Internet-Draft OPAQUE July 2022
Bourdrez, et al. Expires 7 January 2023 [Page]
Workgroup:
Network Working Group
Internet-Draft:
draft-irtf-cfrg-opaque-09
Published:
Intended Status:
Informational
Expires:
Authors:
D. Bourdrez
H. Krawczyk
Algorand Foundation
K. Lewi
Novi Research
C. A. Wood
Cloudflare, Inc.

The OPAQUE Asymmetric PAKE Protocol

Abstract

This document describes the OPAQUE protocol, a secure asymmetric password-authenticated key exchange (aPAKE) 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.

Discussion Venues

This note is to be removed before publishing as an RFC.

Source for this draft and an issue tracker can be found at https://github.com/cfrg/draft-irtf-cfrg-opaque.

Status of This Memo

This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79.

Internet-Drafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as Internet-Drafts. The list of current Internet-Drafts is at https://datatracker.ietf.org/drafts/current/.

Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress."

This Internet-Draft will expire on 7 January 2023.

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Table of Contents

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 to cases where TLS may fail, including PKI attacks, certificate mishandling, termination outside the security perimeter, visibility to TLS-terminating intermediaries, and more.

Asymmetric (or Augmented) 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. Crucially important is 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, a PKI-free secure aPAKE that is secure against pre-computation attacks. 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 [_3DH]. Other instantiations are possible, as discussed in Appendix C, 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, and with future ones 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 draft complies with the requirements for PAKE protocols set forth in [RFC8125].

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): Generate a cryptographically secure pseudorandom byte string of length n bytes.
  • 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 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 [RFC8446], Section 3.

2. Cryptographic Dependencies

OPAQUE depends on the following cryptographic protocols and primitives:

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 PRF such that the client learns the PRF output and neither party learns the input of the other. This specification depends on the prime-order OPRF construction specified in [OPRF], draft version -10, using the OPRF mode (0x00) from [OPRF], Section 3.1.

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.
  • Finalize(element, blind, evaluated_element): Finalize the OPRF evaluation using input element, random inverter blind, and evaluation output evaluated_element, yielding output oprf_output.

Moreover, the following OPRF server APIs are used:

  • Evaluate(k, blinded_element): Evaluate blinded input element blinded_element using input key k, yielding output element evaluated_element. This is equivalent to the Evaluate function described in [OPRF], Section 3.3.1, where k is the private key parameter.
  • DeriveKeyPair(seed, info): Derive a private and public key pair deterministically from a seed and info parameter, as described in [OPRF], Section 3.2.

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

  • SerializeElement(element): Map input element to a fixed-length byte array buf.
  • DeserializeElement(buf): Attempt to map input byte array buf to an OPRF group element. This function can raise a DeserializeError upon failure; see [OPRF], Section 2.1 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 optional 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 collision-resistant Message Authentication Code (MAC) with the following API and parameters:

  • 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, params): Apply a key stretching function with parameters params 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. 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" 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, and chooses a seed (oprf_seed) of Nh bytes for the OPRF. The server can use this single pair of keys with multiple clients and can opt to use multiple seeds (so long as they are kept consistent for each client).

3.2. Offline Registration

Registration is the only stage in OPAQUE that requires a server-authenticated and confidential channel: 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., 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 below:

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

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 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 below:

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

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), and their corresponding wire formats are specified in Section 6.1.

The rest of this document describes the details 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.

Future variants of OPAQUE may use different key recovery mechanisms. See Section 4.1 for details.

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:

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

  Create CleartextCredentials cleartext_credentials with
    (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 DeriveAuthKeyPair() function defined in Section 6.4.1.

4.1.1. Envelope Structure

The key recovery mechanism defines its Envelope as follows:

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

nonce: A unique nonce of length Nn, used to protect this Envelope.

auth_tag: An authentication tag protecting the contents of the envelope, covering the envelope nonce and CleartextCredentials.

4.1.2. Envelope Creation

Clients create an Envelope at registration with the function Store defined below. Note that DeriveAuthKeyPair 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_pwd, 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_pwd, server_public_key, server_identity, client_identity):
  envelope_nonce = random(Nn)
  masking_key = Expand(randomized_pwd, "MaskingKey", Nh)
  auth_key = Expand(randomized_pwd, concat(envelope_nonce, "AuthKey"), Nh)
  export_key = Expand(randomized_pwd, concat(envelope_nonce, "ExportKey"), Nh)
  seed = Expand(randomized_pwd, concat(envelope_nonce, "PrivateKey"), Nseed)
  (_, client_public_key) = DeriveAuthKeyPair(seed)

  cleartext_creds =
    CreateCleartextCredentials(server_public_key, client_public_key,
                               server_identity, client_identity)
  auth_tag = MAC(auth_key, concat(envelope_nonce, cleartext_creds))

  Create Envelope envelope with (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_pwd, 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.
- export_key, an additional client key.

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

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

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

5. Offline Registration

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

The server inputs the following values:

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(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. 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 10.5.

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 to preserve confidentiality of the envelope during login to defend 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 a InvalidInputError error with negligibile 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)
  Create RegistrationRequest request with 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, application can choose a new credential_identifier for this registration record and re-run 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 = Evaluate(oprf_key, blinded_element)
  evaluated_message = SerializeElement(evaluated_element)

  Create RegistrationResponse response with (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, params)
  randomized_pwd = Extract("", concat(oprf_output, stretched_oprf_output))

  (envelope, client_public_key, masking_key, export_key) =
    Store(randomized_pwd, response.server_public_key,
          server_identity, client_identity)
  Create RegistrationRecord record with (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, where KE1 and KE2 include key exchange shares, e.g., DH values, sent by the client and server, respectively, and 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:

The server inputs the following values:

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 10.5. The output export_key MUST NOT be used in any way before the protocol completes successfully. See Appendix B 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 = ClientInit(password)

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

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

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

    (ke3,
    session_key,
    export_key) = ClientFinish(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:

The server state ServerState may have the following fields:

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_keyshare[Npk];
} AuthRequest;

client_nonce: A fresh randomly generated nonce of length Nn.

client_keyshare: A serialized client ephemeral key share 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_keyshare[Npk];
  uint8 server_mac[Nm];
} AuthResponse;

server_nonce: A fresh randomly generated nonce of length Nn.

server_keyshare: Server ephemeral key share 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, 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, RecoverCredentials defined in Section 6.3
  • AuthClientStart, AuthServerRespond, AuthClientFinalize, and AuthServerFinalize defined in Section 6.4.3 and Section 6.4.4

6.2.1. ClientInit

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

ClientInit

State:
- state, a ClientState structure.

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

Output:
- ke1, a KE1 message structure.

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

6.2.2. ServerInit

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

ServerInit

State:
- state, a ServerState structure.

Input:
- server_identity, the optional encoded server identity, which is set to
  server_public_key if nil.
- 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 encoded client identity.

Output:
- ke2, a KE2 structure.

def ServerInit(server_identity, server_private_key, server_public_key,
               record, credential_identifier, oprf_seed, ke1, client_identity):
  credential_response = CreateCredentialResponse(ke1.request, server_public_key, record,
    credential_identifier, oprf_seed)
  auth_response = AuthServerRespond(server_identity, server_private_key,
    client_identity, record.client_public_key, ke1, credential_response)
  Create KE2 ke2 with (credential_response, auth_response)

6.2.3. ClientFinish

The ClientFinish 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.

ClientFinish

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 ClientFinish(client_identity, server_identity, ke2):
  (client_private_key, server_public_key, export_key) =
    RecoverCredentials(state.password, state.blind, ke2.credential_response,
                       server_identity, client_identity)
  (ke3, session_key) =
    AuthClientFinalize(client_identity, client_private_key, server_identity,
                       server_public_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, and it, therefore, 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, 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 + Ne];
} 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 a InvalidInputError error with negligibile 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)
  Create CredentialRequest request with 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 a client that has yet to register).

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 = Evaluate(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 + Ne)
  masked_response = xor(credential_response_pad,
                        concat(server_public_key, record.envelope))
  Create CredentialResponse response with (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 Ne

It is RECOMMENDED that a fake client record is created once (e.g. as the first user record of the application) and stored alongside legitimate client records. This allows servers to locate 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 client's private key for the AKE protocol.
- server_public_key, the public key of the server.
- 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, params)
  randomized_pwd = Extract("", concat(oprf_output, stretched_oprf_output))

  masking_key = Expand(randomized_pwd, "MaskingKey", Nh)
  credential_response_pad = Expand(masking_key,
                                   concat(response.masking_nonce, "CredentialResponsePad"),
                                   Npk + Ne)
  concat(server_public_key, envelope) = xor(credential_response_pad,
                                              response.masked_response)
  (client_private_key, export_key) =
    Recover(randomized_pwd, server_public_key, envelope,
            server_identity, client_identity)

  return (client_private_key, server_public_key, export_key)

6.4. AKE Protocol

This section describes the authenticated key exchange protocol for OPAQUE using 3DH, a 3-message AKE which 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.

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

6.4.1. Key Creation

We assume the following functions to exist for all candidate groups in this setting:

  • DeriveAuthKeyPair(seed): Derive a private and public authentication key pair deterministically from the input seed. This function is implemented as DeriveKeyPair(seed, "OPAQUE-DeriveAuthKeyPair"), where DeriveKeyPair is as specified in [OPRF], Section 3.2.
  • GenerateAuthKeyPair(): Return a randomly generated private and public key pair. This can be implemented by invoking DeriveAuthKeyPair with Nseed random bytes as input.
  • SerializeElement(element): A member function of the underlying group that maps element to a unique byte array, mirrored from the definition of the similarly-named function of the OPRF group described in [OPRF], Section 2.1.

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, re-purposed from TLS 1.3 [RFC8446].

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

Where CustomLabel is specified 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 shared context information in the transcript, such as configuration parameters or application-specific info, 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_keyshare, the corresponding field on the AuthResponse structure.

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

def Preamble(client_identity, ke1, server_identity, ke2):
  preamble = concat("RFCXXXX",
                     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_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_secret, client_keyshare) = GenerateAuthKeyPair()
  Create AuthRequest auth_request with (client_nonce, client_keyshare)
  Create KE1 ke1 with (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:
- client_identity, the optional encoded client identity, which is
  set to client_public_key if not specified.
- client_private_key, the client's private key.
- server_identity, the optional encoded server identity, which is
  set to server_public_key if not specified.
- server_public_key, the server's public key.
- ke2, a KE2 message structure.

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

Exceptions:
- ServerAuthenticationError, the handshake fails.

def AuthClientFinalize(client_identity, client_private_key, server_identity,
                       server_public_key, ke2):

  dh1 = SerializeElement(state.client_secret * ke2.auth_response.server_keyshare)
  dh2 = SerializeElement(state.client_secret * server_public_key)
  dh3 = SerializeElement(client_private_key  * ke2.auth_response.server_keyshare)
  ikm = concat(dh1, dh2, dh3)

  preamble = Preamble(client_identity,
                      state.ke1,
                      server_identity,
                      ke2.credential_response,
                      ke2.auth_response.server_nonce,
                      ke2.auth_response.server_keyshare)
  Km2, Km3, session_key = DeriveKeys(ikm, preamble)
  expected_server_mac = MAC(Km2, Hash(preamble))
  if !ct_equal(ke2.server_mac, expected_server_mac),
    raise ServerAuthenticationError
  client_mac = MAC(Km3, Hash(concat(preamble, expected_server_mac))
  Create KE3 ke3 with 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:
- 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.
- client_identity, the optional encoded client identity, which is set to
  client_public_key if not specified.
- client_public_key, the client's public key.
- ke1, a KE1 message structure.

Output:
- auth_response, an AuthResponse structure.

def AuthServerRespond(server_identity, server_private_key, client_identity,
                      client_public_key, ke1, credential_response):
  server_nonce = random(Nn)
  (server_private_keyshare, server_keyshare) = GenerateAuthKeyPair()
  preamble = Preamble(client_identity,
                      ke1,
                      server_identity,
                      credential_response,
                      server_nonce,
                      server_keyshare)

  dh1 = SerializeElement(server_private_keyshare * ke1.auth_request.client_keyshare)
  dh2 = SerializeElement(server_private_key * ke1.auth_request.client_keyshare)
  dh3 = SerializeElement(server_private_keyshare * client_public_key)
  ikm = concat(dh1, dh2, dh3)

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

  state.expected_client_mac = MAC(Km3, Hash(concat(preamble, server_mac))
  state.session_key = session_key
  Create AuthResponse auth_response with (server_nonce, server_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:

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:

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 which applications can use to control OPAQUE:

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, implementations can incorporate the client identity alongside the password as input to the OPRF. This provides additional client-side entropy which can supplement the entropy that should be introduced by the server during an honest execution of the protocol. This also provides domain separation between different clients that might otherwise share the same password.

Finally, note that 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. Implementations and deployments of OPAQUE SHOULD consider additional checks to mitigate this type of attack: for instance, by ensuring that there is a server-authenticated channel over which OPAQUE registration and login are run.

9.2. 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 KCI security and forward secrecy; see below) into a secure aPAKE protocol. In OPAQUE, the client stores a secret private key at the server 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

[[RFC EDITOR: Please delete this section before publication.]]

The specification as written here differs from the original cryptographic design in [JKX18] and the corresponding CFRG document [I-D.krawczyk-cfrg-opaque-03], 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 UC modeling. An upcoming paper 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 variant is also analyzed in the new paper referred to in the previous item. This change improves the assumption behind the protocol by getting rid of equivocality and random key robustness for the encryption function. The latter property is only required for authentication and achieved by a collision-resistant MAC. 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 authentication 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 but 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.
  • 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 independent of other values in the protocol 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 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 password file 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 OPRF has different PRF inputs, the protocol is secure. The choice of such inputs is up to the application.

The following list enumerates notable differences and refinements from the original cryptographic design in [JKX18] and the corresponding CFRG document [I-D.krawczyk-cfrg-opaque-03] 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 [I-D.irtf-cfrg-voprf], Section 2.1. This specification also delegates OPRF group choice and operations to [I-D.irtf-cfrg-voprf]. As such, the prime-order group as used in the OPRF and 3DH as specified in this document both adhere to the requirements as [JKX18].
  • [JKX18] specified DH-OPRF (see Appendix B) 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 the [I-D.irtf-cfrg-voprf], which is identical to the DH-OPRF functionality in [JKX18] and, concretely, uses the hash-to-curve functions in [I-D.irtf-cfrg-hash-to-curve]. All hash-to-curve methods in [I-D.irtf-cfrg-hash-to-curve] are compliant with the requirement in [JKX18], namely, that the output be a member of the prime-order group.
  • [JKX18] and [I-D.krawczyk-cfrg-opaque-03] 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 C.1). Since 3DH satisfies the essential requirements for the AKE as described in [JKX18] and [I-D.krawczyk-cfrg-opaque-03], 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 in [JKX18], Figure 12 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, such as is done for key exchange protocols like TLS 1.3 [RFC8446].
  • Neither [JKX18] nor [I-D.krawczyk-cfrg-opaque-03] 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 D.

10.2. Security Analysis

Jarecki et al. [JKX18] proved the security of OPAQUE in a strong aPAKE model that ensures security against pre-computation attacks and is formulated in the Universal Composability (UC) framework [Canetti01] under the random oracle model. This assumes security of the OPRF function and the underlying key exchange protocol. In turn, the security of the OPRF protocol from [OPRF] is proven in the random oracle model under the One-More Diffie-Hellman assumption [JKKX16].

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 pre-computation 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).

10.4. 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.

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 (hence determined during password registration) and to 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 be 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.

10.5. Export Key Usage

The export key can be used (separately from the OPAQUE protocol) to provide confidentiality and integrity to other data which only the client should be able to process. For instance, if the server is expected to maintain any client-side secrets which require a password to access, then this export key can be used to encrypt these secrets so that they remain hidden from the server.

10.6. 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 [OPRF] 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.

10.7. 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 offline 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 at the server. Applications SHOULD select parameters that balance cost and complexity. Note that in OPAQUE, the key stretching function is executed by the client, as opposed to the server. 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 extra information about the behavior of clients that have registered with the server. There are two types of attacks we consider:

1) An attacker tries to learn whether a given client identity is registered with a server, and 2) An attacker tries to learn whether a given client identity has recently completed registration, re-registered (e.g. after a password change), or changed its identity.

OPAQUE prevents these attacks during the authentication flow. The first is prevented 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 also encrypting both 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.

Preventing the second type of attack requires the server to supply a credential_identifier value for a given client identity, consistently between the registration response and credential response; see Section 5.2.2 and Section 6.3.2.2. Note that credential_identifier can be set to client_identity for simplicity.

In the event of a server compromise that results in a re-registration of credentials for all compromised clients, the oprf_seed value MUST be resampled, resulting in a change in the oprf_key value for each client. Although this change can be detected by an adversary, it is only leaked upon password rotation after the exposure of the credential files, and equally affects all registered clients.

Finally, applications must use the same key recovery mechanism when using this prevention throughout their lifecycle. The envelope size may vary between mechanisms, so a switch could then be detected.

OPAQUE does not prevent either 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.

10.10. Protecting the Registration Masking Key

The user enumeration prevention method described in this documents uses a symmetric encryption key generated by the client on registration that is sent to the server over an authenticated channel, such as one provided by TLS [RFC8446]. In the event that this channel is compromised, this encryption key could be leaked to an attacker.

One mitigation against this threat is to additionally encrypt the RegistrationRecord sent from client to server at the application layer using public key encryption, e.g., with HPKE [RFC9180]. However, the details of this mechanism are out of scope of 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 pre-computation 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 aPAKE protocol. (OPAQUE enables defense against such offline dictionary attacks by distributing the server so that an offline attack is only possible if all - or a minimal number of - servers are compromised [JKX18].) Furthermore, if the server does not sample this OPRF key 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 for enforcing password rules. Doing so invalidates this important security property of OPAQUE and is NOT RECOMMENDED. Applications should move such checks to the client. Note that limited checks at the server are possible to implement, e.g., detecting repeated passwords.

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 the 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.

11. IANA Considerations

This document makes no IANA requests.

12. References

12.1. Normative References

[I-D.irtf-cfrg-voprf]
Davidson, A., Faz-Hernandez, A., Sullivan, N., and C. A. Wood, "Oblivious Pseudorandom Functions (OPRFs) using Prime-Order Groups", Work in Progress, Internet-Draft, draft-irtf-cfrg-voprf-11, , <https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-voprf-11>.
[OPRF]
Davidson, A., Faz-Hernandez, A., Sullivan, N., and C. A. Wood, "Oblivious Pseudorandom Functions (OPRFs) using Prime-Order Groups", Work in Progress, Internet-Draft, draft-irtf-cfrg-voprf-11, , <https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-voprf-11>.
[RFC2104]
Krawczyk, H., Bellare, M., and R. Canetti, "HMAC: Keyed-Hashing for Message Authentication", RFC 2104, DOI 10.17487/RFC2104, , <https://www.rfc-editor.org/rfc/rfc2104>.
[RFC2119]
Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, , <https://www.rfc-editor.org/rfc/rfc2119>.
[RFC4086]
Eastlake 3rd, D., Schiller, J., and S. Crocker, "Randomness Requirements for Security", BCP 106, RFC 4086, DOI 10.17487/RFC4086, , <https://www.rfc-editor.org/rfc/rfc4086>.
[RFC8174]
Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, , <https://www.rfc-editor.org/rfc/rfc8174>.

12.2. Informative References

[ARGON2]
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, , <https://www.rfc-editor.org/rfc/rfc9106>.
[AuCPace]
Haase, B. and B. Labrique, "AuCPace: Efficient verifier-based PAKE protocol tailored for the IIoT", http://eprint.iacr.org/2018/286 , .
[BG04]
Brown, D. and R. Galant, "The static Diffie-Hellman problem", http://eprint.iacr.org/2004/306 , .
[Boyen09]
Boyen, X., "HPAKE: Password Authentication Secure against Cross-Site User Impersonation", Cryptology and Network Security (CANS) , .
[Canetti01]
Canetti, R., "Universally composable security: A new paradigm for cryptographic protocols", IEEE Symposium on Foundations of Computer Science (FOCS) , .
[Cheon06]
Cheon, J. H., "Security analysis of the strong Diffie-Hellman problem", Euroctypt 2006 , .
[FIPS202]
National Institute of Standards and Technology (NIST), "SHA-3 Standard: Permutation-Based Hash and Extendable-Output Functions", , <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", WETICE , .
[GMR06]
Gentry, C., MacKenzie, P., and Z, Ramzan, "A method for making password-based key exchange resilient to server compromise", CRYPTO , .
[HMQV]
Krawczyk, H., "HMQV: A high-performance secure Diffie-Hellman protocol", CRYPTO , .
[I-D.irtf-cfrg-hash-to-curve]
Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R. S., and C. A. Wood, "Hashing to Elliptic Curves", Work in Progress, Internet-Draft, draft-irtf-cfrg-hash-to-curve-16, , <https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-16>.
[I-D.krawczyk-cfrg-opaque-03]
"The OPAQUE Asymmetric PAKE Protocol", n.d., <https://datatracker.ietf.org/doc/html/draft-krawczyk-cfrg-opaque-03>.
[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)", IEEE European Symposium on Security and Privacy , .
[JKX18]
Jarecki, S., Krawczyk, H., and J. Xu, "OPAQUE: An Asymmetric PAKE Protocol Secure Against Pre-Computation Attacks", Eurocrypt , .
[keyagreement]
Barker, E., Chen, L., Roginsky, A., Vassilev, A., and R. Davis, "Recommendation for pair-wise key-establishment schemes using discrete logarithm cryptography", National Institute of Standards and Technology report, DOI 10.6028/nist.sp.800-56ar3, , <https://doi.org/10.6028/nist.sp.800-56ar3>.
[LGR20]
Len, J., Grubbs, P., and T. Ristenpart, "Partitioning Oracle Attacks", n.d., <https://eprint.iacr.org/2020/1491.pdf>.
[PAKE-Selection]
"CFRG PAKE selection process repository", n.d., <https://github.com/cfrg/pake-selection>.
[PBKDF2]
Kaliski, B., "PKCS #5: Password-Based Cryptography Specification Version 2.0", RFC 2898, DOI 10.17487/RFC2898, , <https://www.rfc-editor.org/rfc/rfc2898>.
[RFC2945]
Wu, T., "The SRP Authentication and Key Exchange System", RFC 2945, DOI 10.17487/RFC2945, , <https://www.rfc-editor.org/rfc/rfc2945>.
[RFC5869]
Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand Key Derivation Function (HKDF)", RFC 5869, DOI 10.17487/RFC5869, , <https://www.rfc-editor.org/rfc/rfc5869>.
[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, , <https://www.rfc-editor.org/rfc/rfc8017>.
[RFC8125]
Schmidt, J., "Requirements for Password-Authenticated Key Agreement (PAKE) Schemes", RFC 8125, DOI 10.17487/RFC8125, , <https://www.rfc-editor.org/rfc/rfc8125>.
[RFC8446]
Rescorla, E., "The Transport Layer Security (TLS) Protocol Version 1.3", RFC 8446, DOI 10.17487/RFC8446, , <https://www.rfc-editor.org/rfc/rfc8446>.
[RFC9180]
Barnes, R., Bhargavan, K., Lipp, B., and C. Wood, "Hybrid Public Key Encryption", RFC 9180, DOI 10.17487/RFC9180, , <https://www.rfc-editor.org/rfc/rfc9180>.
[SCRYPT]
Percival, C. and S. Josefsson, "The scrypt Password-Based Key Derivation Function", RFC 7914, DOI 10.17487/RFC7914, , <https://www.rfc-editor.org/rfc/rfc7914>.
[SPAKE2plus]
Shoup, V., "Security Analysis of SPAKE2+", http://eprint.iacr.org/2020/313 , .
[WhatsAppE2E]
"Security of End-to-End Encrypted Backups", n.d., <https://www.whatsapp.com/security/WhatsApp_Security_Encrypted_Backups_Whitepaper.pdf>.
[_3DH]
"Simplifying OTR deniability", https://signal.org/blog/simplifying-otr-deniability , .

Appendix A. Acknowledgments

The OPAQUE protocol and its analysis is the joint work of the author with Stanislaw Jarecki and Jiayu Xu. We are indebted to the OPAQUE reviewers during CFRG's aPAKE selection process, particularly Julia Hesse and Bjorn Tackmann. This draft has benefited from comments by multiple people. Special thanks to Richard Barnes, Dan Brown, Eric Crockett, Paul Grubbs, Fredrik Kuivinen, Payman Mohassel, Jason Resch, Greg Rubin, and Nick Sullivan.

Appendix B. 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 C. Alternate AKE Instantiations

It is possible to instantiate OPAQUE with other AKEs, such as HMQV [HMQV] and 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 is quantum-safe, but the OPRF is not, would still ensure the confidentiality of application data encrypted under session_key (or a key derived from it) with a quantum-safe encryption function.

C.1. HMQV Instantiation Sketch

An HMQV instantiation would work similar 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_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.

C.2. SIGMA-I Instantiation Sketch

A 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.

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 key shares exchanged between client and server.

Appendix D. Test Vectors

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

Similarly, each fake test vector in Appendix D.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 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. These test vectors were generated using draft-10 of [OPRF].

D.1. Real Test Vectors

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

D.1.1.1. Configuration 
OPRF: 0001
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
D.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
server_keyshare: c8c39f573135474c51660b02425bca633e339cec4e1acc69c94d
d48497fe4028
client_keyshare: 0c3a00c961fead8a16f818929cc976f0475e4f723519318b96f4
947a7a5f9663
server_private_keyshare: 2e842960258a95e28bcfef489cffd19d8ec99cc1375d
840f96936da7dbb0b40d
client_private_keyshare: 22c919134c9bdd9dc0c5ef3450f18b54820f43f646a9
5223bf4a85b2018c2001
blind_registration: 76cfbfe758db884bebb33582331ba9f159720ca8784a2a070
a265d9c2d6abe01
blind_login: 6ecc102d2e7a7cf49617aad7bbe188556792d4acd60a1a8a8d2b65d4
b0790308
D.1.1.3. Intermediate Values 
client_public_key: 8e5e5c04b2154336fa52ac691eb6df5f59ec7315b8467b0bba
1ed4f413043b44
auth_key: e1ff65c196e1c4b4bf46361798eec479b318831329680f33b4f77ad49d8
c6e6ef49d87082d654d21f2e36454582353fefc23c07637bd8ca4aa88a4461ea96d6c
randomized_pwd: 4386bf4b83db06f47672fd60b4cface554558da7be3c616c56b2e
d29b544d1b50bc45893b1c05d8d6866a9bbe91395e4704740be58728e8872352f56d5
319f8f
envelope: ac13171b2f17bc2c74997f0fce1e1f35bec6b91fe2e12dbd323d23ba7a3
8dfec8e8bde8d4eb9e171240b3d2dfb43ef93efe5cd15412614b3df11ecb58890047e
2fa31c283e7c58c40495226cfa0ed7756e493431b85c464aad7fdaaf1ab41ac7
handshake_secret: 885a0a7bd8e704d8fc26f62b8657f8c5d01ffb35b27ad538493
968dcf6dba7a2d42d404d6ed6a87805a030ffafe791fb69fd044c1ac152ee0ee78853
cebb0700
server_mac_key: d29e33eb506fbf199c818d1300e7253404a7d5de9c660a90f79af
e4cc15da2ae31e511c6eb1c4df95f47c9759606732781a3d1884a4d53cba690bdb9e9
ac4d7c
client_mac_key: 4d4d4c4b8b35501876ed01d07f5718357ff720163b84813b1bde4
f3b6ca3e1de744a267e3d145e6095a0e5b1617714e10af7e10093d0ba8dd115e6bdb1
f5ccd9
oprf_key: 6c246eaa55e47d0490ffa8a6f784e803eed9384a250458def36a2acebf1
5c905
D.1.1.4. Output Values 
registration_request: 62235332ae15911d69812e9eeb6ac8fe4fa0ffc7590831d
5c5e1631e01049276
registration_response: 6268d13fea98ebc8e6b88d0b3cc8a78d2ac8fa8efc741c
d2e966940c52c31c71b2fe7af9f48cc502d016729d2fe25cdd433f2c4bc904660b2a3
82c9b79df1a78
registration_upload: 8e5e5c04b2154336fa52ac691eb6df5f59ec7315b8467b0b
ba1ed4f413043b449afea0ddedbbce5c083c5d5d02aa5218bcc7100f541d841bb5974
f084f7aa0b929399feb39efd17e13ce1035cbb23251da3b5126a574b239c7b73519d8
847e2fac13171b2f17bc2c74997f0fce1e1f35bec6b91fe2e12dbd323d23ba7a38dfe
c8e8bde8d4eb9e171240b3d2dfb43ef93efe5cd15412614b3df11ecb58890047e2fa3
1c283e7c58c40495226cfa0ed7756e493431b85c464aad7fdaaf1ab41ac7
KE1: 1670c409ebb699a6012629451d218d42a34eddba1d2978536c45e199c60a0b4e
da7e07376d6d6f034cfa9bb537d11b8c6b4238c334333d1f0aebb380cae6a6cc0c3a0
0c961fead8a16f818929cc976f0475e4f723519318b96f4947a7a5f9663
KE2: 36b4d06f413b72004392d7359cd6a998c667533203d6a671afe81ca09a282f72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KE3: 4e23f0f84a5261918a7fc23bf1978a935cf4e320d56984079f8c7f4a54847b9e
979f519928c5898927cf6aa8d51ac42dc2d0f5840956caa3a34dbc55ce74415f
export_key: 403a270110164ae0de7ea77c6824343211e8c1663ccaedde908dc9acf
661039a379c8ac7e4b0cb23a8d1375ae94a772f91536de131d9d86633cb9445f773df
ac
session_key: d2dea308255aa3cecf72bcd6ac96ff7ab2e8bad0494b90180ad340b7
d8942a36ee358e76c372790d4a5c1ac900997ea2abbf35f2d65510f8dfd668e593b8e
1fe

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

D.1.2.1. Configuration 
OPRF: 0001
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
D.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
server_keyshare: c8c39f573135474c51660b02425bca633e339cec4e1acc69c94d
d48497fe4028
client_keyshare: 0c3a00c961fead8a16f818929cc976f0475e4f723519318b96f4
947a7a5f9663
server_private_keyshare: 2e842960258a95e28bcfef489cffd19d8ec99cc1375d
840f96936da7dbb0b40d
client_private_keyshare: 22c919134c9bdd9dc0c5ef3450f18b54820f43f646a9
5223bf4a85b2018c2001
blind_registration: 76cfbfe758db884bebb33582331ba9f159720ca8784a2a070
a265d9c2d6abe01
blind_login: 6ecc102d2e7a7cf49617aad7bbe188556792d4acd60a1a8a8d2b65d4
b0790308
D.1.2.3. Intermediate Values 
client_public_key: 8e5e5c04b2154336fa52ac691eb6df5f59ec7315b8467b0bba
1ed4f413043b44
auth_key: e1ff65c196e1c4b4bf46361798eec479b318831329680f33b4f77ad49d8
c6e6ef49d87082d654d21f2e36454582353fefc23c07637bd8ca4aa88a4461ea96d6c
randomized_pwd: 4386bf4b83db06f47672fd60b4cface554558da7be3c616c56b2e
d29b544d1b50bc45893b1c05d8d6866a9bbe91395e4704740be58728e8872352f56d5
319f8f
envelope: ac13171b2f17bc2c74997f0fce1e1f35bec6b91fe2e12dbd323d23ba7a3
8dfec43084457c1ffa561c8f37fbad1b8de6c41e6df200e6ebe15d5ce4243fa973ef3
e480644e56a6de865cc4d3d9e20e0510e63474e2b11f4b4c8f665cc439cc2d7d
handshake_secret: 19d0d9f286f44f573dd61435690b0359c3a70e5c363ba4819ac
fa113b0ddeab603f322185812ddcdd2abbfba77933cd5c3430ea6591e99c30a19884a
80d25dab
server_mac_key: 5096c1f1b295521bc8c5aeba462fc11e123eb710899f164dab737
45f55f42b27a31f810efb06fc56890f3635a18f3f8c9ef7881f32a251a5f5a7354c82
70f257
client_mac_key: 1c284c2a22bfb415a5091c94726dd02ae9adb12d28db5207a87be
0c3f75c1c37df549315f51e0dd2053271a477a45bf0adbc246f7f7e47e201785b6429
e93a84
oprf_key: 6c246eaa55e47d0490ffa8a6f784e803eed9384a250458def36a2acebf1
5c905
D.1.2.4. Output Values 
registration_request: 62235332ae15911d69812e9eeb6ac8fe4fa0ffc7590831d
5c5e1631e01049276
registration_response: 6268d13fea98ebc8e6b88d0b3cc8a78d2ac8fa8efc741c
d2e966940c52c31c71b2fe7af9f48cc502d016729d2fe25cdd433f2c4bc904660b2a3
82c9b79df1a78
registration_upload: 8e5e5c04b2154336fa52ac691eb6df5f59ec7315b8467b0b
ba1ed4f413043b449afea0ddedbbce5c083c5d5d02aa5218bcc7100f541d841bb5974
f084f7aa0b929399feb39efd17e13ce1035cbb23251da3b5126a574b239c7b73519d8
847e2fac13171b2f17bc2c74997f0fce1e1f35bec6b91fe2e12dbd323d23ba7a38dfe
c43084457c1ffa561c8f37fbad1b8de6c41e6df200e6ebe15d5ce4243fa973ef3e480
644e56a6de865cc4d3d9e20e0510e63474e2b11f4b4c8f665cc439cc2d7d
KE1: 1670c409ebb699a6012629451d218d42a34eddba1d2978536c45e199c60a0b4e
da7e07376d6d6f034cfa9bb537d11b8c6b4238c334333d1f0aebb380cae6a6cc0c3a0
0c961fead8a16f818929cc976f0475e4f723519318b96f4947a7a5f9663
KE2: 36b4d06f413b72004392d7359cd6a998c667533203d6a671afe81ca09a282f72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KE3: 824fe89731cd47062819165662cd1c42c4b2d2321bd062e637fdd0361b0dad03
02bd5e9a9d02c72452dc65298bf330071e061b8bb4e1c8762a350d99c8c003ac
export_key: 403a270110164ae0de7ea77c6824343211e8c1663ccaedde908dc9acf
661039a379c8ac7e4b0cb23a8d1375ae94a772f91536de131d9d86633cb9445f773df
ac
session_key: 5ea9a76f5f5cc59ba7871012836947c946f8c303cc94e048cdc83ada
c89db7187cf5c718ffdd7cb6d8c3005dc0f77814d5f26011b584f9622c649a357cb17
a4c

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

D.1.3.1. Configuration 
OPRF: 0003
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
D.1.3.2. 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
server_keyshare: 020e67941e94deba835214421d2d8c90de9b0f7f925d11e2032c
e19b1832ae8e0f
client_keyshare: 03493f36ca12467d1f5eaaabea67ca31377c4869c1e9a62346b6
f01a991624b95d
server_private_keyshare: 9addab838c920fa7044f3a46b91ecaea24b0e7203992
8ee7d4c37a5b9bc17349
client_private_keyshare: 89d5a7e18567f255748a86beac13913df755a5adf776
d69e143147b545d22134
blind_registration: 411bf1a62d119afe30df682b91a0a33d777972d4f2daa4b34
ca527d597078153
blind_login: c497fddf6056d241e6cf9fb7ac37c384f49b357a221eb0a802c989b9
942256c1
D.1.3.3. Intermediate Values 
client_public_key: 03763748cc2dfe4f6f80f8e4f3087b2d2222a7c9ba7d3c3aa8
e89c4975eed0999f
auth_key: 1fa6020180e18dde869f4f8363fc1b6841dbbc9fc9d258ece830af7efc2
5abdb
randomized_pwd: 4138e29dc8398d8c83b89129cb29ee5dc962fcb5fb2dca25981cb
351b83e0546
envelope: a921f2a014513bd8a90e477a629794e89fec12d12206dde662ebdcf6567
0e51fc82109537121d7c39d96f3e04732e1f0b8cc55d98bb4e5968ace317de1d42c3d
handshake_secret: 21c9ee3561e6924110d86f99a624fe2fdc1aeea03f1b17c279f
b94da851e3686
server_mac_key: 87cab7092d3219b613459ea1ec2973be054367b331937d6973181
2f418425082
client_mac_key: 9dffe56b53981e86b37553beedb5d2226465a02d75d577bacef82
9775494bd93
oprf_key: 59984c44639e303cd46912ce722fc7d042023f25e264a3775667ea63c30
add69
D.1.3.4. Output Values 
registration_request: 0271e8fd723a873d16ddbda1d3700b9a42eca179ba09a8f
c2a2e40a8142fa35fe0
registration_response: 03c6fe2c086fa5333a15c5718ddda1f15a61e9ea9a0c4a
36f5f0dfe4f090250a70035f40ff9cf88aa1f5cd4fe5fd3da9ea65a4923a5594f84fd
9f2092d6067784874
registration_upload: 03763748cc2dfe4f6f80f8e4f3087b2d2222a7c9ba7d3c3a
a8e89c4975eed0999f5b042a53415b5db1161dacf9f9ef0c30ed6b0179038e5e8e5a0
aa087c8bc0753a921f2a014513bd8a90e477a629794e89fec12d12206dde662ebdcf6
5670e51fc82109537121d7c39d96f3e04732e1f0b8cc55d98bb4e5968ace317de1d42
c3d
KE1: 036514cf26a2578f1a45ea8faf540e52b237236ee97dc54948eca7b7f71ba9e1
29ab3d33bde0e93eda72392346a7a73051110674bbf6b1b7ffab8be4f91fdaeeb1034
93f36ca12467d1f5eaaabea67ca31377c4869c1e9a62346b6f01a991624b95d
KE2: 036ebcb79716cf2ecd0b3e5f3141709f72feb7369d2de41c61e0fa5695e78385
3e38fe59af0df2c79f57b8780278f5ae47355fe1f817119041951c80f612fdfc6d286
5751562662eea8de000fdfd4cd1bf506b137d12f28bffaf11a0d720c6ddfe532b2aff
31acb0a8fbb89de1e29cc5a93a33f2e259cf59ad6c88a473d5f056aeb2b6b5eb03a0e
21e32a309373ed45506c3f58bf3d9978925cbf35b337e8ae220be71cd9960ecef2fe0
d0f7494986fa3d8b2bb01963537e60efb13981e138e3d4a1020e67941e94deba83521
4421d2d8c90de9b0f7f925d11e2032ce19b1832ae8e0fb6eda25f9a67e3930e862860
02b8dd8b6339ddfdbaebaefe205fe474fb66884d
KE3: 4fd2178c39492f816796db05aa2400204944d6bc5ed4a1e4d7b8b24b9f1894bc
export_key: 00e1f2a1613c78183ec5127f805d320f31ce5dfef70d78f64d327d6c6
e325ae1
session_key: e39ed0c2a0b551bad5e9e8bb7017c66918d514b6412a4e30d4cac7a7
08d35646

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

D.1.4.1. Configuration 
OPRF: 0003
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
D.1.4.2. 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
server_keyshare: 020e67941e94deba835214421d2d8c90de9b0f7f925d11e2032c
e19b1832ae8e0f
client_keyshare: 03493f36ca12467d1f5eaaabea67ca31377c4869c1e9a62346b6
f01a991624b95d
server_private_keyshare: 9addab838c920fa7044f3a46b91ecaea24b0e7203992
8ee7d4c37a5b9bc17349
client_private_keyshare: 89d5a7e18567f255748a86beac13913df755a5adf776
d69e143147b545d22134
blind_registration: 411bf1a62d119afe30df682b91a0a33d777972d4f2daa4b34
ca527d597078153
blind_login: c497fddf6056d241e6cf9fb7ac37c384f49b357a221eb0a802c989b9
942256c1
D.1.4.3. Intermediate Values 
client_public_key: 03763748cc2dfe4f6f80f8e4f3087b2d2222a7c9ba7d3c3aa8
e89c4975eed0999f
auth_key: 1fa6020180e18dde869f4f8363fc1b6841dbbc9fc9d258ece830af7efc2
5abdb
randomized_pwd: 4138e29dc8398d8c83b89129cb29ee5dc962fcb5fb2dca25981cb
351b83e0546
envelope: a921f2a014513bd8a90e477a629794e89fec12d12206dde662ebdcf6567
0e51f6f7b04d6f92795c9bdb72da5ebe7745b8a6c38fc64c391b1be60b4f49ff2ce67
handshake_secret: 2bbe0da5102418c041884e9d42e62c946255138d74ea3d69acd
013bf2240c849
server_mac_key: 2b23b08101bbecc22352f1580cd73c1678affdca160ec8cfccbe0
e808029d192
client_mac_key: e279a0b44ae7c1ffb57e7cf179369c6282a18e38e6d1d070eee81
a44062d59e5
oprf_key: 59984c44639e303cd46912ce722fc7d042023f25e264a3775667ea63c30
add69
D.1.4.4. Output Values 
registration_request: 0271e8fd723a873d16ddbda1d3700b9a42eca179ba09a8f
c2a2e40a8142fa35fe0
registration_response: 03c6fe2c086fa5333a15c5718ddda1f15a61e9ea9a0c4a
36f5f0dfe4f090250a70035f40ff9cf88aa1f5cd4fe5fd3da9ea65a4923a5594f84fd
9f2092d6067784874
registration_upload: 03763748cc2dfe4f6f80f8e4f3087b2d2222a7c9ba7d3c3a
a8e89c4975eed0999f5b042a53415b5db1161dacf9f9ef0c30ed6b0179038e5e8e5a0
aa087c8bc0753a921f2a014513bd8a90e477a629794e89fec12d12206dde662ebdcf6
5670e51f6f7b04d6f92795c9bdb72da5ebe7745b8a6c38fc64c391b1be60b4f49ff2c
e67
KE1: 036514cf26a2578f1a45ea8faf540e52b237236ee97dc54948eca7b7f71ba9e1
29ab3d33bde0e93eda72392346a7a73051110674bbf6b1b7ffab8be4f91fdaeeb1034
93f36ca12467d1f5eaaabea67ca31377c4869c1e9a62346b6f01a991624b95d
KE2: 036ebcb79716cf2ecd0b3e5f3141709f72feb7369d2de41c61e0fa5695e78385
3e38fe59af0df2c79f57b8780278f5ae47355fe1f817119041951c80f612fdfc6d286
5751562662eea8de000fdfd4cd1bf506b137d12f28bffaf11a0d720c6ddfe532b2aff
31acb0a8fbb89de1e29cc5a93a33f2e259cf59ad6c88a473d5f056aeb211efe68628e
45c388328e97b78809368c72b9efc78fe51ecc7f5b6f7f4c4c2e471cd9960ecef2fe0
d0f7494986fa3d8b2bb01963537e60efb13981e138e3d4a1020e67941e94deba83521
4421d2d8c90de9b0f7f925d11e2032ce19b1832ae8e0f182fa038ada128f4440131f9
8adc14cfbdf9045d95b6a55db9b38ffd0aa539f7
KE3: a9a61a2442845e83b86c22d56ff038893208fcb0e2026d65e2a04f87497e873f
export_key: 00e1f2a1613c78183ec5127f805d320f31ce5dfef70d78f64d327d6c6
e325ae1
session_key: 9d15a7020c089b7c7ab7d6341e34a16260279b59dda8d63cabd3da0b
a14da32c

D.2. Fake Test Vectors

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

D.2.1.1. Configuration 
OPRF: 0001
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
D.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
server_keyshare: 5236e2e06d49f0b496db2a786f6ee1016f15b4fd6c0dbd95d6b1
17055d914157
server_private_keyshare: 6d8fba9741a357584770f85294430bce2252fe212a8a
372152a73c7ffe414503
masking_key: 39ebd51f0e39a07a1c2d2431995b0399bca9996c5d10014d6ebab445
3dc10ce5cef38ed3df6e56bfff40c2d8dd4671c2b4cf63c3d54860f31fe40220d690b
b71
KE1: 20098d3321812eab08e9f3ccd5640d26194cb5cf73f4c5d551f9fea8f5a5765f
42d4e61ed3f8d64cdd3b9d153343eca15b9b0d5e388232793c6376bd2d9cfd0a0e4ed
8bcc15f3dd01a30365c97c0c0de0a3dd3fbf5d3cbec55fb6ac1d3bf740f
D.2.1.3. Output Values 
KE2: e891a2527f657f5a72d723c735e9c3ae9179275f8e74f89a81418561b1db5670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D.2.2. OPAQUE-3DH Fake Test Vector 2

D.2.2.1. Configuration 
OPRF: 0003
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
D.2.2.2. 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
server_keyshare: 03f42965d5bcba2a590a49eb2418061effe40b5c29a34b8e5163
e0ef32044b2e4c
server_private_keyshare: 1a2a0ff27f3ca75221378a2a21fe5222ce0b439452f8
70475857a34197ba8f6d
masking_key: caecc6ccb4cae27cb54d8f3a1af1bac52a3d53107ce08497cdd362b1
992e4e5e
KE1: 0223afb7e2362271bdf2e20c62e25819e65d379308dfa4d9911f2fc7ada2296f
7f42d4e61ed3f8d64cdd3b9d153343eca15b9b0d5e388232793c6376bd2d9cfd0a039
94d4f1221bfd205063469e92ea4d492f7cc76a327223633ab74590c30cf7285
D.2.2.3. Output Values 
KE2: 029c5324a734851923b27ea573dce1c2ed10c497ee222c5500763c96c5209db0
cd9c035896a043e70f897d87180c543e7a063b83c1bb728fbd189c619e27b6e5a6fac
da65ce0a97b9085e7af07f61fd3fdd046d257cbf2183ce8766090b8041a8bf28d79dd
4c9031ddc75bb6ddb4c291e639937840e3d39fc0d5a3d6e7723c09f7945df485bcf9a
efe3fe82d149e84049e259bb5b33d6a2ff3b25e4bfb7eff0962821e10f6eeab2a7a42
0bf09da9b27a4639645622c46358de9cf7ae813055ae2d1203f42965d5bcba2a590a4
9eb2418061effe40b5c29a34b8e5163e0ef32044b2e4c1bf93ad07640bc9ed22e2a33
8734d55d0d22f5cc16d179e5aa4cce845b9a04a8

Authors' Addresses

Daniel Bourdrez
Hugo Krawczyk
Algorand Foundation
Kevin Lewi
Novi Research
Christopher A. Wood
Cloudflare, Inc.