| RFC 9382 | SPAKE2 | September 2023 |
| Ladd | Informational | [Page] |
This document describes SPAKE2, which is a protocol for two parties that share a password to derive a strong shared key without disclosing the password. This method is compatible with any group, is computationally efficient, and has a security proof. This document predated the Crypto Forum Research Group (CFRG) password-authenticated key exchange (PAKE) competition, and it was not selected; however, given existing use of variants in Kerberos and other applications, it was felt that publication was beneficial. Applications that need a symmetric PAKE, but are unable to hash onto an elliptic curve at execution time, can use SPAKE2. This document is a product of the Crypto Forum Research Group in the Internet Research Task Force (IRTF).¶
This document is not an Internet Standards Track specification; it is published for informational purposes.¶
This document is a product of the Internet Research Task Force (IRTF). The IRTF publishes the results of Internet-related research and development activities. These results might not be suitable for deployment. This RFC represents the individual opinion(s) of one or more members of the Crypto Forum Research Group of the Internet Research Task Force (IRTF). Documents approved for publication by the IRSG are not candidates for any level of Internet Standard; see Section 2 of RFC 7841.¶
Information about the current status of this document, any errata, and how to provide feedback on it may be obtained athttps://www.rfc-editor.org/info/rfc9382.¶
Copyright (c) 2023 IETF Trust and the persons identified as the document authors. All rights reserved.¶
This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (https://trustee.ietf.org/license-info) in effect on the date of publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect to this document.¶
This document describes SPAKE2, which is a means for two parties that share a password to derive a strong shared key without disclosing the password. This password-based key exchange protocol is compatible with any group (requiring only a scheme to map a random input of a fixed length per group to a random group element), is computationally efficient, and has a security proof. Predetermined parameters for a selection of commonly used groups are also provided for use by other protocols.¶
SPAKE2 was not selected as the result of the CFRG PAKE selection competition. However, given existing use of variants in Kerberos and other applications, it was felt that publication was beneficial. This RFC represents the individual opinion(s) of one or more members of the Crypto Forum Research Group of the IRTF.¶
Many of these applications predated methods to hash to elliptic curves being available or predated the publication of the PAKEs that were chosen as an outcome of the PAKE selection competition. In cases where a symmetric PAKE is needed and hashing onto an elliptic curve at protocol execution time is not available, SPAKE2 is useful.¶
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.¶
SPAKE2 is a two-round protocol, wherein the first round establishes a shared secret between A and B, and the second round serves as key confirmation. Prior to invocation, A and B are provisioned with information, such as the input password needed to run the protocol. We assume that the roles of A and B are agreed upon by both sides: A goes first and uses M, and B goes second and uses N. If this assignment of roles is not possible, a symmetric variantMUST be used, as described laterSection 5. For instance, A may be the client when using TCP or TLS as an underlying protocol, and B may be the server. Most protocols have such a distinction. During the first round, A sends a public value pA to B, and B responds with its own public value pB. Both A and B then derive a shared secret used to produce encryption and authentication keys. The latter are used during the second round for key confirmation. (Section 4 details the key derivation and confirmation steps.) In particular, A sends a key confirmation message cA to B, and B responds with its own key confirmation message cB. AMUST NOT consider the protocol complete until it receives and verifies cB. Likewise, BMUST NOT consider the protocol complete until it receives and verifies cA.¶
This sample flow is shown below.¶
A B | | | | (compute pA) | pA | |---------------------->| | pB | (compute pB) |<----------------------| | | | (derive secrets) | | | (compute cA) | cA | |---------------------->| | cB | (compute cB) | | (check cA) |<----------------------| (check cB) | |¶
Let G be a group in which the gap Diffie-Hellman (GDH) problem is hard. Suppose G has order p*h, where p is a large prime and h will be called the cofactor. Let I be the unit element in G, e.g., the point at infinity if G is an elliptic curve group. We denote the operations in the group additively. We assume there is a representation of elements of G as byte strings: common choices would be SEC1[SEC1] uncompressed or compressed for elliptic curve groups or big-endian integers of a fixed (per-group) length for prime field DH. ApplicationsMUST specify this encoding, typically by referring to the document defining the group. We fix two elements, M and N, in the prime-order subgroup of G, as defined inTable 1 of this document for common groups, as well as generator P of the (large) prime-order subgroup of G. In the case of a composite order group, we will work in the quotient group. For common groups used in this document, P is specified in the document defining the group, so we do not repeat it here.¶
For elliptic curves other than the ones in this document, the methods described in[RFC9380]SHOULD be used to generate M and N, e.g., via M = hash_to_curve("M SPAKE2 seed OID x") and N = hash_to_curve("N SPAKE2 seed OID x"), where x is an OID for the curve. ApplicationsMAY include a domain separation tag (DST) in this step, as specified in[RFC9380], though this is not required.¶
|| denotes concatenation of byte strings. We also let len(S) denote the length of a string in bytes, represented as an eight-byte little-endian number. Finally, let nil represent an empty string, i.e., len(nil) = 0. Text strings in double quotes are treated as their ASCII encodings throughout this document.¶
KDF(ikm, salt, info, L) is a key-derivation function that takes as input a salt, input keying material (IKM), an info string, and derived key length L to derive a cryptographic key of length L. MAC(key, message) is a Message Authentication Code algorithm that takes a secret key and message as input to produce an output. Let Hash be a hash function from arbitrary strings to bit strings of a fixed length that is at least 256 bits long. Common choices for Hash are SHA-256 or SHA-512[RFC6234]. Let MHF be a memory-hard hash function designed to slow down brute-force attackers. Scrypt[RFC7914] is a common example of this function. The output length of MHF matches that of Hash. Parameter selection for MHF is out of scope for this document.Section 6 specifies variants of KDF, MAC, and Hash that are suitable for use with the protocols contained herein.¶
Let A and B be two parties. A and B may also have digital representations of the parties' identities, such as Media Access Control addresses or other names (hostnames, usernames, etc.). A and B may share additional authenticated data (AAD) of a length that is at most 216 - 128 bits and separate from their identities, which they may want to include in the protocol execution. One example of AAD is a list of supported protocol versions if SPAKE2 were used in a higher-level protocol that negotiates use of a particular PAKE. Including this list would ensure that both parties agree upon the same set of supported protocols and therefore prevents downgrade attacks. We also assume A and B share integer w; typically, w = MHF(pw) mod p for a user-supplied password, pw. Standards, such as[NIST.SP.800-56Ar3], suggest taking mod p of a hash value that is 64 bits longer than that needed to represent p to remove statistical bias introduced by the modulation. Protocols using this specificationMUST define the method used to compute w. In some cases, it may be necessary to carry out various forms of normalization of the password before hashing[RFC8265]. The hashing algorithmSHOULD be an MHF so as to slow down brute-force attackers.¶
To begin, A picks x randomly and uniformly from the integers in [0,p) and calculates X=x*P and pA=w*M+X. Then, it transmits pA to B.¶
B selects y randomly and uniformly from the integers in [0,p) and calculates Y=y*P and pB=w*N+Y. Then, it transmits pB to A.¶
Both A and B calculate group element K. A calculates it as h*x*(pB-w*N), while B calculates it as h*y*(pA-w*M). A knows pB because it has received it, and likewise B knows pA. The multiplication by h prevents small subgroup confinement attacks by computing a unique value in the quotient group.¶
K is a shared value, though itMUST NOT be used or output as a shared secret from the protocol. Both A and B must derive two additional shared secrets from the protocol transcript, which includes K. This use of the transcript ensures any manipulation of the messagessent is reflected in the keys. The transcript TT is encoded as follows:¶
TT = len(A) || A || len(B) || B || len(pA) || pA || len(pB) || pB || len(K) || K || len(w) || w¶
Here, w is encoded as a big-endian number padded to the length of p. This representation prevents timing attacks that otherwise would reveal the length of w. len(w) is thus a constant for a given group. We include it for consistency.¶
If an identity is absent, it is encoded as a zero-length string. ThisMUST only be done for applications in which identities are implicit. Otherwise, the protocol risks unknown key-share attacks, where both sides of a connection disagree over who is authenticated.¶
Upon completion of this protocol, A and B compute shared secrets Ke, KcA, and KcB, as specified inSection 4. AMUST send B a key confirmation message so that both parties agree upon these shared secrets. The confirmation message cA is computed as a MAC over the protocol transcript TT, using KcA as follows: cA = MAC(KcA, TT). Similarly, BMUST send A a confirmation message using a MAC that is computed equivalently, except with the use of KcB. Key confirmation verification requires computing cA (or cB, respectively) and checking for equality against that which was received.¶
The protocol transcript TT, as defined inSection 3.3, is unique and secret to A and B (though it contains some substrings that are not secret). Both parties use TT to derive shared symmetric secrets Ke and Ka as Ke || Ka = Hash(TT), with |Ke| = |Ka|. The length of each key is equal to half of the digest output, e.g., 128 bits for SHA-256. KeysMUST be at least 128 bits in length.¶
Both endpoints use Ka to derive subsequent MAC keys for key confirmation messages. Specifically, KcA and KcB are the MAC keys used by A and B, respectively. A and B compute them as KcA || KcB = KDF(Ka, nil, "ConfirmationKeys" || AAD, L), where AAD is the associated data given to each endpoint or AAD is nil if none was provided. The length of each of KcA and KcB is equal to half of the underlying hash output length, e.g., |KcA| = |KcB| = 128 bits for HKDF(SHA256), with L=256 bits.¶
The resulting key schedule for this protocol, given transcript TT and AAD, is as follows.¶
TT -> Hash(TT) = Ke || Ka AAD -> KDF(Ka, nil, "ConfirmationKeys" || AAD) = KcA || KcB¶
A and B output Ke as the shared secret from the protocol. Ka and its derived keys are not used for anything except key confirmation.¶
To avoid concerns that an attacker needs to solve a single Elliptic Curve Diffie-Hellman (ECDH) instance to break the authentication of SPAKE2, it is possible to vary M and N using[RFC9380] as follows:¶
M = hash_to_curve(Hash("M SPAKE2" || len(A) || A || len(B) || B)) N = hash_to_curve(Hash("N SPAKE2" || len(A) || A || len(B) || B))¶There is also a symmetric variant where M=N. For this variant, we set:¶
M = hash_to_curve(Hash("M AND N SPAKE2"))¶This variantMUST be used when it is not possible to determine whether A or B should use M (or N), due to asymmetries in the protocol flows or the desire to use only a single shared secret with nil identities for authentication. The security of these variants is examined in[MNVAR]. The variant with per-user M and N may not be suitable for protocols that require the initial messages to be generated by each party at the same time and that do not know the exact identity of the parties before the flow begins.¶
This section documents SPAKE2 ciphersuite configurations. A ciphersuite indicates a group, cryptographic hash function, and pair of KDF and MAC functions, e.g., SPAKE2-P256-SHA256-HKDF-HMAC. This ciphersuite indicates a SPAKE2 protocol instance over P-256 that uses SHA-256, along with HMAC-based Key Derivation Function (HKDF)[RFC5869] and Hashed Message Authentication Code (HMAC)[RFC2104] for G, Hash, KDF, and MAC functions, respectively. For Ed25519, the compressed encoding is used[RFC8032]; all others use the uncompressed SEC1 encoding.¶
| G | Hash | KDF | MAC |
|---|---|---|---|
| P-256 | SHA256[RFC6234] | HKDF[RFC5869] | HMAC[RFC2104] |
| P-256 | SHA512[RFC6234] | HKDF[RFC5869] | HMAC[RFC2104] |
| P-384 | SHA256[RFC6234] | HKDF[RFC5869] | HMAC[RFC2104] |
| P-384 | SHA512[RFC6234] | HKDF[RFC5869] | HMAC[RFC2104] |
| P-521 | SHA512[RFC6234] | HKDF[RFC5869] | HMAC[RFC2104] |
| edwards25519[RFC8032] | SHA256[RFC6234] | HKDF[RFC5869] | HMAC[RFC2104] |
| edwards448[RFC8032] | SHA512[RFC6234] | HKDF[RFC5869] | HMAC[RFC2104] |
| P-256 | SHA256[RFC6234] | HKDF[RFC5869] | CMAC-AES-128[RFC4493] |
| P-256 | SHA512[RFC6234] | HKDF[RFC5869] | CMAC-AES-128[RFC4493] |
The following points represent permissible point generation seeds for the groups listed inTable 1, using the algorithm presented inAppendix A. These byte strings are compressed points, as in[SEC1], for curves from[SEC1].¶
For P-256:¶
M =02886e2f97ace46e55ba9dd7242579f2993b64e16ef3dcab95afd497333d8fa12fseed: 1.2.840.10045.3.1.7 point generation seed (M)N =03d8bbd6c639c62937b04d997f38c3770719c629d7014d49a24b4f98baa1292b49seed: 1.2.840.10045.3.1.7 point generation seed (N)¶
For P-384:¶
M =030ff0895ae5ebf6187080a82d82b42e2765e3b2f8749c7e05eba366434b363d3dc36f15314739074d2eb8613fceec2853seed: 1.3.132.0.34 point generation seed (M)N =02c72cf2e390853a1c1c4ad816a62fd15824f56078918f43f922ca21518f9c543bb252c5490214cf9aa3f0baab4b665c10seed: 1.3.132.0.34 point generation seed (N)¶
For P-521:¶
M =02003f06f38131b2ba2600791e82488e8d20ab889af753a41806c5db18d37d85608cfae06b82e4a72cd744c719193562a653ea1f119eef9356907edc9b56979962d7aaseed: 1.3.132.0.35 point generation seed (M)N =0200c7924b9ec017f3094562894336a53c50167ba8c5963876880542bc669e494b2532d76c5b53dfb349fdf69154b9e0048c58a42e8ed04cef052a3bc349d95575cd25seed: 1.3.132.0.35 point generation seed (N)¶
For edwards25519:¶
M =d048032c6ea0b6d697ddc2e86bda85a33adac920f1bf18e1b0c6d166a5cecdafseed: edwards25519 point generation seed (M)N =d3bfb518f44f3430f29d0c92af503865a1ed3281dc69b35dd868ba85f886c4abseed: edwards25519 point generation seed (N)¶
For edwards448:¶
M =b6221038a775ecd007a4e4dde39fd76ae91d3cf0cc92be8f0c2fa6d6b66f9a12942f5a92646109152292464f3e63d354701c7848d9fc3b8880seed: edwards448 point generation seed (M)N =6034c65b66e4cd7a49b0edec3e3c9ccc4588afd8cf324e29f0a84a072531c4dbf97ff9af195ed714a689251f08f8e06e2d1f24a0ffc0146600seed: edwards448 point generation seed (N)¶
A security proof of SPAKE2 for prime order groups is found in[REF], reducing the security of SPAKE2 to the GDH assumption. Note that the choice of M and N is critical for the security proof. The generation methods specified in this document are designed to eliminate concerns related to knowing discrete logs of M and N.¶
Elements received from a peerMUST be checked for group membership. Failure to properly deserialize and validate group elements can lead to attacks. An endpointMUST abort the protocol if any received public value is not a member of G.¶
The choices of random numbersMUST be uniform. Randomly generated values, e.g., x and y,MUST NOT be reused; such reuse violates the security assumptions of the protocol and results in significant insecurity. It isRECOMMENDED to generate these uniform numbers using rejection sampling.¶
Some implementations of elliptic curve multiplication may leak information about the length of the scalar. TheseMUST NOT be used. All operations on elliptic curve points must take time independent of the inputs. Hashing of the transcript may take time depending only on the length of the transcript but not the contents.¶
SPAKE2 does not support augmentation. As a result, the server has to store a password equivalent. This is considered a significant drawback in some use cases. Applications that need augmented PAKEs should use[CFRG-OPAQUE].¶
The HMAC keys in this document are shorter than recommended in[RFC8032]. This is appropriate, as the difficulty of the discrete logarithm problem is comparable with the difficulty of brute forcing the keys.¶
This document has no IANA actions.¶
This section describes the algorithm that was used to generate points M and N inTable 1.¶
For each curve inTable 1, we construct a string using the curve OID from[RFC5480] (as an ASCII string) or its name, combined with the needed constant, e.g., "1.3.132.0.35 point generation seed (M)" for P-521. This string is turned into a series of blocks by hashing with SHA-256 and hashing that output again to generate the next 32 bytes and so on. This pattern is repeated for each group and value, with the string modified appropriately.¶
A byte string of a length equal to that of an encoded group element is constructed by concatenating as many blocks as are required, starting from the first block and truncating to the desired length. The byte string is then formatted as required for the group. In the case of Weierstrass curves, we take the desired length as the length for representing a compressed point (Section 2.3.4 of[SEC1]) and use the low-order bit of the first byte as the sign bit. In order to obtain the correct format, the value of the first byte is set to 0x02 or 0x03 (clearing the first six bits and setting the seventh bit), leaving the sign bit as it was in the byte string constructed by concatenating hash blocks. For the curves in[RFC8032], a different procedure is used. For edwards448, the 57-byte input has the least-significant 7 bits of the last byte set to zero, and for edwards25519, the 32-byte input is not modified. For both the curves in[RFC8032], the (modified) input is then interpreted as the representation of the group element. If this interpretation yields a valid group element with the correct order (p), the (modified) byte string is the output. Otherwise, the initial hash block is discarded and a new byte string is constructed from the remaining hash blocks. The procedure of constructing a byte string of the appropriate length, formatting it as required for the curve, and checking if it is a valid point of the correct order is repeated until a valid element is found.¶
The following Python snippet generates the above points, assuming an elliptic curve implementation follows the interface of Edwards25519Point.stdbase() and Edwards448Point.stdbase() inAppendix A of [RFC8032]:¶
def iterated_hash(seed, n): h = seed for i in range(n): h = hashlib.sha256(h).digest() return hdef bighash(seed, start, sz): n = -(-sz // 32) hashes = [iterated_hash(seed, i) for i in range(start, start + n)] return b''.join(hashes)[:sz]def canon_pointstr(ecname, s): if ecname == 'edwards25519': return s elif ecname == 'edwards448': return s[:-1] + bytes([s[-1] & 0x80]) else: return bytes([(s[0] & 1) | 2]) + s[1:]def gen_point(seed, ecname, ec): for i in range(1, 1000): hval = bighash(seed, i, len(ec.encode())) pointstr = canon_pointstr(ecname, hval) try: p = ec.decode(pointstr) if p != ec.zero_elem() and p * p.l() == ec.zero_elem(): return pointstr, i except Exception: pass¶
This section contains test vectors for SPAKE2, using the P256-SHA256-HKDF-HMAC ciphersuite. (The choice of MHF is omitted, and the values for w, x, and y are provided directly.) All points are encoded using the uncompressed format, i.e., with a 0x04 octet prefix, specified in[SEC1]. A and B identity strings are provided in the protocol invocation.¶
Line breaks have been added due to line-length limitations.¶
spake2: A='server', B='client'w = 0x2ee57912099d31560b3a44b1184b9b4866e904c49d12ac5042c97dca461b1a5fx = 0x43dd0fd7215bdcb482879fca3220c6a968e66d70b1356cac18bb26c84a78d729pA = 0x04a56fa807caaa53a4d28dbb9853b9815c61a411118a6fe516a8798434751470f9010153ac33d0d5f2047ffdb1a3e42c9b4e6be662766e1eeb4116988ede5f912cy = 0xdcb60106f276b02606d8ef0a328c02e4b629f84f89786af5befb0bc75b6e66bepB = 0x0406557e482bd03097ad0cbaa5df82115460d951e3451962f1eaf4367a420676d09857ccbc522686c83d1852abfa8ed6e4a1155cf8f1543ceca528afb591a1e0b7K = 0x0412af7e89717850671913e6b469ace67bd90a4df8ce45c2af19010175e37eed69f75897996d539356e2fa6a406d528501f907e04d97515fbe83db277b715d3325TT = 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(TT) = 0x0e0672dc86f8e45565d338b0540abe6915bdf72e2b35b5c9e5663168e960a91bKe = 0x0e0672dc86f8e45565d338b0540abe69Ka = 0x15bdf72e2b35b5c9e5663168e960a91bKcA = 0x00c12546835755c86d8c0db7851ae86fKcB = 0xa9fa3406c3b781b93d804485430ca27aA conf = 0x58ad4aa88e0b60d5061eb6b5dd93e80d9c4f00d127c65b3b35b1b5281fee38f0B conf = 0xd3e2e547f1ae04f2dbdbf0fc4b79f8ecff2dff314b5d32fe9fcef2fb26dc459b¶
spake2: A='', B='client'w = 0x0548d8729f730589e579b0475a582c1608138ddf7054b73b5381c7e883e2efaex = 0x403abbe3b1b4b9ba17e3032849759d723939a27a27b9d921c500edde18ed654bpA = 0x04a897b769e681c62ac1c2357319a3d363f610839c4477720d24cbe32f5fd85f44fb92ba966578c1b712be6962498834078262caa5b441ecfa9d4a9485720e918ay = 0x903023b6598908936ea7c929bd761af6039577a9c3f9581064187c3049d87065pB = 0x04e0f816fd1c35e22065d5556215c097e799390d16661c386e0ecc84593974a61b881a8c82327687d0501862970c64565560cb5671f696048050ca66ca5f8cc7fcK = 0x048f83ec9f6e4f87cc6f9dc740bdc2769725f923364f01c84148c049a39a735ebda82eac03e00112fd6a5710682767cff5361f7e819e53d8d3c3a2922e0d837aa6TT = 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(TT) = 0x642f05c473c2cd79909f9a841e2f30a70bf89b18180af97353ba198789c2b963Ke = 0x642f05c473c2cd79909f9a841e2f30a7Ka = 0x0bf89b18180af97353ba198789c2b963KcA = 0xc6be376fc7cd1301fd0a13adf3e7bffdKcB = 0xb7243f4ae60440a49b3f8cab3c1fba07A conf = 0x47d29e6666af1b7dd450d571233085d7a9866e4d49d2645e2df975489521232bB conf = 0x3313c5cefc361d27fb16847a91c2a73b766ffa90a4839122a9b70a2f6bd1d6df¶
spake2: A='server', B=''w = 0x626e0cdc7b14c9db3e52a0b1b3a768c98e37852d5db30febe0497b14eae8c254x = 0x07adb3db6bc623d3399726bfdbfd3d15a58ea776ab8a308b00392621291f9633pA = 0x04f88fb71c99bfffaea370966b7eb99cd4be0ff1a7d335caac4211c4afd855e2e15a873b298503ad8ba1d9cbb9a392d2ba309b48bfd7879aefd0f2cea6009763b0y = 0xb6a4fc8dbb629d4ba51d6f91ed1532cf87adec98f25dd153a75accafafedec16pB = 0x040c269d6be017dccb15182ac6bfcd9e2a14de019dd587eaf4bdfd353f031101e7cca177f8eb362a6e83e7d5e729c0732e1b528879c086f39ba0f31a9661bd34dbK = 0x0445ee233b8ecb51ebd6e7da3f307e88a1616bae2166121221fdc0dadb986afaf3ec8a988dc9c626fa3b99f58a7ca7c9b844bb3e8dd9554aafc5b53813504c1cbeTT = 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(TT) = 0x005184ff460da2ce59062c87733c299c3521297d736598fc0a1127600efa1afbKe = 0x005184ff460da2ce59062c87733c299cKa = 0x3521297d736598fc0a1127600efa1afbKcA = 0xf3da53604f0aeecea5a33be7bddf6edfKcB = 0x9e3f86848736f159bd92b6e107ec6799A conf = 0xbc9f9bbe99f26d0b2260e6456e05a86196a3307ec6663a18bf6ac825736533b2B conf = 0xc2370e1bf813b086dff0d834e74425a06e6390f48f5411900276dcccc5a297ec¶
spake2: A='', B=''w = 0x7bf46c454b4c1b25799527d896508afd5fc62ef4ec59db1efb49113063d70ccax = 0x8cef65df64bb2d0f83540c53632de911b5b24b3eab6cc74a97609fd659e95473pA = 0x04a65b367a3f613cf9f0654b1b28a1e3a8a40387956c8ba6063e8658563890f46ca1ef6a676598889fc28de2950ab8120b79a5ef1ea4c9f44bc98f585634b46d66y = 0xd7a66f64074a84652d8d623a92e20c9675c61cb5b4f6a0063e4648a2fdc02d53pB = 0x04589f13218822710d98d8b2123a079041052d9941b9cf88c6617ddb2fcc0494662eea8ba6b64692dc318250030c6af045cb738bc81ba35b043c3dcb46adf6f58dK = 0x041a3c03d51b452537ca2a1fea6110353c6d5ed483c4f0f86f4492ca3f378d40a994b4477f93c64d928edbbcd3e85a7c709b7ea73ee97986ce3d1438e135543772TT = 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(TT) = 0xfc6374762ba5cf11f4b2caa08b2cd1b9907ae0e26e8d6234318d91583cd74c86Ke = 0xfc6374762ba5cf11f4b2caa08b2cd1b9Ka = 0x907ae0e26e8d6234318d91583cd74c86KcA = 0x5dbd2f477166b7fb6d61febbd77a5563KcB = 0x7689b4654407a5faeffdc8f18359d8a3A conf = 0xdfb4db8d48ae5a675963ea5e6c19d98d4ea028d8e898dad96ea19a80ade95dcaB conf = 0xd0f0609d1613138d354f7e95f19fb556bf52d751947241e8c7118df5ef0ae175¶
Special thanks toNathaniel McCallum andGreg Hudson for generating M and N andChris Wood for generating test vectors. Thanks toMike Hamburg for advice on how to deal with cofactors.Greg Hudson also suggested the addition of warnings on the reuse of x and y. Thanks toFedor Brunner,Adam Langley,Liliya Akhmetzyanova, and the members of the CFRG for comments and advice. Thanks toScott Fluhrer and those Crypto Panel experts involved in the PAKE selection process (https://github.com/cfrg/pake-selection) who have provided valuable comments.Chris Wood contributed substantial text and reformatting to address the excellent review comments fromKenny Paterson.¶