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SPAKE2, a PAKESealancewatsonbladd@gmail.comAkamai Technologieskaduk@mit.edu
Internet Research Task Force
This document describes SPAKE2 which is a protocol for two
parties that share a password to derive a strong shared key
without 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, however, given existing use of variants
in Kerberos and other applications it was felt publication was
beneficial. Applications that need a symmetric PAKE (password authenticated key exchange) and where hashing onto an elliptic curve
at execution time is not possible can use SPAKE2. This document is a product of the Crypto Forum
Research Group (CFRG) in the IRTF.This document describes SPAKE2, a means for two parties that
share a password to derive a strong shared key without
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.SPAKE2 was not selected as the result of the CFRG PAKE selection
competition. However, given existing use of variants in Kerberos and
other applications it was felt publication was beneficial. This RFC
represents the individual opinion(s) of one or more members of
the Crypto Forum Research Group of the Internet Research Task
Force (IRTF). Many of these applications predated methods to hash to
elliptic curves being available or predated the publication of
the PAKEs that were chosen as an outcome of the PAKE selection
competition. In cases where a symmetric PAKE is needed, and
hashing onto an elliptic curve at protocol execution time is not
available, SPAKE2 is useful.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.SPAKE2 is a two round protocol, wherein the first round establishes a shared secret
between A and B, and the second round serves as key confirmation. Prior to invocation, A and B are provisioned with
information such as the input password needed to run the protocol. We assume that the roles of A and B
are agreed upon by both sides: A goes first and uses M, and B goes second and uses N. If this assignment
of roles is not possible a symmetric variant MUST be used as described later . For instance A may be the client when
using TCP or TLS as an underlying protocol and B the server. Most protocols have such a distinction.
During the first round, A sends a public value pA to B, and
B responds with its own public value 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. A MUST NOT
consider the protocol complete until it receives and
verifies cB. Likewise, B MUST NOT consider the protocol
complete until it receives and verifies cA.This sample flow is shown below.Let G be a group in which the gap Diffie-Hellman (GDH)
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. Applications MUST
specify this encoding, typically by referring to the document defining the group.
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. For common groups used in this document,
P is specified in the document defining the group, and so we do not repeat it here. For elliptic curves other than the ones in this document the methods of
SHOULD be used to generate M and N, e.g. via
M = hash_to_curve("M SPAKE2 seed OID x") and N = hash_to_curve("N SPAKE2 seed OID x"), where
x is an OID for the curve. Applications MAY include a DST tag in this step, as specified
in , though this is not required.|| denotes concatenation of byte 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. Text strings in double quotes are treated as their ASCII encodings
throughout this document.KDF(ikm, salt, info, L) 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(key, message) 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, at least 256 bits long. Common choices for Hash 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 - 128 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. In some cases, 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. To begin, A picks x randomly and uniformly from the integers in [0,p),
and calculates X=x*P and pA=w*M+X, then transmits pA to B.B selects y randomly and uniformly from the integers in [0,p), and calculates
Y=y*P, pB=w*N+Y, then transmits pB to A.Both A and B calculate a group element K. A calculates it
as h*x*(pB-w*N), while B calculates it as h*y*(pA-w*M). A knows pB
because it has received it, and likewise B knows pA. The
multiplication by h prevents small subgroup confinement
attacks by computing a unique value in the quotient
group.K is a shared value, though it MUST NOT be used or output as a shared secret
from the protocol. Both A and B must derive two additional shared secrets from
the protocol transcript, which includes K.
This prevents man-in-the-middle attackers from inserting themselves into
the exchange. The transcript TT is encoded as follows: Here w is encoded as a big endian number padded to the length of p. This representation
prevents timing attacks that otherwise would reveal the length of w. len(w) is thus a constant.
We include it for consistency.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, where both sides of a connection disagree over who is authenticated.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 cA
is computed as a MAC over the protocol transcript TT using KcA, as follows:
cA = 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 cB and checking for equality against that which was received.The protocol transcript TT, as defined in , 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. Keys MUST be at least
128 bits in length.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(Ka, nil, "ConfirmationKeys" || AAD, L), 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 underlying hash output length, e.g.,
|KcA| = |KcB| = 128 bits for HKDF(SHA256), with L=256 bits. 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, it is
possible to vary M and N using as follows:
There is also a symmetric variant where M=N. For this variant we set
This variant MUST be used when it is not possible to determine which of A and B
should use M or N, due to asymmetries in the protocol flows or the desire to use only a single shared secret with
nil identities for authentication. The security of these variants is examined in . The variant with per-user M and N may not be
suitable for protocols that require the initial messages to be
generated by each party at the same time and do not know the
exact identity of the parties before the flow begins.
This section documents SPAKE2 ciphersuite
configurations. A ciphersuite indicates a group,
cryptographic hash function, 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. For Ed25519
the compressed encoding is used ,
all others use the uncompressed SEC1 encoding.GHashKDFMACP-256SHA256 HKDF HMAC P-256SHA512 HKDF HMAC P-384SHA256 HKDF HMAC P-384SHA512 HKDF HMAC P-521SHA512 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 are 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 deserialize and validate group
elements can lead to attacks. An endpoint MUST abort the protocol
if any received public value is not a member of G.The choices of random numbers MUST BE uniform. Randomly
generated values, e.g., x and y, MUST NOT be reused; such reuse
violates the security assumptions of the protocol and results in significant insecurity.
It is RECOMMENDED
to generate these uniform numbers using rejection sampling.Some implementations of elliptic curve multiplication may
leak information about the length of the scalar. These MUST NOT
be used. All operations on elliptic curve points must take time
independent of the inputs. Hashing of the transcript may take
time depending only on the length of the transcript, but not the
contents.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 cases. Applications that need
augmented PAKEs should use .The HMAC keys in this document are shorter than recommended
in . This is appropriate as the
difficulty of the discrete logarithm problem is comparable with
the difficulty of brute forcing the keys.No IANA action is required.Special thanks to Nathaniel McCallum and Greg Hudson for
generation of M and N, and Chris Wood for 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, Liliya
Akhmetzyanova, and the members of the CFRG for comments and
advice. Thanks to Scott Fluhrer and those Crypto Panel experts
involved in the PAKE selection process
(https://github.com/cfrg/pake-selection) who have provided
valuable comments. Chris Wood contributed substantial text and
reformatting to address the excellent review comments from Kenny
Paterson.
&h2c;
&RFC2104;
&RFC2119;
&RFC4493;
&RFC5480;
&RFC5869;
&RFC6234;
&RFC7914;
&RFC8032;
&RFC8174;
SEC 1: Elliptic Curve Cryptography Standards for Efficient Cryptography Group Universally Composable Relaxed Password Authentication Appears in Micciancio D., Ristenpart T. (eds)
Advances in Cryptology -CRYPTO 2020. Crypto 2020. Lecture
notes in Computer Science volume 12170. Springer.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. 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;
&opaque;
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, e.g., "1.3.132.0.35
point generation seed (M)" for P-521. 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, x, and y 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.