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This 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 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 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.This document represents the consensus of the Crypto Forum Research Group (CFRG).

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(ikm, salt, info) 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.

| | pB | (compute pB) |<-----------------| | | | (derive secrets) | (compute cA) | cA | |----------------->| | cB | (compute cB) |<-----------------| ]]>

To begin, A picks x randomly and uniformly from the integers in [0,p), and calculates X=x*P and S=w*M+X, then transmits pA=S to B. B selects y randomly and uniformly from the integers in [0,p), and calculates Y=y*P, T=w*N+Y, then transmits pB=T to A. Both A and B calculate a group element K. A calculates it as h*x*(T-w*N), while B calculates it as h*y*(S-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:

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 (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(Ka,nil, "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 underlying hash output length, 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.

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.

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 variant may not be suitable for protocols that require the messages to be exchanged symmetrically 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 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. G Hash KDF MAC P-256 SHA256 HKDF HMAC P-256 SHA512 HKDF HMAC P-384 SHA256 HKDF HMAC P-384 SHA512 HKDF HMAC P-512 SHA512 HKDF HMAC edwards25519 SHA256 HKDF HMAC edwards448 SHA512 HKDF HMAC P-256 SHA256 HKDF CMAC-AES-128 P-256 SHA512 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 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. To generate these uniform numbers rejection sampling is recommended. Some implementations of elliptic curve multiplication may leak information about the length of the scalar: these MUST NOT be used. 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. 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 Cris 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; &RFC7748; &RFC7914; &RFC8032; &RFC8174; Standards for Efficient Cryptography Group Appears in Micciancio D., Ristenpart T. (eds) Advances in Cryptology -CRYPTO 20202. Crypto 20202. Lecture notes in Computer Science volume 12170. Springer. 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. EUROCRYPT 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,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.