Internet Research Task Force                              W. Ladd
Internet-Draft                                                Cloudflare
Intended status: Informational                             B. Kaduk, Ed.
Expires: November 25, 2021                                        Akamai
                                                            May 24, 2021

                             SPAKE2, a PAKE


   This document describes SPAKE2 which is a protocol for two parties
   that share a password to derive a strong shared key with no risk of
   disclosing the password.  This method is compatible with any group,
   is computationally efficient, and SPAKE2 has a security proof.  This
   document predated the CFRG PAKE competition and it was not selected.
   This document is a product of the Crypto Forum Research Group (CFRG)
   in the IRTF.

Status of This Memo

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   This Internet-Draft will expire on November 25, 2021.

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   document authors.  All rights reserved.

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Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
   2.  Requirements Notation . . . . . . . . . . . . . . . . . . . .   2
   3.  Definition of SPAKE2  . . . . . . . . . . . . . . . . . . . .   2
   4.  Key Schedule and Key Confirmation . . . . . . . . . . . . . .   5
   5.  Per-User M and N  . . . . . . . . . . . . . . . . . . . . . .   6
   6.  Ciphersuites  . . . . . . . . . . . . . . . . . . . . . . . .   6
   7.  Security Considerations . . . . . . . . . . . . . . . . . . .   9
   8.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .   9
   9.  Acknowledgments . . . . . . . . . . . . . . . . . . . . . . .   9
   10. References  . . . . . . . . . . . . . . . . . . . . . . . . .  10
   Appendix A.  Algorithm used for Point Generation  . . . . . . . .  12
   Appendix B.  Test Vectors . . . . . . . . . . . . . . . . . . . .  13
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  16

1.  Introduction

   This document describes SPAKE2, a means for two parties that share a
   password to derive a strong shared key with no risk of disclosing the
   password.  This password-based key exchange protocol is compatible
   with any group (requiring only a scheme to map a random input of
   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.This document represents the consensus of the Crypto Forum
   Research Group (CFRG).

2.  Requirements Notation

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "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.

3.  Definition of SPAKE2

3.1.  Setup

   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; 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

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   DH.  We fix two elements M and N in the prime-order subgroup of G as
   defined in the table in this document for common groups, as well as a
   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.  P is
   specified in the document defining the group, and so we do not repeat
   it here.

   For elliptic curves other than the ones in this document the methods
   of [I-D.irtf-cfrg-hash-to-curve] SHOULD be used to generate M and N,
   e.g. via M = h2c("M SPAKE2 seed OID x"), N= h2c("N SPAKE2 seed OID
   x") where x is an OID for the curve.

   || denotes concatenation of 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.

   KDF(ikm, salt, info) is a key-derivation function that takes as input
   a salt, intermediate keying material (IKM), info string, and derived
   key length L to derive a cryptographic key of length L.  MAC 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.  Common
   choices for H are SHA256 or SHA512 [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 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 length at most
   2^16 - 1 bits that is 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 which negotiates use of a particular PAKE.  Including
   this list would ensure that both parties agree upon the same set of
   supported protocols and therefore prevent downgrade attacks.  We also
   assume A and B share an 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 specification must define the
   method used to compute w: it may be necessary to carry out various
   forms of normalization of the password before hashing [RFC8265].  The

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   hashing algorithm SHOULD be a MHF so as to slow down brute-force

3.2.  Protocol Flow

   SPAKE2 is a one round protocol to establish a shared secret with an
   additional round for key confirmation.  Prior to invocation, A and B
   are provisioned with information such as the input password needed to
   run the protocol.  During the first round, A sends a public share pA
   to B, and B responds with its own public share 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.  Both
   parties MUST NOT consider the protocol complete prior to receipt and
   validation of these key confirmation messages.

   This sample trace is shown below.

                   A                  B
                   | (setup protocol) |
     (compute pA)  |        pA        |
                   |        pB        | (compute pB)
                   |                  |
                   | (derive secrets) |
     (compute cA)  |        cA        |
                   |        cB        | (compute cB)

3.3.  SPAKE2

   To begin, A picks x randomly and uniformly from the integers in
   [0,p), and calculates X=x*P and S=w*M+X, then transmits pA=S to B.

   B selects y randomly and uniformly from the integers in [0,p), and
   calculates Y=y*P, T=w*N+Y, then transmits pB=T to A.

   Both A and B calculate a group element K.  A calculates it as
   h*x*(T-w*N), while B calculates it as h*y*(S-w*M).  A knows S because
   it has received it, and likewise B knows T.  The multiplication by h
   prevents small subgroup confinement attacks by computing a unique
   value in the quotient group.  This is a common mitigation against
   this kind of attack.

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   K is a shared value, though it MUST NOT be used as a shared secret.
   Both A and B must derive two shared secrets from the protocol
   transcript.  This prevents man-in-the-middle attackers from inserting
   themselves into the exchange.  The transcript TT is encoded as

           TT = len(A) || A
             || len(B) || B
             || len(S) || S
             || len(T) || T
             || 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.  We include it for

   If an identity is absent, it is encoded as a zero-length string.
   This MUST only be done for applications in which identities are
   implicit.  Otherwise, the protocol risks Unknown Key Share attacks
   (discussion of Unknown Key Share attacks in a specific protocol is
   given in [I-D.ietf-mmusic-sdp-uks]).

   Upon completion of this protocol, A and B compute shared secrets Ke,
   KcA, and KcB as specified in Section 4.  A MUST send B a key
   confirmation message so both parties agree upon these shared secrets.
   This confirmation message F is computed as a MAC over the protocol
   transcript TT using KcA, as follows: F = MAC(KcA, TT).  Similarly, B
   MUST send A a confirmation message using a MAC computed equivalently
   except with the use of KcB.  Key confirmation verification requires
   computing F and checking for equality against that which was

4.  Key Schedule and Key Confirmation

   The protocol transcript TT, as defined in Section Section 3.3, is
   unique and secret to A and B.  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.

   Both endpoints use Ka to derive subsequent MAC keys for key
   confirmation messages.  Specifically, let KcA and KcB be the MAC keys
   used by A and B, respectively.  A and B compute them as KcA || KcB =
   KDF(Ka,nil, "ConfirmationKeys" || AAD), where AAD is the associated
   data each given to each endpoint, or nil if none was provided.  The

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   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).

   The resulting key schedule for this protocol, given transcript TT and
   additional associated data AAD, is as follows.

       TT  -> Hash(TT) = Ke || Ka
       AAD -> KDF(nil, Ka, "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.

5.  Per-User M and N

   To avoid concerns that an attacker needs to solve a single ECDH
   instance to break the authentication of SPAKE2, a variant based on
   using [I-D.irtf-cfrg-hash-to-curve] is also presented.  In this
   variant, M and N are computed as follows:

       M = h2c(Hash("M for SPAKE2" || len(A) || A || len(B) || B))
       N = h2c(Hash("N for SPAKE2" || len(A) || A || len(B) || B))

   In addition M and N may be equal to have a symmetric variant.  The
   security of these variants is examined in [MNVAR].  This variant may
   not be suitable for protocols that require the messages to be
   exchanged symmetrically and do not know the exact identity of the
   parties before the flow begins.

6.  Ciphersuites

   This section documents SPAKE2 ciphersuite configurations.  A
   ciphersuite indicates a group, cryptographic hash algorithm, 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
   SHA256 along with HKDF [RFC5869] and HMAC [RFC2104] for G, Hash, KDF,
   and MAC functions, respectively.

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   |        G         |      Hash     |     KDF     |       MAC        |
   |      P-256       |     SHA256    |     HKDF    |  HMAC [RFC2104]  |
   |                  |   [RFC6234]   |  [RFC5869]  |                  |
   |                  |               |             |                  |
   |      P-256       |     SHA512    |     HKDF    |  HMAC [RFC2104]  |
   |                  |   [RFC6234]   |  [RFC5869]  |                  |
   |                  |               |             |                  |
   |      P-384       |     SHA256    |     HKDF    |  HMAC [RFC2104]  |
   |                  |   [RFC6234]   |  [RFC5869]  |                  |
   |                  |               |             |                  |
   |      P-384       |     SHA512    |     HKDF    |  HMAC [RFC2104]  |
   |                  |   [RFC6234]   |  [RFC5869]  |                  |
   |                  |               |             |                  |
   |      P-512       |     SHA512    |     HKDF    |  HMAC [RFC2104]  |
   |                  |   [RFC6234]   |  [RFC5869]  |                  |
   |                  |               |             |                  |
   |   edwards25519   |     SHA256    |     HKDF    |  HMAC [RFC2104]  |
   |    [RFC7748]     |   [RFC6234]   |  [RFC5869]  |                  |
   |                  |               |             |                  |
   |    edwards448    |     SHA512    |     HKDF    |  HMAC [RFC2104]  |
   |    [RFC7748]     |   [RFC6234]   |  [RFC5869]  |                  |
   |                  |               |             |                  |
   |      P-256       |     SHA256    |     HKDF    |   CMAC-AES-128   |
   |                  |   [RFC6234]   |  [RFC5869]  |    [RFC4493]     |
   |                  |               |             |                  |
   |      P-256       |     SHA512    |     HKDF    |   CMAC-AES-128   |
   |                  |   [RFC6234]   |  [RFC5869]  |    [RFC4493]     |

                       Table 1: SPAKE2 Ciphersuites

   The following points represent permissible point generation seeds for
   the groups listed in the Table Table 1, using the algorithm presented
   in Appendix A.  These bytestrings are compressed points as in [SEC1]
   for curves from [SEC1].

   For P256:

   M =
   seed: 1.2.840.10045.3.1.7 point generation seed (M)

   N =
   seed: 1.2.840.10045.3.1.7 point generation seed (N)

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   For P384:

   M =
   seed: point generation seed (M)

   N =
   seed: point generation seed (N)

   For P521:

   M =
   seed: point generation seed (M)

   N =
   seed: point generation seed (N)

   For edwards25519:

   M =
   seed: edwards25519 point generation seed (M)

   N =
   seed: edwards25519 point generation seed (N)

   For edwards448:

   M =
   seed: edwards448 point generation seed (M)

   N =
   seed: edwards448 point generation seed (N)

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7.  Security Considerations

   A security proof of SPAKE2 for prime order groups is found in [REF],
   reducing the security of SPAKE2 to the gap Diffie-Hellman 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 peer MUST be checked for group membership:
   failure to properly validate group elements can lead to attacks.  It
   is essential that endpoints verify received points are members of G.

   The choices of random numbers MUST BE uniform.  Randomly generated
   values (e.g., x and y) MUST NOT be reused; such reuse may permit
   dictionary attacks on the password.  To generate these uniform
   numbers rejection sampling is recommended.  Some implementations of
   elliptic curve multiplication may leak information about the length
   of the scalar: these MUST NOT be used.

   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.

   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.

8.  IANA Considerations

   No IANA action is required.

9.  Acknowledgments

   Special thanks to Nathaniel McCallum and Greg Hudson for generation
   of M and N, and Cris Wood for test vectors.  Thanks to Mike 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 to Fedor
   Brunner, Adam Langley,Liliya Akhmetzyanova, and the members of the
   CFRG for comments and advice.  Thanks to Scott Fluhrer and those
   Crypto Panel experts involved in the PAKE selection process
   ( who have provided valuable
   comments.  Chris Wood contributed substantial text and reformatting
   to address the excellent review comments from Kenny Paterson.

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10.  References

10.1.  Normative References

              Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R., and
              C. Wood, "Hashing to Elliptic Curves", draft-irtf-cfrg-
              hash-to-curve-05 (work in progress), November 2019.

   [RFC2104]  Krawczyk, H., Bellare, M., and R. Canetti, "HMAC: Keyed-
              Hashing for Message Authentication", RFC 2104,
              DOI 10.17487/RFC2104, February 1997,

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997,

   [RFC4493]  Song, JH., Poovendran, R., Lee, J., and T. Iwata, "The
              AES-CMAC Algorithm", RFC 4493, DOI 10.17487/RFC4493, June
              2006, <>.

   [RFC5480]  Turner, S., Brown, D., Yiu, K., Housley, R., and T. Polk,
              "Elliptic Curve Cryptography Subject Public Key
              Information", RFC 5480, DOI 10.17487/RFC5480, March 2009,

   [RFC5869]  Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand
              Key Derivation Function (HKDF)", RFC 5869,
              DOI 10.17487/RFC5869, May 2010,

   [RFC6234]  Eastlake 3rd, D. and T. Hansen, "US Secure Hash Algorithms
              (SHA and SHA-based HMAC and HKDF)", RFC 6234,
              DOI 10.17487/RFC6234, May 2011,

   [RFC7748]  Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves
              for Security", RFC 7748, DOI 10.17487/RFC7748, January
              2016, <>.

   [RFC7914]  Percival, C. and S. Josefsson, "The scrypt Password-Based
              Key Derivation Function", RFC 7914, DOI 10.17487/RFC7914,
              August 2016, <>.

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   [RFC8032]  Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital
              Signature Algorithm (EdDSA)", RFC 8032,
              DOI 10.17487/RFC8032, January 2017,

   [RFC8174]  Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
              2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
              May 2017, <>.

10.2.  Informative References

              Thomson, M. and E. Rescorla, "Unknown Key-Share Attacks on
              Uses of TLS with the Session Description Protocol (SDP)",
              draft-ietf-mmusic-sdp-uks-07 (work in progress), August

   [MNVAR]    Abdalla, M., Barbosa, M., Bradley, T., Jarecki, S., Katz,
              J., and J. Xu, "Universally Composable Relaxed Password
              Authentication", August 2020.

              Appears in Micciancio D., Ristenpart T. (eds) Advances in
              Cryptology -CRYPTO 20202.  Crypto 20202.  Lecture notes in
              Computer Science volume 12170.  Springer.

   [REF]      Abdalla, M. and D. Pointcheval, "Simple Password-Based
              Encrypted Key Exchange Protocols.", Feb 2005.

              Appears in A.  Menezes, editor.  Topics in Cryptography-
              CT-RSA 2005, Volume 3376 of Lecture Notes in Computer
              Science, pages 191-208, San Francisco, CA, US.  Springer-
              Verlag, Berlin, Germany.

   [RFC8265]  Saint-Andre, P. and A. Melnikov, "Preparation,
              Enforcement, and Comparison of Internationalized Strings
              Representing Usernames and Passwords", RFC 8265,
              DOI 10.17487/RFC8265, October 2017,

   [SEC1]     Standards for Efficient Cryptography Group, "SEC 1:
              Elliptic Curve Cryptography", May 2009.

   [TDH]      Cash, D., Kiltz, E., and V. Shoup, "The Twin-Diffie
              Hellman Problem and Applications", 2008.

              EUROCRYPT 2008.  Volume 4965 of Lecture notes in Computer
              Science, pages 127-145.  Springer-Verlag, Berlin, Germany.

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Appendix A.  Algorithm used for Point Generation

   This section describes the algorithm that was used to generate the
   points (M) and (N) in the table in Section 6.

   For each curve in the table below, we construct a string using the
   curve OID from [RFC5480] (as an ASCII string) or its name, combined
   with the needed constant, for instance " point generation
   seed (M)" for P-512.  This string is turned into a series of blocks
   by hashing with SHA256, 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 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 [RFC8032] curves 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 [RFC8032] curves 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
   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 following the interface of
   Edwards25519Point.stdbase() and Edwards448Point.stdbase() in
   Appendix&nbsp;A of [RFC8032]:

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  def iterated_hash(seed, n):
      h = seed
      for i in range(n):
          h = hashlib.sha256(h).digest()
      return h

  def 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])
          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)
              p = ec.decode(pointstr)
              if p != ec.zero_elem() and p * p.l() == ec.zero_elem():
                  return pointstr, i
          except Exception:

Appendix B.  Test Vectors

   This section contains test vectors for SPAKE2 using the P256-SHA256-
   HKDF-HMAC ciphersuite.  (Choice of MHF is omitted and 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

B.1.  SPAKE2 Test Vectors

spake2: A='server', B='client'
w = 0x2ee57912099d31560b3a44b1184b9b4866e904c49d12ac5042c97dca461b1a5f
x = 0x43dd0fd7215bdcb482879fca3220c6a968e66d70b1356cac18bb26c84a78d729
S = 0x04a56fa807caaa53a4d28dbb9853b9815c61a411118a6fe516a8798434751470
y = 0xdcb60106f276b02606d8ef0a328c02e4b629f84f89786af5befb0bc75b6e66be
T = 0x0406557e482bd03097ad0cbaa5df82115460d951e3451962f1eaf4367a420676

Ladd & Kaduk            Expires November 25, 2021              [Page 13]

Internet-Draft               SPAKE2, a PAKE                     May 2021

K = 0x0412af7e89717850671913e6b469ace67bd90a4df8ce45c2af19010175e37eed
TT = 0x06000000000000007365727665720600000000000000636c69656e744100000
Hash(TT) = 0x0e0672dc86f8e45565d338b0540abe6915bdf72e2b35b5c9e5663168e960a91bKe = 0x0e0672dc86f8e45565d338b0540abe69
Ka = 0x15bdf72e2b35b5c9e5663168e960a91b
KcA = 0x00c12546835755c86d8c0db7851ae86f
KcB = 0xa9fa3406c3b781b93d804485430ca27a
A conf = 0x58ad4aa88e0b60d5061eb6b5dd93e80d9c4f00d127c65b3b35b1b5281fee38f0
B conf = 0xd3e2e547f1ae04f2dbdbf0fc4b79f8ecff2dff314b5d32fe9fcef2fb26dc459b

spake2: A='', B='client'
w = 0x0548d8729f730589e579b0475a582c1608138ddf7054b73b5381c7e883e2efae
x = 0x403abbe3b1b4b9ba17e3032849759d723939a27a27b9d921c500edde18ed654b
S = 0x04a897b769e681c62ac1c2357319a3d363f610839c4477720d24cbe32f5fd85f
y = 0x903023b6598908936ea7c929bd761af6039577a9c3f9581064187c3049d87065
T = 0x04e0f816fd1c35e22065d5556215c097e799390d16661c386e0ecc84593974a6
K = 0x048f83ec9f6e4f87cc6f9dc740bdc2769725f923364f01c84148c049a39a735e
TT = 0x00000000000000000600000000000000636c69656e74410000000000000004a
Hash(TT) = 0x642f05c473c2cd79909f9a841e2f30a70bf89b18180af97353ba198789c2b963Ke = 0x642f05c473c2cd79909f9a841e2f30a7
Ka = 0x0bf89b18180af97353ba198789c2b963
KcA = 0xc6be376fc7cd1301fd0a13adf3e7bffd
KcB = 0xb7243f4ae60440a49b3f8cab3c1fba07
A conf = 0x47d29e6666af1b7dd450d571233085d7a9866e4d49d2645e2df975489521232b
B conf = 0x3313c5cefc361d27fb16847a91c2a73b766ffa90a4839122a9b70a2f6bd1d6df

spake2: A='server', B=''
w = 0x626e0cdc7b14c9db3e52a0b1b3a768c98e37852d5db30febe0497b14eae8c254

Ladd & Kaduk            Expires November 25, 2021              [Page 14]

Internet-Draft               SPAKE2, a PAKE                     May 2021

x = 0x07adb3db6bc623d3399726bfdbfd3d15a58ea776ab8a308b00392621291f9633
S = 0x04f88fb71c99bfffaea370966b7eb99cd4be0ff1a7d335caac4211c4afd855e2
y = 0xb6a4fc8dbb629d4ba51d6f91ed1532cf87adec98f25dd153a75accafafedec16
T = 0x040c269d6be017dccb15182ac6bfcd9e2a14de019dd587eaf4bdfd353f031101
K = 0x0445ee233b8ecb51ebd6e7da3f307e88a1616bae2166121221fdc0dadb986afa
TT = 0x06000000000000007365727665720000000000000000410000000000000004f
Hash(TT) = 0x005184ff460da2ce59062c87733c299c3521297d736598fc0a1127600efa1afbKe = 0x005184ff460da2ce59062c87733c299c
Ka = 0x3521297d736598fc0a1127600efa1afb
KcA = 0xf3da53604f0aeecea5a33be7bddf6edf
KcB = 0x9e3f86848736f159bd92b6e107ec6799
A conf = 0xbc9f9bbe99f26d0b2260e6456e05a86196a3307ec6663a18bf6ac825736533b2
B conf = 0xc2370e1bf813b086dff0d834e74425a06e6390f48f5411900276dcccc5a297ec

spake2: A='', B=''
w = 0x7bf46c454b4c1b25799527d896508afd5fc62ef4ec59db1efb49113063d70cca
x = 0x8cef65df64bb2d0f83540c53632de911b5b24b3eab6cc74a97609fd659e95473
S = 0x04a65b367a3f613cf9f0654b1b28a1e3a8a40387956c8ba6063e8658563890f4
y = 0xd7a66f64074a84652d8d623a92e20c9675c61cb5b4f6a0063e4648a2fdc02d53
T = 0x04589f13218822710d98d8b2123a079041052d9941b9cf88c6617ddb2fcc0494
K = 0x041a3c03d51b452537ca2a1fea6110353c6d5ed483c4f0f86f4492ca3f378d40
TT = 0x00000000000000000000000000000000410000000000000004a65b367a3f613
Hash(TT) = 0xfc6374762ba5cf11f4b2caa08b2cd1b9907ae0e26e8d6234318d91583cd74c86Ke = 0xfc6374762ba5cf11f4b2caa08b2cd1b9
Ka = 0x907ae0e26e8d6234318d91583cd74c86
KcA = 0x5dbd2f477166b7fb6d61febbd77a5563
KcB = 0x7689b4654407a5faeffdc8f18359d8a3
A conf = 0xdfb4db8d48ae5a675963ea5e6c19d98d4ea028d8e898dad96ea19a80ade95dca
B conf = 0xd0f0609d1613138d354f7e95f19fb556bf52d751947241e8c7118df5ef0ae175

Ladd & Kaduk            Expires November 25, 2021              [Page 15]

Internet-Draft               SPAKE2, a PAKE                     May 2021

Authors' Addresses

   Watson Ladd


   Benjamin Kaduk (editor)
   Akamai Technologies


Ladd & Kaduk            Expires November 25, 2021              [Page 16]