Internet Draft W. Ladd
UC Berkeley
Category: Informational
Expires 26 July 2015 22 January 2015
SPAKE2, a PAKE
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Abstract
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, 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
DH.
|| denotes concatenation of strings. We also let len(S) denote the
length of a string in bytes, represented as an eight-byte big-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.
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.
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.
Both A and B can now calculate a shared key as H(len(A)|| A || len(B)
|| B || len(S) || S || len(T) || T || len(w) || w || len(K) || K).
The encoding of group elements must be decided upon based on
convenience. For elliptic curve groups in short Weierstrass form,
SEC1 uncompressed format is recommended due to wide support.
Note that the calculation of S=wN+yG may be performed more
efficiently then by two separate scalar multiplications via Strauss's
algorithm.
3. Table of points for common groups
This table was generated in the following way: A string S was hashed
with the SHA-2 function matching the curve size repeatedly until a
valid x coordinate for the curve was generated. The points are
presented in hexdecimal SEC1 format. The string was "CURVE point
generation seed (X)" with CURVE the name of the curve and X M or N
accordingly.
For P256:
M =
02004F3886286C3DBEDAABC44EAE84C7D88205289AB3A6F7DFC9B055B41CDC5D71
N =
02004E10BC191275D4AEB183DB6E3385CDE56AE90BEA034FB20FE4D3E0E86B57F9
For P384:
M =
0300D96F8C84B8EB7BE566CA5B8788F6D7B71619F78DCA54C061E75FD0D5353570A
CA36EB3EB16C93C855442B66970A197
N =
020024C63E7770841FA3F1ABCF7469F6822C84F0EFCA2DAC8D7FD4B097C8291DD70
AA1CA824B2DFC4104F0D4FA0301EDFF
For P521:
M =
0200000073962354404088E8407DE57063FE70C5F9B014531CCD09A007509193A60
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F345031F8B1239F754B20CC5946C0257339314D112AFFE96EA880C3EBC074E5FF96
N =
02000000594BAFF0BEF7134EBBCC5D86670777EBC4A473D6797167BBEEFECEC11F8
863AF4CEC3A651E99F0357C59450D8E06124B099D1FBBF498546400AA80F08CFFB8
4. Security Considerations
A security proof for prime order groups is found in [REF]. Note that
the choice of M and N is critical: anyone who is aware of an x such
that xN=M, or xG=N or M can break the scheme above. 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 recieved points are valid compressions of
points on an elliptic curve when using elliptic curves. This can be
done by a quadratic character computation. It is not necessary to
validate prime order.
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
drawback.
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.
Thanks to Fedor Brunner and the members of the CFRG for comments and
advice.
7. References
[REF] Abdalla, M. and Pointcheval, D. Simple Password-Based Encrypted
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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.
Author Addresses
Watson Ladd
watsonbladd@gmail.com
Berkeley, CA
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