Internet Draft                                                   W. Ladd
<draft-irtf-cfrg-spake2-01.txt>                                UC Berkeley
Category: Informational
Expires 20 August 2015                                  16 February 2015

                             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

   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.

   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),

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   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.

   Note that the calculation of S=wN+yG may be performed more
   efficiently then by two separate scalar multiplications via Strauss's

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 is taken, and the first byte has all
   bits but the low-order one cleared, and the second-order bit set.
   This string of bytes is then interpreted as a SEC1 compressed point.
   If this is impossible, then the next non-overlapping segment of
   sufficient length is taken.

   For P256:

   M =

   N =

   For P384:

   M =

   N =

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   For P521:

   M =

   N =

4. Security Considerations

   A security proof for prime order groups is found in [REF]. Note that
   the choice of M and N is critical for the security proof. The points
   in the table of points were generated via the use of a hash function
   to mitigate this risk.

   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 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.

   This PAKE does not support augmentation. As a result, the server has
   to store a password equivalent. This is considered a significant

   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.

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6. Acknowledgments

   Special thanks to Nathaniel McCallum for generation of test vectors.
   Thanks to Mike Hamburg for advice on how to deal with cofactors.
   Thanks to Fedor Brunner and the members of the CFRG for comments and

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.

   [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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