Internet Draft                                                   W. Ladd
<draft-irtf-cfrg-spake2-02.txt>                              UC Berkeley
Category: Informational
Expires 17 February 2015                                  17 August 2015

                             SPAKE2, a PAKE

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

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   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 for elliptic curve
   groups or big endian integers of a particular length for prime field

   || denotes concatenation of strings. We also let len(S) denote the
   length of a string in bytes, represented as an eight-byte big-endian

   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.

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

2.2 SPAKE2

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   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.
   It is essential to combine K 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) 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+, Alice 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 Bob. Bob selects y uniformly at random from the
   numbers in [0, ph) divisible by h, then computes Y=yG+w0*N, and
   transmits it to Alice.

   Alice computes Z as x(Y-w0*N), and V as w1(Y-w0*N). Bob 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.

   The initial segment of bytes of length equal to that of an encoded

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   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. 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.

   These bytes appear in the format commonly associated with each group.

   For P256:

   M =

   N =

   For P384:

   M =

   N =

   For P521:

   M =

   N =

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

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   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 should be uniformly at random. 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 should 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 should 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' should be used instead. This is critical for security.

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 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.

   [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.

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   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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