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SPAKE2, a PAKECloudflarewatsonbladd@gmail.comAkamai Technologieskaduk@mit.eduThis document describes SPAKE2 which is a protocol 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 group, is computationally efficient, and
SPAKE2 has a security proof. This document predated the CFRG PAKE competition
and it was not selected.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.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
when, and only when, they
appear in all capitals, as shown here.Let G be a group in which the gap 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
uncompressed or compressed for elliptic curve groups or big
endian integers of a fixed (per-group) length for prime field DH.
We fix 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. In the case of a composite order group
we will work in the quotient group. 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 . Let MHF be a memory-hard
hash function designed to slow down brute-force attackers. Scrypt
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.
specifies variants of KDF, MAC, and Hash
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 as 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 .
The hashing algorithm SHOULD be a MHF so as to slow down brute-force
attackers. SPAKE2 is a one round protocol to establish a shared secret with an
additional round for key confirmation. Prior to invocation, A and B are provisioned with
information such as the input password needed to run the protocol.
During the first round, A sends a public share pA
to B, and B responds with its own public share pB. Both A and B then derive a shared secret
used to produce encryption and authentication keys. The latter are used during the second
round for key confirmation. ( details the key derivation and
confirmation steps.) In particular, A sends a key confirmation message cA to B, and B responds
with its own key confirmation message cB. Both parties MUST NOT consider the protocol complete
prior to receipt and validation of these key confirmation messages.This sample trace is shown below.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 pA=T to B.B selects y randomly and uniformly from the integers in [0,p), and calculates
Y=y*P, S=w*N+Y, then transmits pB=S to A.Both A and B calculate a group element K. A calculates it
as h*x*(S-w*N), while B calculates it as h*y*(T-w*M). A knows S
because it has received it, and likewise B knows T. The
multiplication by h prevents small subgroup confinement
attacks by computing a unique value in the quotient
group. This is a common mitigation against this kind of attack.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 the protocol transcript.
This prevents man-in-the-middle attackers from inserting themselves into
the exchange. The transcript TT is encoded as follows:If an identity is absent, it is encoded as a zero-length string.
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 protocol is given in ).Upon completion of this protocol, A and B compute shared secrets Ke, KcA, and KcB as
specified in . 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.The protocol transcript TT, as defined in Section , 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), with |Ke| = |Ka|.
The length of each key is equal to half of the digest output, e.g., 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 = 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.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.
To avoid concerns that an attacker needs to solve a single ECDH instance to break the authentication of SPAKE2, a variant
based on using is also presented. In this variant, M and N are computed as follows:
In addition M and N may be equal to have a symmetric variant. The security of these variants is examined in .
This section documents 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 and HMAC
for G, Hash, KDF, and MAC functions, respectively.GHashKDFMACP-256SHA256 HKDF HMAC P-256SHA512 HKDF HMAC P-384SHA256 HKDF HMAC P-384SHA512 HKDF HMAC P-512SHA512 HKDF HMAC edwards25519 SHA256 HKDF HMAC edwards448 SHA512 HKDF HMAC P-256SHA256 HKDF CMAC-AES-128 P-256SHA512 HKDF CMAC-AES-128 The following points represent permissible point generation seeds
for the groups listed in the Table ,
using the algorithm presented in .
These bytestrings are compressed points as in
for curves from .For P256:For P384:For P521:For edwards25519:For edwards448:A security proof of SPAKE2 for prime order groups is found in
, reducing the security of SPAKE2 to the
gap Diffie-Hellman assumption. Note that the choice of M and N
is critical for the security proof. The generation methods
specified in this document is designed to eliminate concerns
related to knowing discrete logs of M and N.Elements received from a peer MUST be checked for group
membership: failure to properly validate group elements can lead
to attacks. 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 in
some use casesNo IANA action is required.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.
&h2c;
&RFC2104;
&RFC2119;
&RFC4493;
&RFC5480;
&RFC5869;
&RFC6234;
&RFC7748;
&RFC7914;
&RFC8032;
&RFC8174;
SEC 1: Elliptic Curve Cryptography Standards for Efficient Cryptography Group Perfect Forward Security of SPAKE2 IACR eprint 2019/1194Simple 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. Springer-Verlag,
Berlin, Germany.
The Twin-Diffie Hellman Problem and ApplicationsEUROCRYPT 2008. Volume 4965 of Lecture notes in Computer
Science, pages 127-145. Springer-Verlag, Berlin, Germany.
&RFC8265;
&uks;
This section describes the algorithm that was used to generate
the points (M) and (N) in the table in .For each curve in the table below, we construct a string
using the curve OID from (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 ),
and use the low-order bit of the first byte as the sign bit.
In order to 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 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 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 :This section contains test vectors for SPAKE2 using
the P256-SHA256-HKDF-HMAC ciphersuite. (Choice of MHF is omitted
and values for w and w0,w1 are provided directly.) All points are
encoded using the uncompressed format, i.e., with a 0x04 octet
prefix, specified in A and B identity strings
are provided in the protocol invocation.