Internet Draft W. Ladd
UC Berkeley
Category: Informational
Expires 16 August 2016 13 February 2016
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, 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
attackers.
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.
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
Alice.
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
H(len(A)||A||len(B)||B||len(X)||X||len(Y)||Y||len(Z)||Z||len(V)||V)
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 "1.3.132.0.35 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
[SEC1].
For P256:
M =
02886e2f97ace46e55ba9dd7242579f2993b64e16ef3dcab95afd497333d8fa12f
N =
03d8bbd6c639c62937b04d997f38c3770719c629d7014d49a24b4f98baa1292b49
For P384:
M =
030ff0895ae5ebf6187080a82d82b42e2765e3b2f8749c7e05eba366434b363d3dc
36f15314739074d2eb8613fceec2853
N =
02c72cf2e390853a1c1c4ad816a62fd15824f56078918f43f922ca21518f9c543bb
252c5490214cf9aa3f0baab4b665c10
For P521:
M =
02003f06f38131b2ba2600791e82488e8d20ab889af753a41806c5db18d37d85608
cfae06b82e4a72cd744c719193562a653ea1f119eef9356907edc9b56979962d7aa
N =
0200c7924b9ec017f3094562894336a53c50167ba8c5963876880542bc669e494b25
32d76c5b53dfb349fdf69154b9e0048c58a42e8ed04cef052a3bc349d95575cd25
The following python snippet generates the above points:
def canon_pointstr(self, s): return chr(ord(s[0]) & 1 | 2) +
s[1:]
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def iterated_hash(seed, n): h = seed for i in xrange(n):
h = SHA256.new(h).digest() 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.
[SEC1] STANDARDS FOR EFFICIENT CRYPTOGRAPHY, "SEC 1: Elliptic Curve
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
watsonbladd@gmail.com
Berkeley, CA
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