Internet Draft                                                   W. Ladd
<draft-irtf-cfrg-spake2-03.txt>                              UC Berkeley
Category: Informational
Expires 16 August 2016                                  13 February 2016

                             SPAKE2, a PAKE

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   This Internet-Draft describes SPAKE2, a secure, efficient password

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   based key exchange protocol.

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Table of Contents

   1. Introduction ....................................................3
   2. Definition of SPAKE2.............................................3
   3. Table of points .................................................4
   4. Security considerations .........................................5
   5. IANA actions ....................................................5
   6. Acknowledgements.................................................5
   7. References.......................................................5

1. Introduction

   This document describes a means for two parties that share a password
   to derive a shared key. This method is compatible with any group, is
   computationally efficient, and has a strong security proof.

2. Definition of SPAKE2
2.1 Setup

   Let G be a group in which the Diffie-Hellman problem is hard of order
   ph, with p a big prime and h a cofactor. We denote the operations in
   the group additively. Let H be a hash function from arbitrary strings
   to bit strings of a fixed length. Common choices for H are SHA256 or
   SHA512. We assume there is a representation of elements of G as byte
   strings: common choices would be SEC1 uncompressed [SEC1] for
   elliptic curve groups or big endian integers of a particular length
   for prime field DH.

   || 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.

   We fix two elements M and N as defined in the table in this document
   for common groups, as well as a generator G of the group. G is
   specified in the document defining the group, and so we do not recall
   it here.

   Let A and B be two parties. We will assume that A and B are also
   representations of the parties such as MAC addresses or other names
   (hostnames, usernames, etc). We assume they share an integer w.
   Typically w will be the hash of a user-supplied password, truncated
   and taken mod p. Protocols using this protocol must define the method
   used to compute w: it may be necessary to carry out normalization.
   The hashing algorithm SHOULD be designed to slow down brute force

   We present two protocols below. Note that it is insecure to use the
   same password with both protocols, this MUST NOT be done.

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2.2 SPAKE2

   A picks x randomly and uniformly from the integers in [0,ph)
   divisible by h, and calculates X=xG and T=wM+X, then transmits T to

   B selects y randomly and uniformly from the integers in [0,ph),
   divisible by h and calculates Y=yG, S=wN+Y, then transmits S to A.

   Both A and B calculate a group element K. A calculates it as x(S-wN),
   while B calculates it as y(T-wM). A knows S because it has received
   it, and likewise B knows T.

   This K is a shared secret, but the scheme as described is not secure.
   K  MUST be combined with the values transmitted and received via a
   hash function to have a secure protocol. If higher-level protocols
   prescribe a method for doing so, that SHOULD be used. Otherwise we
   can compute K' as H(len(A)||A||len(B)||B||len(S)||S||
   len(T)||T||len(K)||K || len(w) || w) and use K' as the key.

2.3 SPAKE2+

   This protocol and security proof appear in [TDH]. We use the same
   setup as for SPAKE2, except that we have two secrets, w0 and w1. The
   server, here Bob, stores L=w1*g and w0.

   When executing SPAKE2+, A selects x uniformly at random from the
   numbers in the range [0, ph) divisible by h, and lets X=xG+w0*M, then
   transmits X to B. B selects y uniformly at random from the numbers in
   [0, ph) divisible by h, then computes Y=yG+w0*N, and transmits it to

   A computes Z as x(Y-w0*N), and V as w1(Y-w0*N). B computes Z as y(X-
   w0*M) and V as yL. Both share Z and V as common keys. It is essential
   that both Z and V be used in combination with the transcript to
   derive the keying material. For higher-level protocols without
   sufficient transcript hashing, let K' be
   and use K' as the established key.

3. Table of points for common groups

   Every curve presented in the table below has an OID from [OID]. We
   construct a string using the OID and the needed constant, for
   instance " point generation seed (M)" for P-512.  This
   string is turned into an infinite sequence of bytes 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

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   the string modified appropriately.

   The initial segment of bytes of length equal to that of an encoded
   group element is taken, and is then formatted as required for the
   group.  In the case of Weierstrass points, this means setting the
   first byte to 0x02 or 0x03 depending on the low-order bit. For
   Ed25519 style formats this means taking all the bytes as the
   representation of the group element.  This string of bytes is then
   interpreted as a point in the group. If this is impossible, then the
   next non-overlapping segment of sufficient length is taken. We
   multiply that point by the cofactor h, and if that is not the
   identity, output it.

   These bytestrings are compressed points as in [SEC1] for curves from

   For P256:

   M =

   N =

   For P384:

   M =

   N =

   For P521:

   M =

   N =

   The following python snippet generates the above points:

   def canon_pointstr(self, s):        return chr(ord(s[0]) & 1 | 2) +

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   def iterated_hash(seed, n):      h = seed      for i in xrange(n):
         h =      return h

     def bighash(seed, start, sz):      n = -(-sz // 32)      hashes =
   [iterated_hash(seed, i) for i in xrange(start, start + n)]
   return ''.join(hashes)[:sz]

     def gen_point(seed, ec, order):      for i in xrange(1, 1000):
       pointstr = ec.canon_pointstr(bighash(seed, i, ec.nbytes_point()))
            try:              p = ec.decode_point(pointstr)
   if ec.mul(p, order) == ec.identity():                  return
   pointstr, i          except Exception:              pass

4. Security Considerations

   A security proof of SPAKE2 for prime order groups is found in [REF].
   Note that the choice of M and N is critical for the security proof.
   The generation method specified in this document is designed to
   eliminate concerns related to knowing discrete logs of M and N.

   SPAKE2+ appears in [TDH], along with proof.

   There is no key-confirmation as this is a one round protocol. It is
   expected that a protocol using this key exchange mechanism provides
   key confirmation separately if desired.

   Elements should be checked for group membership: failure to properly
   validate group elements can lead to attacks. In particular it is
   essential to verify that received points are valid compressions of
   points on an elliptic curve when using elliptic curves. It is not
   necessary to validate membership in the prime order subgroup: the
   multiplication by cofactors eliminates this issue.

   The choices of random numbers MUST BE uniform. Note that to pick a
   random multiple of h in [0, ph) one can pick a random integer in
   [0,p) and multiply by h. Reuse of ephemerals results in dictionary
   attacks and MUST NOT be done.

   SPAKE2 does not support augmentation. As a result, the server has to
   store a password equivalent. This is considered a significant
   drawback, and so SPAKE2+ also appears in this document.

   As specified the shared secret K is not suitable for use as a shared
   key.  It MUST be passed to a hash function along with the public
   values used to derive it and the party identities to avoid attacks.
   In protocols which do not perform this separately, the value denoted
   K' MUST be used instead.

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5. IANA Considerations

   No IANA action is required.

6. Acknowledgments

   Special thanks to Nathaniel McCallum for generation of test vectors.
   Thanks to Mike Hamburg for advice on how to deal with cofactors. Greg
   Hudson suggested addition of warnings on the reuse of x and y. Thanks
   to Fedor Brunner, Adam Langley, and the members of the CFRG for
   comments and advice. Trevor Perrin informed me of SPAKE2+.

7. References

   [REF] Abdalla, M. and Pointcheval, D. Simple Password-Based Encrypted
   Key Exchange Protocols. 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 Feb. 14-18, 2005.
   Springer-Verlag, Berlin, Germany.

   Cryptography", version 2.0, May 2009,           <

   [TDH] Cash, D. Kiltz, E. and Shoup, V. The Twin-Diffie Hellman
   Problem and Applications. Advances in Cryptology--EUROCRYPT 2008.
   Volume 4965 of Lecture notes in Computer Science, pages 127-145.
   Springer-Verlag, Berlin, Germany.

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

Author Addresses
   Watson Ladd
   Berkeley, CA

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