Network Working Group W. Ladd
Internet-Draft UC Berkeley
Intended status: Informational B. Kaduk, Ed.
Expires: September 12, 2019 Akamai
March 11, 2019
SPAKE2, a PAKE
draft-irtf-cfrg-spake2-08
Abstract
This document describes SPAKE2 and its augmented variant SPAKE2+,
which are protocols for two parties that share a password to derive a
strong shared key with no risk of disclosing the password. This
method is compatible with any prime order group, is computationally
efficient, and SPAKE2 (but not SPAKE2+) has a security proof.
Status of This Memo
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provisions of BCP 78 and BCP 79.
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This Internet-Draft will expire on September 12, 2019.
Copyright Notice
Copyright (c) 2019 IETF Trust and the persons identified as the
document authors. All rights reserved.
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include Simplified BSD License text as described in Section 4.e of
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the Trust Legal Provisions and are provided without warranty as
described in the Simplified BSD License.
Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2
2. Requirements Notation . . . . . . . . . . . . . . . . . . . . 2
3. Definition of SPAKE2 . . . . . . . . . . . . . . . . . . . . 2
4. Key Schedule and Key Confirmation . . . . . . . . . . . . . . 5
5. Ciphersuites . . . . . . . . . . . . . . . . . . . . . . . . 6
6. Security Considerations . . . . . . . . . . . . . . . . . . . 9
7. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 9
8. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 9
9. References . . . . . . . . . . . . . . . . . . . . . . . . . 9
Appendix A. Algorithm used for Point Generation . . . . . . . . 11
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 13
1. Introduction
This document describes SPAKE2, a means for two parties that share a
password to derive a strong shared key with no risk of disclosing the
password. This password-based key exchange protocol is compatible
with any group (requiring only a scheme to map a random input of
fixed length per group to a random group element), is computationally
efficient, and has a security proof. Predetermined parameters for a
selection of commonly used groups are also provided for use by other
protocols.
2. Requirements Notation
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
"OPTIONAL" in this document are to be interpreted as described in BCP
14 [RFC2119] [RFC8174] when, and only when, they appear in all
capitals, as shown here.
3. Definition of SPAKE2
3.1. Setup
Let G be a group in which the computational Diffie-Hellman (CDH)
problem is hard. Suppose G has order p*h where p is a large prime; h
will be called the cofactor. Let I be the unit element in G, e.g.,
the point at infinity if G is an elliptic curve group. We denote the
operations in the group additively. We assume there is a
representation of elements of G as byte strings: common choices would
be SEC1 compressed [SEC1] for elliptic curve groups or big endian
integers of a fixed (per-group) length for prime field DH. We fix
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two elements M and N in the prime-order subgroup of G as defined in
the table in this document for common groups, as well as a generator
P of the (large) prime-order subgroup of G. P is specified in the
document defining the group, and so we do not repeat it here.
|| 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. Finally, let nil represent an empty string, i.e.,
len(nil) = 0.
KDF is a key-derivation function that takes as input a salt,
intermediate keying material (IKM), info string, and derived key
length L to derive a cryptographic key of length L. MAC is a Message
Authentication Code algorithm that takes a secret key and message as
input to produce an output. Let Hash be a hash function from
arbitrary strings to bit strings of a fixed length. Common choices
for H are SHA256 or SHA512 [RFC6234]. Let MHF be a memory-hard hash
function designed to slow down brute-force attackers. Scrypt
[RFC7914] is a common example of this function. The output length of
MHF matches that of Hash. Parameter selection for MHF is out of
scope for this document. Section 5 specifies variants of KDF, MAC,
Hash, and MHF suitable for use with the protocols contained herein.
Let A and B be two parties. A and B may also have digital
representations of the parties' identities such as Media Access
Control addresses or other names (hostnames, usernames, etc). A and
B may share Additional Authenticated Data (AAD) of length at most
2^16 - 1 bits that is separate from their identities which they may
want to include in the protocol execution. One example of AAD is a
list of supported protocol versions if SPAKE2(+) were used in a
higher-level protocol which negotiates use of a particular PAKE.
Including this list would ensure that both parties agree upon the
same set of supported protocols and therefore prevent downgrade
attacks. We also assume A and B share an integer w; typically w =
MHF(pw) mod p, for a user-supplied password pw. Standards such
NIST.SP.800-56Ar3 suggest taking mod p of a hash value that is 64
bits longer than that needed to represent p to remove statistical
bias introduced by the modulation. Protocols using this
specification must define the method used to compute w: it may be
necessary to carry out various forms of normalization of the password
before hashing [RFC8265]. The hashing algorithm SHOULD be a MHF so
as to slow down brute-force attackers.
We present two protocols below. Note that it is insecure to use the
same password with both protocols; passwords MUST NOT be used for
both SPAKE2 and SPAKE2+.
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3.2. SPAKE2
To begin, A picks x randomly and uniformly from the integers in
[0,p), and calculates X=x*P and T=w*M+X, then transmits T to B. Upon
receipt of T, B computes T*h and aborts if the result is equal to I.
(This ensures T is in the prime order subgroup of G.)
B selects y randomly and uniformly from the integers in [0,p), and
calculates Y=y*P, S=w*N+Y, then transmits S to A. Upon receipt of S,
A computes S*h and aborts if the result is equal to I.
Both A and B calculate a group element K. A calculates it as
x*(S-wN), while B calculates it as y*(T-w*M). A knows S because it
has received it, and likewise B knows T. A and B multiply protocol
messages from each peer by h so as to avoid small subgroup attacks,
but the result of the multiplication is not used for operations other
than the comparison against I and the non-multiplied value is used in
subsequent calculations.
K is a shared value, though it MUST NOT be used as a shared secret.
Both A and B must derive two shared secrets from K and the protocol
transcript. This prevents man-in-the-middle attackers from inserting
themselves into the exchange. The transcript TT is encoded as
follows:
TT = len(A) || A || len(B) || B || len(S) || S || len(T) || T
|| len(K) || K || len(w) || w
If an identity is absent, it is omitted from the transcript entirely.
For example, if both A and B are absent, then TT = len(S) || S ||
len(T) || T || len(K) || K || len(w) || w. Likewise, if only A is
absent, TT = len(B) || B || len(S) || S || len(T) || T || len(K) ||
K || len(w) || w. This must only be done for applications in which
identities are implicit. Otherwise, the protocol risks Unknown Key
Share attacks (discussion of Unknown Key Share attacks in a specific
protocl is given in [I-D.ietf-mmusic-sdp-uks].
Upon completion of this protocol, A and B compute shared secrets Ke,
KcA, and KcB as specified in Section 4. A MUST send B a key
confirmation message so both parties agree upon these shared secrets.
This confirmation message F is computed as a MAC over the protocol
transcript TT using KcA, as follows: F = MAC(KcA, TT). Similarly, B
MUST send A a confirmation message using a MAC computed equivalently
except with the use of KcB. Key confirmation verification requires
computing F and checking for equality against that which was
received.
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3.3. SPAKE2+
This protocol appears in [TDH]. We use the same setup as for SPAKE2,
except that we have two secrets, w0 and w1, derived by hashing the
password pw with the identities of the two participants, A and B.
Specifically, w0s || w1s = MHF(len(pw) || pw || len(A) || A ||
len(B) || B), and then computing w0 = w0s mod p and w1 = w1s mod p.
The length of each of w0s and w1s is equal to half of the MHF output,
e.g., |w0s| = |w1s| = 128 bits for scrypt. w0 and w1 MUST NOT equal
I. If they are, they MUST be iteratively regenerated by computing
w0s || w1s = MHF(len(pw) || pw || len(A) || A || len(B) || B ||
0x0000), where 0x0000 is 16-bit increasing counter. This process
must repeat until valid w0 and w1 are produced. B stores L=w1*P and
w0.
When executing SPAKE2+, A selects x uniformly at random from the
numbers in the range [0, p), and lets X=x*P+w0*M, then transmits X to
B. Upon receipt of X, A computes h*X and aborts if the result is
equal to I. B then selects y uniformly at random from the numbers in
[0, p), then computes Y=y*P+w0*N, and transmits Y to A. Upon receipt
of Y, A computes Y*h and aborts if the result is equal to I.
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 y*L. 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. The protocol transcript
encoding is shown below.
TT = len(A) || A || len(B) || B || len(X) || X || len(Y) || Y
|| len(Z) || Z || len(V) || V || len(w0) || w0
As in Section 3.2, inclusion of A and B in the transcript is optional
depending on whether or not the identities are implicit.
Upon completion of this protocol, A and B follow the same key
derivation and confirmation steps as outlined in Section 3.2.
4. Key Schedule and Key Confirmation
The protocol transcript TT, as defined in Sections Section 3.3 and
Section 3.2, is unique and secret to A and B. Both parties use TT to
derive shared symmetric secrets Ke and Ka as Ke || Ka = Hash(TT).
The length of each key is equal to half of the digest output,
e.g., |Ke| = |Ka| = 128 bits for SHA-256.
Both endpoints use Ka to derive subsequent MAC keys for key
confirmation messages. Specifically, let KcA and KcB be the MAC keys
used by A and B, respectively. A and B compute them as KcA || KcB =
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KDF(nil, Ka, "ConfirmationKeys" || AAD), where AAD is the associated
data each given to each endpoint, or nil if none was provided. The
length of each of KcA and KcB is equal to half of the KDF output,
e.g., |KcA| = |KcB| = 128 bits for HKDF(SHA256).
The resulting key schedule for this protocol, given transcript TT and
additional associated data AAD, is as follows.
TT -> Hash(TT) = Ke || Ka
AAD -> KDF(nil, Ka, "ConfirmationKeys" || AAD) = KcA || KcB
A and B output Ke as the shared secret from the protocol. Ka and its
derived keys are not used for anything except key confirmation.
5. Ciphersuites
This section documents SPAKE2 and SPAKE2+ ciphersuite configurations.
A ciphersuite indicates a group, cryptographic hash algorithm, and
pair of KDF and MAC functions, e.g., SPAKE2-P256-SHA256-HKDF-HMAC.
This ciphersuite indicates a SPAKE2 protocol instance over P-256 that
uses SHA256 along with HKDF [RFC5869] and HMAC [RFC2104] for G, Hash,
KDF, and MAC functions, respectively.
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+--------------+-----------+-----------+---------------+------------+
| G | Hash | KDF | MAC | MHF |
+--------------+-----------+-----------+---------------+------------+
| P-256 | SHA256 | HKDF | HMAC | scrypt |
| | [RFC6234] | [RFC5869] | [RFC2104] | [RFC7914] |
| | | | | |
| P-256 | SHA512 | HKDF | HMAC | scrypt |
| | [RFC6234] | [RFC5869] | [RFC2104] | [RFC7914] |
| | | | | |
| P-384 | SHA256 | HKDF | HMAC | scrypt |
| | [RFC6234] | [RFC5869] | [RFC2104] | [RFC7914] |
| | | | | |
| P-384 | SHA512 | HKDF | HMAC | scrypt |
| | [RFC6234] | [RFC5869] | [RFC2104] | [RFC7914] |
| | | | | |
| P-512 | SHA512 | HKDF | HMAC | scrypt |
| | [RFC6234] | [RFC5869] | [RFC2104] | [RFC7914] |
| | | | | |
| edwards25519 | SHA256 | HKDF | HMAC | scrypt |
| [RFC7748] | [RFC6234] | [RFC5869] | [RFC2104] | [RFC7914] |
| | | | | |
| edwards448 | SHA512 | HKDF | HMAC | scrypt |
| [RFC7748] | [RFC6234] | [RFC5869] | [RFC2104] | [RFC7914] |
| | | | | |
| P-256 | SHA256 | HKDF | CMAC-AES-128 | scrypt |
| | [RFC6234] | [RFC5869] | [RFC4493] | [RFC7914] |
| | | | | |
| P-256 | SHA512 | HKDF | CMAC-AES-128 | scrypt |
| | [RFC6234] | [RFC5869] | [RFC4493] | [RFC7914] |
+--------------+-----------+-----------+---------------+------------+
Table 1: SPAKE2(+) Ciphersuites
The following points represent permissible point generation seeds for
the groups listed in the Table Table 1, using the algorithm presented
in Appendix A. These bytestrings are compressed points as in [SEC1]
for curves from [SEC1].
For P256:
M =
02886e2f97ace46e55ba9dd7242579f2993b64e16ef3dcab95afd497333d8fa12f
seed: 1.2.840.10045.3.1.7 point generation seed (M)
N =
03d8bbd6c639c62937b04d997f38c3770719c629d7014d49a24b4f98baa1292b49
seed: 1.2.840.10045.3.1.7 point generation seed (N)
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For P384:
M =
030ff0895ae5ebf6187080a82d82b42e2765e3b2f8749c7e05eba366434b363d3dc
36f15314739074d2eb8613fceec2853
seed: 1.3.132.0.34 point generation seed (M)
N =
02c72cf2e390853a1c1c4ad816a62fd15824f56078918f43f922ca21518f9c543bb
252c5490214cf9aa3f0baab4b665c10
seed: 1.3.132.0.34 point generation seed (N)
For P521:
M =
02003f06f38131b2ba2600791e82488e8d20ab889af753a41806c5db18d37d85608
cfae06b82e4a72cd744c719193562a653ea1f119eef9356907edc9b56979962d7aa
seed: 1.3.132.0.35 point generation seed (M)
N =
0200c7924b9ec017f3094562894336a53c50167ba8c5963876880542bc669e494b25
32d76c5b53dfb349fdf69154b9e0048c58a42e8ed04cef052a3bc349d95575cd25
seed: 1.3.132.0.35 point generation seed (N)
For edwards25519:
M =
d048032c6ea0b6d697ddc2e86bda85a33adac920f1bf18e1b0c6d166a5cecdaf
seed: edwards25519 point generation seed (M)
N =
d3bfb518f44f3430f29d0c92af503865a1ed3281dc69b35dd868ba85f886c4ab
seed: edwards25519 point generation seed (N)
For edwards448:
M =
b6221038a775ecd007a4e4dde39fd76ae91d3cf0cc92be8f0c2fa6d6b66f9a12
942f5a92646109152292464f3e63d354701c7848d9fc3b8880
seed: edwards448 point generation seed (M)
N =
6034c65b66e4cd7a49b0edec3e3c9ccc4588afd8cf324e29f0a84a072531c4db
f97ff9af195ed714a689251f08f8e06e2d1f24a0ffc0146600
seed: edwards448 point generation seed (N)
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6. 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 a path to a proof that server
compromise does not lead to password compromise under the DH
assumption (though the corresponding model excludes precomputation
attacks).
Elements received from a peer MUST be checked for group membership:
failure to properly validate group elements can lead to attacks.
Beyond the cofactor multiplication checks to ensure that these
elements are in the prime order subgroup of G, it is essential that
endpoints verify received points are members of G.
The choices of random numbers MUST BE uniform. Randomly generated
values (e.g., x and y) MUST NOT be reused; such reuse may permit
dictionary attacks on the password.
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.
7. IANA Considerations
No IANA action is required.
8. Acknowledgments
Special thanks to Nathaniel McCallum and Greg Hudson for generation
of test vectors. Thanks to Mike Hamburg for advice on how to deal
with cofactors. Greg Hudson also suggested the 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. Chris Wood
contributed substantial text and reformatting to address the
excellent review comments from Kenny Paterson. Trevor Perrin
informed me of SPAKE2+.
9. References
9.1. Normative References
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[RFC2104] Krawczyk, H., Bellare, M., and R. Canetti, "HMAC: Keyed-
Hashing for Message Authentication", RFC 2104,
DOI 10.17487/RFC2104, February 1997,
.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
.
[RFC4493] Song, JH., Poovendran, R., Lee, J., and T. Iwata, "The
AES-CMAC Algorithm", RFC 4493, DOI 10.17487/RFC4493, June
2006, .
[RFC5480] Turner, S., Brown, D., Yiu, K., Housley, R., and T. Polk,
"Elliptic Curve Cryptography Subject Public Key
Information", RFC 5480, DOI 10.17487/RFC5480, March 2009,
.
[RFC5869] Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand
Key Derivation Function (HKDF)", RFC 5869,
DOI 10.17487/RFC5869, May 2010,
.
[RFC6234] Eastlake 3rd, D. and T. Hansen, "US Secure Hash Algorithms
(SHA and SHA-based HMAC and HKDF)", RFC 6234,
DOI 10.17487/RFC6234, May 2011,
.
[RFC7748] Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves
for Security", RFC 7748, DOI 10.17487/RFC7748, January
2016, .
[RFC7914] Percival, C. and S. Josefsson, "The scrypt Password-Based
Key Derivation Function", RFC 7914, DOI 10.17487/RFC7914,
August 2016, .
[RFC8032] Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital
Signature Algorithm (EdDSA)", RFC 8032,
DOI 10.17487/RFC8032, January 2017,
.
[RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
May 2017, .
[SEC1] SEC, "STANDARDS FOR EFFICIENT CRYPTOGRAPHY, "SEC 1:
Elliptic Curve Cryptography", version 2.0", May 2009.
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9.2. Informative References
[I-D.ietf-mmusic-sdp-uks]
Thomson, M. and E. Rescorla, "Unknown Key Share Attacks on
uses of TLS with the Session Description Protocol (SDP)",
draft-ietf-mmusic-sdp-uks-03 (work in progress), January
2019.
[REF] Abdalla, M. and D. Pointcheval, "Simple Password-Based
Encrypted Key Exchange Protocols.", Feb 2005.
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. Springer-
Verlag, Berlin, Germany.
[RFC8265] Saint-Andre, P. and A. Melnikov, "Preparation,
Enforcement, and Comparison of Internationalized Strings
Representing Usernames and Passwords", RFC 8265,
DOI 10.17487/RFC8265, October 2017,
.
[TDH] Cash, D., Kiltz, E., and V. Shoup, "The Twin-Diffie
Hellman Problem and Applications", 2008.
EUROCRYPT 2008. Volume 4965 of Lecture notes in Computer
Science, pages 127-145. Springer-Verlag, Berlin, Germany.
Appendix A. Algorithm used for Point Generation
This section describes the algorithm that was used to generate the
points (M) and (N) in the table in Section 5.
For each curve in the table below, we construct a string using the
curve OID from [RFC5480] (as an ASCII string) or its name, combined
with the needed constant, for instance "1.3.132.0.35 point generation
seed (M)" for P-512. This string is turned into a series of blocks
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 the string modified appropriately.
A byte string of length equal to that of an encoded group element is
constructed by concatenating as many blocks as are required, starting
from the first block, and truncating to the desired length. The byte
string is then formatted as required for the group. In the case of
Weierstrass curves, we take the desired length as the length for
representing a compressed point (section 2.3.4 of [SEC1]), and use
the low-order bit of the first byte as the sign bit. In order to
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obtain the correct format, the value of the first byte is set to 0x02
or 0x03 (clearing the first six bits and setting the seventh bit),
leaving the sign bit as it was in the byte string constructed by
concatenating hash blocks. For the [RFC8032] curves a different
procedure is used. For edwards448 the 57-byte input has the least-
significant 7 bits of the last byte set to zero, and for edwards25519
the 32-byte input is not modified. For both the [RFC8032] curves the
(modified) input is then interpreted as the representation of the
group element. If this interpretation yields a valid group element
with the correct order (p), the (modified) byte string is the output.
Otherwise, the initial hash block is discarded and a new byte string
constructed from the remaining hash blocks. The procedure of
constructing a byte string of the appropriate length, formatting it
as required for the curve, and checking if it is a valid point of the
correct order, is repeated until a valid element is found.
The following python snippet generates the above points, assuming an
elliptic curve implementation following the interface of
Edwards25519Point.stdbase() and Edwards448Point.stdbase() in
Appendix A of [RFC8032]:
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def iterated_hash(seed, n):
h = seed
for i in range(n):
h = hashlib.sha256(h).digest()
return h
def bighash(seed, start, sz):
n = -(-sz // 32)
hashes = [iterated_hash(seed, i) for i in range(start, start + n)]
return b''.join(hashes)[:sz]
def canon_pointstr(ecname, s):
if ecname == 'edwards25519':
return s
elif ecname == 'edwards448':
return s[:-1] + bytes([s[-1] & 0x80])
else:
return bytes([(s[0] & 1) | 2]) + s[1:]
def gen_point(seed, ecname, ec):
for i in range(1, 1000):
hval = bighash(seed, i, len(ec.encode()))
pointstr = canon_pointstr(ecname, hval)
try:
p = ec.decode(pointstr)
if p != ec.zero_elem() and p * p.l() == ec.zero_elem():
return pointstr, i
except Exception:
pass
Authors' Addresses
Watson Ladd
UC Berkeley
Email: watsonbladd@gmail.com
Benjamin Kaduk (editor)
Akamai Technologies
Email: kaduk@mit.edu
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