bf0e79d5e5
   server_public_key: 035f40ff9cf88aa1f5cd4fe5fd3da9ea65a4923a5594f84fd9
   f2092d6067784874
   server_nonce: 71cd9960ecef2fe0d0f7494986fa3d8b2bb01963537e60efb13981e
   138e3d4a1
   client_nonce: ab3d33bde0e93eda72392346a7a73051110674bbf6b1b7ffab8be4f
   91fdaeeb1
   client_keyshare_seed: 633b875d74d1556d2a2789309972b06db21dfcc4f5ad51d
   7e74d783b7cfab8dc
   server_keyshare_seed: 05a4f54206eef1ba2f615bc0aa285cb22f26d1153b5b40a
   1e85ff80da12f982f
   blind_registration: 411bf1a62d119afe30df682b91a0a33d777972d4f2daa4b34
   ca527d597078153
   blind_login: c497fddf6056d241e6cf9fb7ac37c384f49b357a221eb0a802c989b9
   942256c1

C.1.6.3.  Intermediate Values

   client_public_key: 03b218507d978c3db570ca994aaf36695a731ddb2db272c817
   f79746fc37ae5214
   auth_key: 5bd4be1602516092dc5078f8d699f5721dc1720a49fb80d8e5c16377abd
   0987b
   randomized_password: 06be0a1a51d56557a3adad57ba29c5510565dcd8b5078fa3
   19151b9382258fb0
   envelope: a921f2a014513bd8a90e477a629794e89fec12d12206dde662ebdcf6567
   0e51f4d7773a36a208a866301dbb2858e40dc5638017527cf91aef32d3848eebe0971
   handshake_secret: 80bdcc498f22de492e90ee8101fcc7c101e158dd49c77f7c283
   816ae329ed62f
   server_mac_key: 0f82432fbdb5b90daf27a91a3acc42299a9590dba1b77932c2207
   b4cb3d4a157
   client_mac_key: 7f629eb0b1b69979b07ca1f564b3e92ed22f07569fd1d11725d93
   e46731fbe71
   oprf_key: 2dfb5cb9aa1476093be74ca0d43e5b02862a05f5d6972614d7433acdc66
   f7f31

C.1.6.4.  Output Values

   registration_request: 029e949a29cfa0bf7c1287333d2fb3dc586c41aa652f507
   0d26a5315a1b50229f8
   registration_response: 0350d3694c00978f00a5ce7cd08a00547e4ab5fb5fc2b2
   f6717cdaa6c89136efef035f40ff9cf88aa1f5cd4fe5fd3da9ea65a4923a5594f84fd
   9f2092d6067784874
   registration_upload: 03b218507d978c3db570ca994aaf36695a731ddb2db272c8
   17f79746fc37ae52147f0ed53532d3ae8e505ecc70d42d2b814b6b0e48156def71ea0
   29148b2803aafa921f2a014513bd8a90e477a629794e89fec12d12206dde662ebdcf6
   5670e51f4d7773a36a208a866301dbb2858e40dc5638017527cf91aef32d3848eebe0
   971
   KE1: 037342f0bcb3ecea754c1e67576c86aa90c1de3875f390ad599a26686cdfee6e
   07ab3d33bde0e93eda72392346a7a73051110674bbf6b1b7ffab8be4f91fdaeeb1022
   ed3f32f318f81bab80da321fecab3cd9b6eea11a95666dfa6beeaab321280b6
   KE2: 0246da9fe4d41d5ba69faa6c509a1d5bafd49a48615a47a8dd4b0823cc147648
   1138fe59af0df2c79f57b8780278f5ae47355fe1f817119041951c80f612fdfc6d2f0
   c547f70deaeca54d878c14c1aa5e1ab405dec833777132eea905c2fbb12504a67dcbe
   0e66740c76b62c13b04a38a77926e19072953319ec65e41f9bfd2ae268d7f10604202
   1c80300e4c6f585980cf39fc51a4a6bba41b0729f9b240c729e5671cd9960ecef2fe0
   d0f7494986fa3d8b2bb01963537e60efb13981e138e3d4a103c1701353219b53acf33
   7bf6456a83cefed8f563f1040b65afbf3b65d3bc9a19b84922c7e5d074838a8f27859
   2c53f61fb59f031e85ad480c0c71086b871e1b24
   KE3: 46833578cee137775f6be3f01b80748daac5a694101ad0e9e7025480552da56a
   export_key: c3c9a1b0e33ac84dd83d0b7e8af6794e17e7a3caadff289fbd9dc769a
   853c64b
   session_key: 27766fabd8dd88ff37fbd0ef1a491e601d10d9f016c2b28c4bd1b0fb
   7511a3c3

C.2.  Fake Test Vectors

C.2.1.  OPAQUE-3DH Fake Test Vector 1

C.2.1.1.  Configuration

   OPRF: ristretto255-SHA512
   Hash: SHA512
   KSF: Identity
   KDF: HKDF-SHA512
   MAC: HMAC-SHA512
   Group: ristretto255
   Context: 4f50415155452d504f43
   Nh: 64
   Npk: 32
   Nsk: 32
   Nm: 64
   Nx: 64
   Nok: 32

C.2.1.2.  Input Values

   client_identity: 616c696365
   server_identity: 626f62
   oprf_seed: 743fc168d1f826ad43738933e5adb23da6fb95f95a1b069f0daa0522d0
   a78b617f701fc6aa46d3e7981e70de7765dfcd6b1e13e3369a582eb8dc456b10aa53b
   0
   credential_identifier: 31323334
   masking_nonce: 9c035896a043e70f897d87180c543e7a063b83c1bb728fbd189c61
   9e27b6e5a6
   client_private_key: 2b98980aa95ab53a0f39f0291903d2fdf04b00c167f081416
   9922df873002409
   client_public_key: 84f43f9492e19c22d8bdaa4447cc3d4db1cdb5427a9f852c47
   07921212c36251
   server_private_key: c788585ae8b5ba2942b693b849be0c0426384e41977c18d2e
   81fbe30fd7c9f06
   server_public_key: 825f832667480f08b0c9069da5083ac4d0e9ee31b49c4e0310
   031fea04d52966
   server_nonce: 1e10f6eeab2a7a420bf09da9b27a4639645622c46358de9cf7ae813
   055ae2d12
   client_keyshare_seed: a270dc715dc2b4612bc7864312a05c3e9788ee1bad1f276
   d1e15bdeb4c355e94
   server_keyshare_seed: 360b0937f47d45f6123a4d8f0d0c0814b6120d840ebb8bc
   5b4f6b62df07f78c2
   masking_key: 39ebd51f0e39a07a1c2d2431995b0399bca9996c5d10014d6ebab445
   3dc10ce5cef38ed3df6e56bfff40c2d8dd4671c2b4cf63c3d54860f31fe40220d690b
   b71
   KE1: b0a26dcaca2230b8f5e4b1bcab9c84b586140221bb8b2848486874b0be448905
   42d4e61ed3f8d64cdd3b9d153343eca15b9b0d5e388232793c6376bd2d9cfd0ab641d
   7f20a245a09f1d4dbb6e301661af7f352beb0791d055e48d3645232f77f

C.2.1.3.  Output Values

   KE2: 928f79ad8df21963e91411b9f55165ba833dea918f441db967cdc09521d22925
   9c035896a043e70f897d87180c543e7a063b83c1bb728fbd189c619e27b6e5a632b5a
   b1bff96636144faa4f9f9afaac75dd88ea99cf5175902ae3f3b2195693f165f11929b
   a510a5978e64dcdabecbd7ee1e4380ce270e58fea58e6462d92964a1aaef72698bca1
   c673baeb04cc2bf7de5f3c2f5553464552d3a0f7698a9ca7f9c5e70c6cb1f706b2f17
   5ab9d04bbd13926e816b6811a50b4aafa9799d5ed7971e10f6eeab2a7a420bf09da9b
   27a4639645622c46358de9cf7ae813055ae2d1298251c5ba55f6b0b2d58d9ff0c88fe
   4176484be62a96db6e2a8c4d431bd1bf27fe6c1d0537603835217d42ebf7b25819827
   32e74892fd28211b31ed33863f0beaf75ba6f59474c0aaf9d78a60a9b2f4cd24d7ab5
   4131b3c8efa192df6b72db4c

C.2.2.  OPAQUE-3DH Fake Test Vector 2

C.2.2.1.  Configuration

   OPRF: ristretto255-SHA512
   Hash: SHA512
   KSF: Identity
   KDF: HKDF-SHA512
   MAC: HMAC-SHA512
   Group: curve25519
   Context: 4f50415155452d504f43
   Nh: 64
   Npk: 32
   Nsk: 32
   Nm: 64
   Nx: 64
   Nok: 32

C.2.2.2.  Input Values

   client_identity: 616c696365
   server_identity: 626f62
   oprf_seed: 66e650652a8266b2205f31fdd68adeb739a05b5e650b19e7edc75e734a
   1296d6088188ca46c31ae8ccbd42a52ed338c06e53645387a7efbc94b6a0449526155
   e
   credential_identifier: 31323334
   masking_nonce: 9c035896a043e70f897d87180c543e7a063b83c1bb728fbd189c61
   9e27b6e5a6
   client_private_key: 288bf63470199221847bb035d99f96531adf8badd14cb1571
   b48f7a506649660
   client_public_key: 3c64a3153854cc9f0c23aab3c1a19106ec8bab4730736d1d00
   3880a1d5a59005
   server_private_key: 30fbe7e830be1fe8d2187c97414e3826040cbe49b893b6422
   9bab5e85a588846
   server_public_key: 78b3040047ff26572a7619617601a61b9c81899bee92f00cfc
   aa5eed96863555
   server_nonce: 1e10f6eeab2a7a420bf09da9b27a4639645622c46358de9cf7ae813
   055ae2d12
   client_keyshare_seed: a270dc715dc2b4612bc7864312a05c3e9788ee1bad1f276
   d1e15bdeb4c355e94
   server_keyshare_seed: 360b0937f47d45f6123a4d8f0d0c0814b6120d840ebb8bc
   5b4f6b62df07f78c2
   masking_key: 79ad2621b0757a447dff7108a8ae20a068ce67872095620f415ea611
   c9dcc04972fa359538cd2fd6528775ca775487b2b56db642049b8a90526b975a38484
   c6a
   KE1: b0a26dcaca2230b8f5e4b1bcab9c84b586140221bb8b2848486874b0be448905
   42d4e61ed3f8d64cdd3b9d153343eca15b9b0d5e388232793c6376bd2d9cfd0ac059b
   7ba2aec863933ae48816360c7a9022e83d822704f3b0b86c0502a66e574

C.2.2.3.  Output Values

   KE2: 6606b6fedbb33f19a81a1feb5149c600fe77252f58acd3080d7504d3dad4922f
   9c035896a043e70f897d87180c543e7a063b83c1bb728fbd189c619e27b6e5a67db39
   8c0f65d8c298eac430abdae4c80e82b552fb940c00f0cbcea853c0f96c1c15099f3d4
   b0e83ecc249613116d605b8d77bb68bdf76994c2bc507e2dcae4176f00afed68ad25c
   f3040a0e991acece31ca532117f5c12816997372ff031ad04ebcdce06c501da24e7b4
   db95343456e2ed260895ec362694230a1fa20e24a9c71e10f6eeab2a7a420bf09da9b
   27a4639645622c46358de9cf7ae813055ae2d122d9055eb8f83e1b497370adad5cc2a
   417bf9be436a792def0c7b7ccb92b9e275d7c663104ea4655bd70570d975c05351655
   d55fbfb392286edb55600a23b55ce18f8c60e0d1960c960412dd08eabc81ba7ca8ae2
   b04aad65462321f51c298010

C.2.3.  OPAQUE-3DH Fake Test Vector 3

C.2.3.1.  Configuration

   OPRF: P256-SHA256
   Hash: SHA256
   KSF: Identity
   KDF: HKDF-SHA256
   MAC: HMAC-SHA256
   Group: P256_XMD:SHA-256_SSWU_RO_
   Context: 4f50415155452d504f43
   Nh: 32
   Npk: 33
   Nsk: 32
   Nm: 32
   Nx: 32
   Nok: 32

C.2.3.2.  Input Values

   client_identity: 616c696365
   server_identity: 626f62
   oprf_seed: bb1cd59e16ac09bc0cb6d528541695d7eba2239b1613a3db3ade77b362
   80f725
   credential_identifier: 31323334
   masking_nonce: 9c035896a043e70f897d87180c543e7a063b83c1bb728fbd189c61
   9e27b6e5a6
   client_private_key: d423b87899fc61d014fc8330a4e26190fcfa470a3afe59243
   24294af7dbbc1dd
   client_public_key: 03b81708eae026a9370616c22e1e8542fe9dbebd36ce8a2661
   b708e9628f4a57fc
   server_private_key: 34fbe7e830be1fe8d2187c97414e3826040cbe49b893b6422
   9bab5e85a5888c7
   server_public_key: 0221e034c0e202fe883dcfc96802a7624166fed4cfcab4ae30
   cf5f3290d01c88bf
   server_nonce: 1e10f6eeab2a7a420bf09da9b27a4639645622c46358de9cf7ae813
   055ae2d12
   client_keyshare_seed: a270dc715dc2b4612bc7864312a05c3e9788ee1bad1f276
   d1e15bdeb4c355e94
   server_keyshare_seed: 360b0937f47d45f6123a4d8f0d0c0814b6120d840ebb8bc
   5b4f6b62df07f78c2
   masking_key: caecc6ccb4cae27cb54d8f3a1af1bac52a3d53107ce08497cdd362b1
   992e4e5e
   KE1: 0396875da2b4f7749bba411513aea02dc514a48d169d8a9531bd61d3af3fa9ba
   ae42d4e61ed3f8d64cdd3b9d153343eca15b9b0d5e388232793c6376bd2d9cfd0a021
   47a6583983cc9973b5082db5f5070890cb373d70f7ac1b41ed2305361009784

C.2.3.3.  Output Values

   KE2: 0201198dcd13f9792eb75dcfa815f61b049abfe2e3e9456d4bbbceec5f442efd
   049c035896a043e70f897d87180c543e7a063b83c1bb728fbd189c619e27b6e5a6fac
   da65ce0a97b9085e7af07f61fd3fdd046d257cbf2183ce8766090b8041a8bf28d79dd
   4c9031ddc75bb6ddb4c291e639937840e3d39fc0d5a3d6e7723c09f7945df485bcf9a
   efe3fe82d149e84049e259bb5b33d6a2ff3b25e4bfb7eff0962821e10f6eeab2a7a42
   0bf09da9b27a4639645622c46358de9cf7ae813055ae2d12023f82bbb24e75b8683fd
   13b843cd566efae996cd0016cffdcc24ee2bc937d026f80144878749a69565b433c10
   40aff67e94f79345de888a877422b9bbe21ec329

Acknowledgments

   We are indebted to the OPAQUE reviewers during CFRG's aPAKE selection
   process, particularly Julia Hesse and Bjorn Tackmann.  This document
   has benefited from comments by multiple people.  Special thanks to
   Richard Barnes, Dan Brown, Matt Campagna, Eric Crockett, Paul Grubbs,
   Fredrik Kuivinen, Stefan Marsiske, Payman Mohassel, Marta Mularczyk,
   Jason Resch, Greg Rubin, and Nick Sullivan.  Hugo Krawczyk wishes to
   thank his OPAQUE co-authors Stas Jarecki and Jiayu Xu, without whom
   this work would have not been possible.

Authors' Addresses

   Daniel Bourdrez
   Email: d@bytema.re


   Hugo Krawczyk
   AWS
   Email: hugokraw@gmail.com


   Kevin Lewi
   Meta
   Email: lewi.kevin.k@gmail.com


   Christopher A. Wood
   Cloudflare, Inc.
   Email: caw@heapingbits.net
