The OPAQUE Asymmetric PAKE Protocol

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 terms are used throughout this document to describe the operations, roles, and behaviors of OPAQUE:¶

• Client (C): Entity which has knowledge of a password and wishes to authenticate.¶
• Server (S): Entity which authenticates clients using passwords.¶
• password: An opaque byte string containing the client's password.¶
• (skX, pkX): An AKE key pair used in role X; skX is the private key and pkX is the public key. For example, (client_private_key, client_public_key) refers to C's private and public key.¶
• kX: An OPRF private key used for role X. For example, as described in Section 3.3.2, oprf_key refers to the private OPRF key for client C known only to the server.¶
• I2OSP and OS2IP: Convert a byte string to and from a non-negative integer as described in [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 random 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 two arguments of unequal length.¶
• ct_equal(a, b): Return true if a is equal to b, and false otherwise. This function is 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.¶

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

3.1. Credential Storage

OPAQUE makes use of a structure Envelope to store client credentials. The Envelope structure embeds the following types of credentials:¶

• client_private_key: The encoded client private key for the AKE protocol.¶
• server_public_key: The encoded server public key for the AKE protocol.¶
• client_identity: The client identity. This is an application-specific value, e.g., an e-mail address or normal account name.¶
• server_identity: The server identity. This is typically a domain name, e.g., example.com. See Section 6.2 for information about this identity.¶

Each public and private key value is an opaque byte string, specific to the AKE protocol in which OPAQUE is instantiated. For example, if used as raw public keys for TLS 1.3 [RFC8446], they may be RSA or ECDSA keys as per [RFC7250].¶

These credentials are incorporated in the SecretCredentials and CleartextCredentials structs, depending on the mode set by the value of EnvelopeMode:¶

enum {
  base(1),
  custom_identifier(2),
  (255)
} EnvelopeMode;

The base mode defines SecretCredentials and CleartextCredentials as follows:¶

struct {
  opaque client_private_key<1..2^16-1>;
} SecretCredentials;

struct {
  opaque server_public_key<1..2^16-1>;
} CleartextCredentials;

The custom_identifier mode defines SecretCredentials and CleartextCredentials as follows:¶

struct {
  opaque client_private_key<1..2^16-1>;
} SecretCredentials;

struct {
  opaque server_public_key<1..2^16-1>;
  opaque client_identity<0..2^16-1>;
  opaque server_identity<0..2^16-1>;
} CleartextCredentials;

These credentials are embedded into the following Envelope structure with encryption and authentication.¶

struct {
  EnvelopeMode mode;
  opaque nonce[32];
  opaque encrypted_creds<1..2^16-1>;
} InnerEnvelope;

struct {
  InnerEnvelope inner_env;
  opaque auth_tag[Nh];
} Envelope;

mode

The EnvelopeMode value. This MUST be one of base or custom_identifier.¶

nonce

A unique 32-byte nonce used to protect this Envelope.¶

encrypted_creds

Encoding of encrypted and authenticated SecretCredentials.¶

auth_tag

Authentication tag protecting the contents of the envelope, covering InnerEnvelope and CleartextCredentials.¶

The full procedure for constructing Envelope and InnerEnvelope from SecretCredentials and CleartextCredentials is described in Section 3.3.3. Note that only SecretCredentials are stored in the Envelope (in encrypted form).¶

The EnvelopeMode value is specified as part of the configuration (see Section 5).¶

Credential information corresponding to the configuration-specific mode, along with the client public key client_public_key and private key client_private_key, are stored in a Credentials object with the following named fields:¶

• client_private_key, the client's private key¶
• client_public_key, the client's public key corresponding to client_private_key¶
• client_identity, an optional client identity (present only in the custom_identifier mode)¶
• server_identity, an optional server identity (present only in the custom_identifier mode)¶

3.2. Registration Messages

struct {
    SerializedElement data;
} RegistrationRequest;

data

A serialized OPRF group element.¶

struct {
    SerializedElement data;
    opaque server_public_key<1..2^16-1>;
} RegistrationResponse;

data

A serialized OPRF group element.¶

server_public_key

An encoded public key that will be used for the online authenticated key exchange stage.¶

struct {
    opaque client_public_key<1..2^16-1>;
    Envelope envelope;
} RegistrationUpload;

client_public_key

An encoded public key, corresponding to the private key client_private_key.¶

envelope

The client's Envelope structure.¶

3.3. Registration Functions

CreateRegistrationRequest(password)

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

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

Steps:
1. (blind, M) = Blind(password)
2. Create RegistrationRequest request with M
3. Output (request, blind)

CreateRegistrationResponse(request, server_public_key)

Input:
- request, a RegistrationRequest structure
- server_public_key, the server's public key

Output:
- response, a RegistrationResponse structure
- oprf_key, the per-client OPRF key known only to the server

Steps:
1. (oprf_key, _) = KeyGen()
2. Z = Evaluate(oprf_key, request.data)
3. Create RegistrationResponse response with (Z, server_public_key)
4. Output (response, oprf_key)

FinalizeRequest(password, creds, blind, response)

Parameters:
- params, the MHF parameters established out of band
- mode, the InnerEnvelope mode
- Nh, the output size of the Hash function

Input:
- password, an opaque byte string containing the client's password
- creds, a Credentials structure
- blind, an OPRF scalar value
- response, a RegistrationResponse structure

Output:
- record, a RegistrationUpload structure
- export_key, an additional key

Steps:
1. N = Unblind(blind, response.data)
2. y = Finalize(password, N, "OPAQUE")
3. envelope_nonce = random(32)
4. prk = HKDF-Extract(envelope_nonce, Harden(y, params))
5. Create SecretCredentials secret_creds with creds.client_private_key
6. Create CleartextCredentials cleartext_creds with response.server_public_key
   and custom identifiers creds.client_identity and creds.server_identity if
   mode is custom_identifier
7. pseudorandom_pad =
     HKDF-Expand(prk, "Pad", len(secret_creds))
8. auth_key = HKDF-Expand(prk, "AuthKey", Nh)
9. export_key = HKDF-Expand(prk, "ExportKey", Nh)
10. encrypted_creds = xor(secret_creds, pseudorandom_pad)
11. Create InnerEnvelope inner_env
      with (mode, envelope_nonce, encrypted_creds)
12. auth_tag = HMAC(auth_key, concat(inner_env, cleartext_creds))
13. Create Envelope envelope with (inner_env, auth_tag)
14. Create RegistrationUpload record with (envelope, creds.client_public_key)
15. Output (record, export_key)

struct {
    SerializedScalar oprf_key;
    opaque client_public_key<1..2^16-1>;
    Envelope envelope;
} credential_file;

4.1. Credential Retrieval

The online AKE stage of the protocol requires clients to obtain and decrypt their credentials from the server-stored envelope. This process is similar to the offline registration stage, as shown below.¶

 Client (password)             Server (server_private_key, server_public_key, credential_file)
 --------------------------------------------------------------------
 (request, blind) = CreateCredentialRequest(password)

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

    response = CreateCredentialResponse(request, server_public_key, credential_file)

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

 (client_private_key, server_public_key, export_key) =
     RecoverCredentials(password, blind, response)

The rest of this section describes these credential retrieval functions in more detail.¶

struct {
    SerializedElement data;
} CredentialRequest;

struct {
    SerializedElement data;
    opaque server_public_key<1..2^16-1>;
    Envelope envelope;
} CredentialResponse;

CreateCredentialRequest(password)

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

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

Steps:
1. (blind, M) = Blind(password)
2. Create CredentialRequest request with M
3. Output (request, blind)

CreateCredentialResponse(request, server_public_key, credential_file)

Input:
- request, a CredentialRequest structure
- server_public_key, the public key of the server
- credential_file, the server's output from registration
  (see {{credential-file}})

Output:
- response, a CredentialResponse structure

Steps:
1. Z = Evaluate(DeserializeScalar(credential_file.oprf_key), request.data)
2. Create CredentialResponse response
    with (Z, server_public_key, credential_file.envelope)
3. Output response

RecoverCredentials(password, blind, response)

Parameters:
- params, the MHF parameters established out of band
- Nh, the output size of the Hash function

Input:
- password, an opaque byte string containing the client's password
- blind, an OPRF scalar value
- response, a CredentialResponse structure

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 key

Steps:
1. N = Unblind(blind, response.data)
2. y = Finalize(password, N, "OPAQUE")
3. contents = response.envelope.contents
4. envelope_nonce = contents.nonce
5. prk = HKDF-Extract(envelope_nonce, Harden(y, params))
6. pseudorandom_pad =
    HKDF-Expand(prk, "Pad", len(contents.encrypted_creds))
7. auth_key = HKDF-Expand(prk, "AuthKey", Nh)
8. export_key = HKDF-Expand(prk, "ExportKey", Nh)
9. Create CleartextCredentials cleartext_creds with response.server_public_key
   and custom identifiers creds.client_identity and creds.server_identity if mode is
   custom_identifier
10. expected_tag = HMAC(auth_key, concat(contents, cleartext_creds))
11. If !ct_equal(response.envelope.auth_tag, expected_tag),
    raise DecryptionError
12. secret_creds = xor(contents.encrypted_creds, pseudorandom_pad)
13. Output (secret_creds.client_private_key, response.server_public_key, export_key)

4.2. AKE Instantiations

This section describes instantiations of OPAQUE using 3-message AKEs which satisfies the forward secrecy and KCI properties discussed in Section 6. As shown in [OPAQUE], OPAQUE cannot use less than three messages so the 3-message instantiations presented here are optimal in terms of number of messages. On the other hand, there is no impediment of using OPAQUE with protocols with more than 3 messages as in the case of IKEv2 (or the underlying SIGMA-R protocol [SIGMA]).¶

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 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).¶

The output of the authentication phase is a session secret session_key and export key export_key. Applications can use session_key to derive additional keying material as needed. Key derivation and other details of the protocol are specified by the AKE scheme. We note that by the results in [OPAQUE], KE2 and KE3 must authenticate credential_request and credential_response, respectively, for binding between the underlying OPRF protocol messages and the KE session.¶

The rest of this section includes key schedule utility functions used by OPAQUE-3DH, and then provides a detailed specification for OPAQUE-3DH, including its wire format messages.¶

HKDF-Expand-Label(Secret, Label, Context, Length) =
  HKDF-Expand(Secret, HkdfLabel, Length)

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

Derive-Secret(Secret, Label, Transcript) =
    HKDF-Expand-Label(Secret, Label, Hash(Transcript), Nh)

struct {
  CredentialRequest request;
  uint8 client_nonce[32];
  opaque client_info<0..2^16-1>;
  uint8 client_keyshare[Npk];
} KE1;

struct {
  CredentialResponse response;
  uint8 server_nonce[32];
  uint8 server_keyshare[Npk];
  opaque enc_server_info<0..2^16-1>;
  uint8 mac[Nh];
} KE2;

struct {
  uint8 mac[Nh];
} KE3;

HKDF-Extract(salt=0, IKM)
    |
    +-> Derive-Secret(., "handshake secret", info) = handshake_secret
    |
    +-> Derive-Secret(., "session secret", info) = session_key

server_mac_key =
  HKDF-Expand-Label(handshake_secret, "server mac", "", Nh)
client_mac_key =
  HKDF-Expand-Label(handshake_secret, "client mac", "", Nh)
handshake_encrypt_key =
  HKDF-Expand-Label(handshake_secret, "handshake enc", "", Nh)

info = "3DH keys" || I2OSP(len(client_nonce), 2) || client_nonce
                  || I2OSP(len(server_nonce), 2) || server_nonce
                  || I2OSP(len(client_identity), 2) || client_identity
                  || I2OSP(len(server_identity), 2) || server_identity

IKM = epkS^eskU || pkS^eskU || epkS^skU

IKM = epkU^eskS || epkU^skS || pkU^eskS

info_pad = HKDF-Expand(Ke2, "encryption pad", len(server_info))
enc_server_info = xor(info_pad, server_info)

6.1. Related Analysis

Jarecki et al. [OPAQUE] 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 of the underlying key exchange protocol. In turn, the security of the OPRF protocol from [I-D.irtf-cfrg-voprf] is proven in the random oracle model under the One-More Diffie-Hellman assumption [JKKX16].¶

Very few aPAKE protocols have been proven formally, and those proven were analyzed in a weak security model that allows for pre-computation attacks (e.g., [GMR06]). This is not just a formal issue: these protocols are actually vulnerable to such attacks. This includes protocols that have recent analyses in the UC model such as AuCPace [AuCPace] and SPAKE2+ [SPAKE2plus]. We note that as shown in [OPAQUE], these protocols, and any aPAKE in the model from [GMR06], can be converted into an aPAKE secure against pre-computation attacks at the expense of an additional OPRF execution.¶

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).¶

6.2. 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 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 envelope or be tied to the parties' public keys. In principle, it is possible that identities 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 time of password registration.¶

The client identity (client_identity) and server identity (server_identity) are optional parameters which are left to the application to designate as monikers 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 to 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 matches 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.¶

6.3. Envelope Encryption

The analysis of OPAQUE from [OPAQUE] requires the authenticated encryption scheme used to produce envelope to have a special property called random key-robustness (or key-committing). This specification enforces this property by utilizing encrypt-then-HMAC in the construction of envelope. There is no option to use another authenticated-encryption scheme with this specification. (Deviating from the key-robustness requirement may open the protocol to attacks, e.g., [LGR20].) 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 the envelope is validated, i.e., after the HMAC in RecoverCredentials passes verification.¶

6.4. 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.¶

6.5. Static Diffie-Hellman Oracles

While one can expect the practical security of the OPRF function (namely, the hardness of computing the function without knowing the key) to be in the order of computing discrete logarithms or solving Diffie-Hellman, Brown and Gallant [BG04] and Cheon [Cheon06] show an attack that slightly improves on generic attacks. For the case that q-1 or q+1, where q is the order of the group G, has a t-bit divisor, they show an attack that calls the OPRF on 2^t chosen inputs and reduces security by t/2 bits, i.e., it can find the OPRF key in time 2^{q/2-t/2} and 2^{q/2-t/2} memory. For typical curves, the attack requires an infeasible number of calls and/or results in insignificant security loss (*). Moreover, in the OPAQUE application, these attacks are completely impractical as the number of calls to the function translates to an equal number of failed authentication attempts by a single client. For example, one would need a billion impersonation attempts to reduce security by 15 bits and a trillion to reduce it by 20 bits - and most curves will not even allow for such attacks in the first place (note that this theoretical loss of security is with respect to computing discrete logarithms, not in reducing the password strength).¶

(*) Some examples (courtesy of Dan Brown): For P-384, 2^90 calls reduce security from 192 to 147 bits; for NIST P-256 the options are 6-bit reduction with 2153 OPRF calls, about 14 bit reduction with 187 million calls and 20 bits with a trillion calls. For Curve25519, attacks are completely infeasible (require over 2^100 calls) but its twist form allows an attack with 25759 calls that reduces security by 7 bits and one with 117223 calls that reduces security by 8.4 bits.¶

6.6. 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.¶

6.7. OPRF Hardening

Hardening the output of the OPRF greatly increases the cost of an offline attack upon the compromise of the password file at the server. Applications SHOULD select parameters that balance cost and complexity.¶

6.8. Client Enumeration

Client enumeration refers to attacks where the attacker tries to learn whether a given client identity is registered with a server. Preventing such attacks requires the server to act with unknown client identities in a way that is indistinguishable from its behavior with existing clients. Here we suggest a way to implement such defense, namely, a way for simulating a CredentialResponse for non-existing clients. Note that if the same CredentialRequest is received twice by the server, the response needs to be the same in both cases (since this would be the case for real clients). For protection against this attack, one would apply the encryption function in the construction of envelope to all the key material in envelope. The server S will have two keys MK, MK' for a pseudorandom function f. f refers to a regular pseudorandom function such as HMAC or CMAC. Upon receiving a CredentialRequest for a non-existing client client_identity, S computes oprf_key=f(MK; client_identity) and oprf_key'=f(MK'; client_identity) and responds with CredentialResponse carrying Z=M^oprf_key and envelope, where the latter is computed as follows. prk is set to oprf_key' and secret_creds is set to the all-zero string (of the length of a regular envelope plaintext). Care needs to be taken to avoid side channel leakage (e.g., timing) from helping differentiate these operations from a regular server response. The above requires changes to the server-side implementation but not to the protocol itself or the client side.¶

There is one form of leakage that the above allows and whose prevention would require a change in OPAQUE. An attacker that attempts authentication with the same CredentialRequest twice and receives different responses can conclude that either the client registered with the service between these two activations or that the client was registered before but changed its password in between the activations (assuming the server changes oprf_key at the time of a password change). In any case, this indicates that client_identity is a registered client at the time of the second activation. To conceal this information, S can implement the derivation of oprf_key as oprf_key=f(MK; client_identity) also for registered clients. Hiding changes in envelope, however, requires a change in the protocol. Instead of sending envelope as is, S would send an encryption of envelope under a key that the client derives from the OPRF result (similarly to prk) and that S stores during password registration. During the authenticated key exchange stage, the client will derive this key from the OPRF result, will use it to decrypt envelope, and continue with the regular protocol. If S uses a randomized encryption, the encrypted envelope will look each time as a fresh random string, hence S can simulate the encrypted envelope also for non-existing clients.¶

Note that the first case above does not change the protocol so its implementation is a server's decision (the client side is not changed). The second case, requires changes on the client side so it changes OPAQUE itself.¶

[[https://github.com/cfrg/draft-irtf-cfrg-opaque/issues/22: Should this variant be documented/standardized?]]¶

6.9. 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 password 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 a 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 [OPAQUE].)¶

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.¶

B.1. HMQV Instantiation Sketch

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

info = "HMQV keys" || I2OSP(len(client_nonce), 2) || client_nonce
                   || I2OSP(len(server_nonce), 2) || server_nonce
                   || I2OSP(len(client_identity), 2) || client_identity
                   || I2OSP(len(server_identity), 2) || server_identity

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 = 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 = 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.¶

B.2. SIGMA-I Instantiation Sketch

A SIGMA-I instantiation differs more drastically from OPAQUE-3DH, since authentication uses digital signatures in lieu 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 info value would use a different constant prefix -- "SIGMA-I keys" instead of "3DH keys" -- and the IKM computation would use only the ephemeral key shares exchanged between client and server.¶

C.1. OPAQUE-3DH Test Vector 1

Group: ristretto255
EnvelopeMode: 01
OPRF: 0001
SlowHash: Identity
Hash: SHA512

server_nonce: a4997137a8fa0d4baf7052a499bf877057f9404e03c889d641a0d7c
807b6a518
oprf_key: 5ed895206bfc53316d307b23e46ecc6623afb3086da74189a416012be03
7e50b
password: 436f7272656374486f72736542617474657279537461706c65
blind_login: ed8366feb6b1d05d1f46acb727061e43aadfafe9c10e5a64e7518d63
e3263503
server_private_keyshare: 31587dff30b8001d9d43584decc22e358fa7f9d6e606
29fb1223081c3bae7103
client_nonce: 75a1ad27ab77578bc08b44c4318f09b31d53145c9ba3b42abf0ea08
a781277a6
server_info: 6772656574696e677320616c696365
client_info: 68656c6c6f20626f62
client_private_keyshare: fbbf4ad24119f08a35bf999f8ae0c779ed7b3e266bf3
3f793f6bf9ebf4578005
envelope_nonce: 6c6bb9021b9833c788dd25994d3ce7f3811338744fd6dc556e037
585783444c0
server_keyshare: 82be40ef93bf7c6edd43d4ed9f52fa19827649b819de39c52a22
43e985b75d62
blind_registration: c604c785ada70d77a5256ae21767de8c3304115237d262134
f5e46e512cf8e03
client_public_key: b2d1b10f3741a8efef3139f4d2889f2b11f776b79b9aa2c3ef
caba4f1c4ae861
client_private_key: f0f56cfb488649fe28691dd9aa5dc9ff4c0e6028075baa3c5
615398a2cb12304
client_keyshare: 484e47e31b3132f4ee512e41805a1690891111a7b885bc526198
22c14cabf360
server_public_key: 5442a6f57333a332b4c6f07308f6fa846bde3ed27425820cdc
eeeb935924a903
server_private_key: d63d709e3a739a128929a9f289ff263fdcbc457f2f47f7c43
ccafbbfee72290c

auth_key: 22198da4ad73b1d35cd8bb875e64ce1a9fc2edeb073d760e114d1d7a2f8
6d47411caf1787907e2ff96cd3190b14d101101d74cff234259d9f19a18f2cfe29d0b
server_mac_key: 9de28e2f7107afe266570934c033dd6a403fb2b09a9f1a1357a81
9fb072d25e6651626638d77bd7f0adf4b2b715d0a0bee2ac531fd0f7da699aba6e9ed
717466
envelope: 016c6bb9021b9833c788dd25994d3ce7f3811338744fd6dc556e0375857
83444c00022e9b28440a1dc99cca7513e8eb754d427caf8515642b45a92c73838c043
fc316deeb83deface81589bc8e5c2c767b5ab79218834fbf4fa4f87cf23775aa54633
ed75665cd88f451e044cac1b282d890269476d1ac18ff9a4c0cb832e1143ffa7d447a
prk: 732303f65e76c39f30876aec31af1f5bcd8861626c922baa2e842209c7d5bf2e
024912db2d6cac00b1260c437d34ec588a664a55fc7a0a40251915d2b15d8ec9
client_mac_key: 1536168d48218d08dcbed3438e897d98eff566894240d8d136be0
4ba46b2c788883fd165ce7614c52a6a9c926bf55d249e6c29bc4a23d5a1775ad294a7
858922
pseudorandom_pad: e99274b5cd27d14aeeaf16e7aa8d7e7a03071d58229c5dc96d0
46ed57a761ddccdbc
handshake_encrypt_key: c646def7a8ca75282ac1a0ad15630834cba2772837d7a3
a26d00b36923e55b3f5d065679c715e4c799654f261c865f1a55bc94cabc29a6e2e72
296dfc494a984
handshake_secret: cc887e128c28064106d4101ac3de633e9096e170f2c4a9913d3
50f306274b1665a5b251f761672c12db4c403a615c22cee96adb3539fa62662a17f2e
18cd5fed

registration_response: 1867301bcc67bdf8e640b7d6edcbe2a65488446417b50d
30cdba66ccb379e57200205442a6f57333a332b4c6f07308f6fa846bde3ed27425820
cdceeeb935924a903
export_key: 6ca2c344763e5bc9e3d2bbfe3d982b826b709da597e28e85f9594ec54
2a20c697d55de277ccce1d1af7c48ab7fea1467ac1e3a99c71dcf6326a909d280bd2f
6f
registration_upload: 0020b2d1b10f3741a8efef3139f4d2889f2b11f776b79b9a
a2c3efcaba4f1c4ae861016c6bb9021b9833c788dd25994d3ce7f3811338744fd6dc5
56e037585783444c00022e9b28440a1dc99cca7513e8eb754d427caf8515642b45a92
c73838c043fc316deeb83deface81589bc8e5c2c767b5ab79218834fbf4fa4f87cf23
775aa54633ed75665cd88f451e044cac1b282d890269476d1ac18ff9a4c0cb832e114
3ffa7d447a
registration_request: 241b621c417c0705b5ea7a8b7cdd5039fd61e6b63effe2a
44418164c4d49003e
session_key: ee9f1ef224d498858f6c9b3a121016a38bad7816055c452b1c7edf3d
d439c42a4cc78cbd672e985a20910df14f8f1af4ce5793303ffe6954ff5f1a264e3fd
515
KE3: e771daf28bc5e8068dead67c3db19f9ad03ee919e52f6c7a6e79cf1085bd7448
1e76512c77f37762578eb2faff8fe98e4185ca2d01957216c556d33a6fba3028
KE2: e83812f06568d57b8cdfdcc90fe91454e21bd25dd2a1c32dd1599a2e4a4b6c35
00205442a6f57333a332b4c6f07308f6fa846bde3ed27425820cdceeeb935924a9030
16c6bb9021b9833c788dd25994d3ce7f3811338744fd6dc556e037585783444c00022
e9b28440a1dc99cca7513e8eb754d427caf8515642b45a92c73838c043fc316deeb83
deface81589bc8e5c2c767b5ab79218834fbf4fa4f87cf23775aa54633ed75665cd88
f451e044cac1b282d890269476d1ac18ff9a4c0cb832e1143ffa7d447aa4997137a8f
a0d4baf7052a499bf877057f9404e03c889d641a0d7c807b6a51882be40ef93bf7c6e
dd43d4ed9f52fa19827649b819de39c52a2243e985b75d62000f13aed85ae30aee2f8
9ac5e1c5dd53609b890267ef2d765bb56000bc704c2ba4e4256107befee6655cdb084
64e6c75d8b251642e0d721b3fce38f245568b50e7f6c98825d54cc1a26d6eafadcc1d
344
KE1: b68e0e356f8490fa9c3bed952e16cc02db21eda686b3c484f3d9d912caa41f76
75a1ad27ab77578bc08b44c4318f09b31d53145c9ba3b42abf0ea08a781277a600096
8656c6c6f20626f62484e47e31b3132f4ee512e41805a1690891111a7b885bc526198
22c14cabf360

C.2. OPAQUE-3DH Test Vector 2

Group: ristretto255
EnvelopeMode: 02
OPRF: 0001
SlowHash: Identity
Hash: SHA512

server_nonce: 0f3a6da8b667bc7a383c987586bee749c5f2787691baca68757e78b
6128b0a0f
oprf_key: 89c61a42c8191a5ca41f2fe959843d333bcf43173b7de4c5c119e0e0d8b
0e707
password: 436f7272656374486f72736542617474657279537461706c65
blind_login: e6d0f1d89ad552e383d6c6f4e8598cc3037d6e274d22da3089e7afbd
4171ea02
server_private_keyshare: 70c944dcb7f4dddde168ecb48dd9488c62b6fc7e9bb4
2a16d291afca9dd25b07
client_nonce: 480917b09c6720680b4a7a0ba9f54b69d870f640a4a7994b47ad07d
1a95c984f
server_info: 6772656574696e677320616c696365
server_identity: 5442a6f57333a332b4c6f07308f6fa846bde3ed27425820cdcee
eb935924a903
client_info: 68656c6c6f20626f62
client_private_keyshare: a4ba6cb7e16ab76eccdb4c0b9261eedd426d7863f00b
fc4a0e09476d3121e70c
envelope_nonce: e38fb444afe3df13ae05e6876d10eca7661196375518eb66d7dee
b8cfb42d13f
server_public_key: 5442a6f57333a332b4c6f07308f6fa846bde3ed27425820cdc
eeeb935924a903
client_identity: b2d1b10f3741a8efef3139f4d2889f2b11f776b79b9aa2c3efca
ba4f1c4ae861
blind_registration: 019cbd1d7420292528f8cdd62f339fdabb602f04a95dac9db
cec831b8c681a09
client_public_key: b2d1b10f3741a8efef3139f4d2889f2b11f776b79b9aa2c3ef
caba4f1c4ae861
client_private_key: f0f56cfb488649fe28691dd9aa5dc9ff4c0e6028075baa3c5
615398a2cb12304
client_keyshare: 4ae7d50bb80cc8f5034d36c1c27edc30caca0983677a941bc0ac
e5e10b18300a
server_keyshare: a2e9e0809b270a1d5c8208f3498a3188538265e6a9e6b274cb38
c4c5d9b1792d
server_private_key: d63d709e3a739a128929a9f289ff263fdcbc457f2f47f7c43
ccafbbfee72290c

auth_key: 4c7c0ae950ad7e9c518266f953d4a01bd2232695f92fb028aa6c124996e
31a205f621305fd4997edf8a5fce04a51252ba8430227c134b81d7093a58e01c70752
server_mac_key: 974bde939ef1d30ba29f9ec2addcaff1eeb105a1f534e04113f9f
0b5c3b9a0797e3fa5f7006e63dfb2b0ce74d002bd1161767c361507bfb1fa0fed5063
1cdcb7
envelope: 02e38fb444afe3df13ae05e6876d10eca7661196375518eb66d7deeb8cf
b42d13f00227ed4654d4452340f26c6304dceeeb1fddb142fcd91b3b26ecd566d7688
095f9478a721e5a592be9dd3bda76ed97421819aca4a30752813223d33bc7ea443be9
76754605116278c7fffda4de142674ed154f540bd3285080637eb4b929e1378336b11
prk: e3bb74ca5f88a95571578e921489e1b6119b438e4efce6e955ae9b6453f24aa4
e3a34fa22bc5f470cfc134ca0784a9cd7df64be46ff3b325fc19f4009979bf2e
client_mac_key: 1c55ecba64d98dc2db8f45faa72b8fdd63cad8677b8665cc575cf
8ad36ef38ec63c9dfc5552007573215983f55d8f5cfa1651da8417e3f27094dde5254
7f5b89
pseudorandom_pad: 7ef495b828a97c896f381824d3371ba012eb63c3f19bb535676
a3b63b18373255ba3
handshake_encrypt_key: 0852fb826109073be6e7c065b6e2d1f872c16f9977c177
cd2f945b26bb933fe6252f226f40483f7aa52f545f801d1e4430f30f80fd42070494e
a77fdcecd6595
handshake_secret: bfd3bfbe451520880975d0e568cbd3b5155b23c02de504fccb9
dd5a8195266cd94d49d040530b3d6b0a585d542eb24da708a2b6f6dc34dee4652d0c1
c62c4e59

registration_response: 088ac01ebf5700f0c96bc2988509343cb7e2dd6f0df820
d0fb807faa11a26f5600205442a6f57333a332b4c6f07308f6fa846bde3ed27425820
cdceeeb935924a903
export_key: 0effa605dc47ba4fe565c423b782b8b6697b26ee2ede7059b0e17510d
f8b11554ce053409671480a56ffbe77b91edc95205c213caeaf9dcb0841790ff834a0
09
registration_upload: 0020b2d1b10f3741a8efef3139f4d2889f2b11f776b79b9a
a2c3efcaba4f1c4ae86102e38fb444afe3df13ae05e6876d10eca7661196375518eb6
6d7deeb8cfb42d13f00227ed4654d4452340f26c6304dceeeb1fddb142fcd91b3b26e
cd566d7688095f9478a721e5a592be9dd3bda76ed97421819aca4a30752813223d33b
c7ea443be976754605116278c7fffda4de142674ed154f540bd3285080637eb4b929e
1378336b11
registration_request: c8d2e9ba503bf3f8821226653314427edb1ec8a3ecc94a5
dfbbe33d59d07b645
session_key: 8dc21ff264f2774de95955d35544ba314e92d07f4a3b32c89ead5e70
c83ac4c0221deadd34ed11d43fc4d3651aec612d696c63979c96bf1ddd1ee44da5d0c
d68
KE3: c178e45ed9b314653685cdbf5f7730e3e40f8652ceb9b10f47d1c784fdd75dd1
07c4ae3f7683de5a692359178c8f13f41a043fc1dcfc14b1fb7cb411514efc6c
KE2: 5079b16709b195b3b63257b419efb752bd0603170160fa72b828ce9ff9209c0c
00205442a6f57333a332b4c6f07308f6fa846bde3ed27425820cdceeeb935924a9030
2e38fb444afe3df13ae05e6876d10eca7661196375518eb66d7deeb8cfb42d13f0022
7ed4654d4452340f26c6304dceeeb1fddb142fcd91b3b26ecd566d7688095f9478a72
1e5a592be9dd3bda76ed97421819aca4a30752813223d33bc7ea443be976754605116
278c7fffda4de142674ed154f540bd3285080637eb4b929e1378336b110f3a6da8b66
7bc7a383c987586bee749c5f2787691baca68757e78b6128b0a0fa2e9e0809b270a1d
5c8208f3498a3188538265e6a9e6b274cb38c4c5d9b1792d000f5a1a0b34573bf728b
14f53485f3bf62fd91154dc9ca01b21945b2204f96adc87bd80e8283ecf3522b6d893
7527f2e9b9782a94fedab0fa304590d1d4f03d72fb1664fca1523a3be95f6c3a97d0a
2ed
KE1: 7024ca0d5423176294fbb9ca968d8ce3fc879a231f1ceef69e672c89e02ded59
480917b09c6720680b4a7a0ba9f54b69d870f640a4a7994b47ad07d1a95c984f00096
8656c6c6f20626f624ae7d50bb80cc8f5034d36c1c27edc30caca0983677a941bc0ac
e5e10b18300a