rfc9382.original   rfc9382.txt 
Internet Research Task Force W. Ladd Internet Research Task Force (IRTF) W. Ladd
Internet-Draft Sealance Request for Comments: 9382 Akamai
Intended status: Informational B. Kaduk, Ed. Category: Informational September 2023
Expires: 12 August 2022 Akamai ISSN: 2070-1721
February 2022
SPAKE2, a PAKE SPAKE2, a Password-Authenticated Key Exchange
draft-irtf-cfrg-spake2-26
Abstract Abstract
This document describes SPAKE2 which is a protocol for two parties This document describes SPAKE2, which is a protocol for two parties
that share a password to derive a strong shared key without that share a password to derive a strong shared key without
disclosing the password. This method is compatible with any group, disclosing the password. This method is compatible with any group,
is computationally efficient, and SPAKE2 has a security proof. This is computationally efficient, and has a security proof. This
document predated the CFRG PAKE competition and it was not selected, document predated the Crypto Forum Research Group (CFRG) password-
however, given existing use of variants in Kerberos and other authenticated key exchange (PAKE) competition, and it was not
applications it was felt publication was beneficial. Applications selected; however, given existing use of variants in Kerberos and
that need a symmetric PAKE (password authenticated key exchange) and other applications, it was felt that publication was beneficial.
where hashing onto an elliptic curve at execution time is not Applications that need a symmetric PAKE, but are unable to hash onto
possible can use SPAKE2. This document is a product of the Crypto an elliptic curve at execution time, can use SPAKE2. This document
Forum Research Group (CFRG) in the IRTF. is a product of the Crypto Forum Research Group in the Internet
Research Task Force (IRTF).
Status of This Memo Status of This Memo
This Internet-Draft is submitted in full conformance with the This document is not an Internet Standards Track specification; it is
provisions of BCP 78 and BCP 79. published for informational purposes.
Internet-Drafts are working documents of the Internet Engineering
Task Force (IETF). Note that other groups may also distribute
working documents as Internet-Drafts. The list of current Internet-
Drafts is at https://datatracker.ietf.org/drafts/current/.
Internet-Drafts are draft documents valid for a maximum of six months This document is a product of the Internet Research Task Force
and may be updated, replaced, or obsoleted by other documents at any (IRTF). The IRTF publishes the results of Internet-related research
time. It is inappropriate to use Internet-Drafts as reference and development activities. These results might not be suitable for
material or to cite them other than as "work in progress." deployment. 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). Documents approved for publication by
the IRSG are not candidates for any level of Internet Standard; see
Section 2 of RFC 7841.
This Internet-Draft will expire on 5 August 2022. Information about the current status of this document, any errata,
and how to provide feedback on it may be obtained at
https://www.rfc-editor.org/info/rfc9382.
Copyright Notice Copyright Notice
Copyright (c) 2022 IETF Trust and the persons identified as the Copyright (c) 2023 IETF Trust and the persons identified as the
document authors. All rights reserved. document authors. All rights reserved.
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Table of Contents Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2 1. Introduction
2. Requirements Notation . . . . . . . . . . . . . . . . . . . . 3 2. Requirements Notation
3. Definition of SPAKE2 . . . . . . . . . . . . . . . . . . . . 3 3. Definition of SPAKE2
4. Key Schedule and Key Confirmation . . . . . . . . . . . . . . 6 3.1. Protocol Flow
5. Per-User M and N and M=N . . . . . . . . . . . . . . . . . . 7 3.2. Setup
6. Ciphersuites . . . . . . . . . . . . . . . . . . . . . . . . 7 3.3. SPAKE2
7. Security Considerations . . . . . . . . . . . . . . . . . . . 10 4. Key Schedule and Key Confirmation
8. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 11 5. Per-User M and N and M=N
9. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 11 6. Ciphersuites
10. References . . . . . . . . . . . . . . . . . . . . . . . . . 11 7. Security Considerations
10.1. Normative References . . . . . . . . . . . . . . . . . . 11 8. IANA Considerations
10.2. Informative References . . . . . . . . . . . . . . . . . 12 9. References
Appendix A. Algorithm used for Point Generation . . . . . . . . 13 9.1. Normative References
Appendix B. Test Vectors . . . . . . . . . . . . . . . . . . . . 14 9.2. Informative References
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 17 Appendix A. Algorithm Used for Point Generation
Appendix B. SPAKE2 Test Vectors
Acknowledgements
Contributors
Author's Address
1. Introduction 1. Introduction
This document describes SPAKE2, a means for two parties that share a This document describes SPAKE2, which is a means for two parties that
password to derive a strong shared key without disclosing the share a password to derive a strong shared key without disclosing the
password. This password-based key exchange protocol is compatible password. This password-based key exchange protocol is compatible
with any group (requiring only a scheme to map a random input of with any group (requiring only a scheme to map a random input of a
fixed length per group to a random group element), is computationally fixed length per group to a random group element), is computationally
efficient, and has a security proof. Predetermined parameters for a efficient, and has a security proof. Predetermined parameters for a
selection of commonly used groups are also provided for use by other selection of commonly used groups are also provided for use by other
protocols. protocols.
SPAKE2 was not selected as the result of the CFRG PAKE selection SPAKE2 was not selected as the result of the CFRG PAKE selection
competition. However, given existing use of variants in Kerberos and competition. However, given existing use of variants in Kerberos and
other applications it was felt publication was beneficial. This RFC other applications, it was felt that publication was beneficial.
represents the individual opinion(s) of one or more members of the This RFC represents the individual opinion(s) of one or more members
Crypto Forum Research Group of the Internet Research Task Force of the Crypto Forum Research Group of the IRTF.
(IRTF).
Many of these applications predated methods to hash to elliptic Many of these applications predated methods to hash to elliptic
curves being available or predated the publication of the PAKEs that curves being available or predated the publication of the PAKEs that
were chosen as an outcome of the PAKE selection competition. In were chosen as an outcome of the PAKE selection competition. In
cases where a symmetric PAKE is needed, and hashing onto an elliptic cases where a symmetric PAKE is needed and hashing onto an elliptic
curve at protocol execution time is not available, SPAKE2 is useful. curve at protocol execution time is not available, SPAKE2 is useful.
2. Requirements Notation 2. Requirements Notation
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
"OPTIONAL" in this document are to be interpreted as described in BCP "OPTIONAL" in this document are to be interpreted as described in
14 [RFC2119] [RFC8174] when, and only when, they appear in all BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all
capitals, as shown here. capitals, as shown here.
3. Definition of SPAKE2 3. Definition of SPAKE2
3.1. Protocol Flow 3.1. Protocol Flow
SPAKE2 is a two round protocol, wherein the first round establishes a 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 shared secret between A and B, and the second round serves as key
confirmation. Prior to invocation, A and B are provisioned with confirmation. Prior to invocation, A and B are provisioned with
information such as the input password needed to run the protocol. 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 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 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 assignment of roles is not possible, a symmetric variant MUST be
as described later Section 5. For instance A may be the client when used, as described later Section 5. For instance, A may be the
using TCP or TLS as an underlying protocol and B the server. Most client when using TCP or TLS as an underlying protocol, and B may be
protocols have such a distinction. During the first round, A sends a the server. Most protocols have such a distinction. During the
public value pA to B, and B responds with its own public value pB. first round, A sends a public value pA to B, and B responds with its
Both A and B then derive a shared secret used to produce encryption own public value pB. Both A and B then derive a shared secret used
and authentication keys. The latter are used during the second round to produce encryption and authentication keys. The latter are used
for key confirmation. (Section 4 details the key derivation and during the second round for key confirmation. (Section 4 details the
confirmation steps.) In particular, A sends a key confirmation key derivation and confirmation steps.) In particular, A sends a key
message cA to B, and B responds with its own key confirmation message confirmation message cA to B, and B responds with its own key
cB. A MUST NOT consider the protocol complete until it receives and confirmation message cB. A MUST NOT consider the protocol complete
verifies cB. Likewise, B MUST NOT consider the protocol complete until it receives and verifies cB. Likewise, B MUST NOT consider the
until it receives and verifies cA. protocol complete until it receives and verifies cA.
This sample flow is shown below. This sample flow is shown below.
A B A B
| (setup protocol) | | |
| | | |
(compute pA) | pA | (compute pA) | pA |
|----------------->| |---------------------->|
| pB | (compute pB) | pB | (compute pB)
|<-----------------| |<----------------------|
| | | |
| (derive secrets) | | (derive secrets) |
| | | |
(compute cA) | cA | (compute cA) | cA |
|----------------->| |---------------------->|
| cB | (compute cB) | cB | (compute cB)
| | (check cA) | | (check cA)
|<-----------------| |<----------------------|
(check cB) | | (check cB) | |
3.2. Setup 3.2. Setup
Let G be a group in which the gap Diffie-Hellman (GDH) problem is 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 hard. Suppose G has order p*h, where p is a large prime and h will
called the cofactor. Let I be the unit element in G, e.g., the point be called the cofactor. Let I be the unit element in G, e.g., the
at infinity if G is an elliptic curve group. We denote the point at infinity if G is an elliptic curve group. We denote the
operations in the group additively. We assume there is a operations in the group additively. We assume there is a
representation of elements of G as byte strings: common choices would representation of elements of G as byte strings: common choices would
be SEC1 [SEC1] uncompressed or compressed for elliptic curve groups be SEC1 [SEC1] uncompressed or compressed for elliptic curve groups
or big endian integers of a fixed (per-group) length for prime field or big-endian integers of a fixed (per-group) length for prime field
DH. Applications MUST specify this encoding, typically by referring DH. Applications MUST specify this encoding, typically by referring
to the document defining the group. We fix two elements M and N in 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 the prime-order subgroup of G, as defined in Table 1 of this document
document for common groups, as well as a generator P of the (large) for common groups, as well as generator P of the (large) prime-order
prime-order subgroup of G. In the case of a composite order group we subgroup of G. In the case of a composite order group, we will work
will work in the quotient group. For common groups used in this in the quotient group. For common groups used in this document, P is
document, P is specified in the document defining the group, and so specified in the document defining the group, so we do not repeat it
we do not repeat it here. here.
For elliptic curves other than the ones in this document the methods For elliptic curves other than the ones in this document, the methods
of [I-D.irtf-cfrg-hash-to-curve] SHOULD be used to generate M and N, described in [RFC9380] SHOULD be used to generate M and N, e.g., via
e.g. via M = hash_to_curve("M SPAKE2 seed OID x") and N = M = hash_to_curve("M SPAKE2 seed OID x") and N = hash_to_curve("N
hash_to_curve("N SPAKE2 seed OID x"), where x is an OID for the SPAKE2 seed OID x"), where x is an OID for the curve. Applications
curve. Applications MAY include a DST tag in this step, as specified MAY include a domain separation tag (DST) in this step, as specified
in [I-D.irtf-cfrg-hash-to-curve], though this is not required. in [RFC9380], though this is not required.
|| denotes concatenation of byte strings. We also let len(S) denote || denotes concatenation of byte strings. We also let len(S) denote
the length of a string in bytes, represented as an eight-byte little- the length of a string in bytes, represented as an eight-byte little-
endian number. Finally, let nil represent an empty string, i.e., endian number. Finally, let nil represent an empty string, i.e.,
len(nil) = 0. Text strings in double quotes are treated as their len(nil) = 0. Text strings in double quotes are treated as their
ASCII encodings throughout this document. ASCII encodings throughout this document.
KDF(ikm, salt, info, L) is a key-derivation function that takes as KDF(ikm, salt, info, L) is a key-derivation function that takes as
input a salt, intermediate keying material (IKM), info string, and input a salt, input keying material (IKM), an info string, and
derived key length L to derive a cryptographic key of length L. derived key length L to derive a cryptographic key of length L.
MAC(key, message) is a Message Authentication Code algorithm that MAC(key, message) is a Message Authentication Code algorithm that
takes a secret key and message as input to produce an output. Let 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 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 fixed length that is at least 256 bits long. Common choices for Hash
SHA256 or SHA512 [RFC6234]. Let MHF be a memory-hard hash function are SHA-256 or SHA-512 [RFC6234]. Let MHF be a memory-hard hash
designed to slow down brute-force attackers. Scrypt [RFC7914] is a function designed to slow down brute-force attackers. Scrypt
common example of this function. The output length of MHF matches [RFC7914] is a common example of this function. The output length of
that of Hash. Parameter selection for MHF is out of scope for this MHF matches that of Hash. Parameter selection for MHF is out of
document. Section 6 specifies variants of KDF, MAC, and Hash scope for this document. Section 6 specifies variants of KDF, MAC,
suitable for use with the protocols contained herein. and Hash that are suitable for use with the protocols contained
herein.
Let A and B be two parties. A and B may also have digital Let A and B be two parties. A and B may also have digital
representations of the parties' identities such as Media Access representations of the parties' identities, such as Media Access
Control addresses or other names (hostnames, usernames, etc). A and Control addresses or other names (hostnames, usernames, etc.). A and
B may share Additional Authenticated Data (AAD) of length at most B may share additional authenticated data (AAD) of a length that is
2^16 - 128 bits that is separate from their identities which they may at most 2^16 - 128 bits and separate from their identities, which
want to include in the protocol execution. One example of AAD is a they may want to include in the protocol execution. One example of
list of supported protocol versions if SPAKE2 were used in a higher- AAD is a list of supported protocol versions if SPAKE2 were used in a
level protocol which negotiates use of a particular PAKE. Including higher-level protocol that negotiates use of a particular PAKE.
this list would ensure that both parties agree upon the same set of Including this list would ensure that both parties agree upon the
supported protocols and therefore prevent downgrade attacks. We also same set of supported protocols and therefore prevents downgrade
assume A and B share an integer w; typically w = MHF(pw) mod p, for a attacks. We also assume A and B share integer w; typically, w =
user-supplied password pw. Standards such as NIST.SP.800-56Ar3 MHF(pw) mod p for a user-supplied password, pw. Standards, such as
suggest taking mod p of a hash value that is 64 bits longer than that [NIST.SP.800-56Ar3], suggest taking mod p of a hash value that is 64
needed to represent p to remove statistical bias introduced by the bits longer than that needed to represent p to remove statistical
modulation. Protocols using this specification MUST define the bias introduced by the modulation. Protocols using this
method used to compute w. In some cases, it may be necessary to specification MUST define the method used to compute w. In some
carry out various forms of normalization of the password before cases, it may be necessary to carry out various forms of
hashing [RFC8265]. The hashing algorithm SHOULD be a MHF so as to normalization of the password before hashing [RFC8265]. The hashing
slow down brute-force attackers. algorithm SHOULD be an MHF so as to slow down brute-force attackers.
3.3. SPAKE2 3.3. SPAKE2
To begin, A picks x randomly and uniformly from the integers in To begin, A picks x randomly and uniformly from the integers in [0,p)
[0,p), and calculates X=x*P and pA=w*M+X, then transmits pA to B. and calculates X=x*P and pA=w*M+X. Then, it transmits pA to B.
B selects y randomly and uniformly from the integers in [0,p), and 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. calculates Y=y*P and pB=w*N+Y. Then, it transmits pB to A.
Both A and B calculate a group element K. A calculates it as Both A and B calculate group element K. A calculates it as h*x*(pB-
h*x*(pB-w*N), while B calculates it as h*y*(pA-w*M). A knows pB w*N), while B calculates it as h*y*(pA-w*M). A knows pB because it
because it has received it, and likewise B knows pA. The has received it, and likewise B knows pA. The multiplication by h
multiplication by h prevents small subgroup confinement attacks by prevents small subgroup confinement attacks by computing a unique
computing a unique value in the quotient group. value in the quotient group.
K is a shared value, though it MUST NOT be used or output as a shared 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 secret from the protocol. Both A and B must derive two additional
shared secrets from the protocol transcript, which includes K. This shared secrets from the protocol transcript, which includes K. This
prevents man-in-the-middle attackers from inserting themselves into use of the transcript ensures any manipulation of the messages sent
the exchange. The transcript TT is encoded as follows: is reflected in the keys. The transcript TT is encoded as follows:
TT = len(A) || A TT = len(A) || A
|| len(B) || B || len(B) || B
|| len(pA) || pA || len(pA) || pA
|| len(pB) || pB || len(pB) || pB
|| len(K) || K || len(K) || K
|| len(w) || w || len(w) || w
Here w is encoded as a big endian number padded to the length of p. Here, w is encoded as a big-endian number padded to the length of p.
This representation prevents timing attacks that otherwise would This representation prevents timing attacks that otherwise would
reveal the length of w. len(w) is thus a constant. We include it for reveal the length of w. len(w) is thus a constant for a given group.
consistency. We include it for consistency.
If an identity is absent, it is encoded as a zero-length string. If an identity is absent, it is encoded as a zero-length string.
This MUST only be done for applications in which identities are This MUST only be done for applications in which identities are
implicit. Otherwise, the protocol risks unknown key share attacks, implicit. Otherwise, the protocol risks unknown key-share attacks,
where both sides of a connection disagree over who is authenticated. where both sides of a connection disagree over who is authenticated.
Upon completion of this protocol, A and B compute shared secrets Ke, 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 KcA, and KcB, as specified in Section 4. A MUST send B a key
confirmation message so both parties agree upon these shared secrets. confirmation message so that both parties agree upon these shared
This confirmation message cA is computed as a MAC over the protocol secrets. The confirmation message cA is computed as a MAC over the
transcript TT using KcA, as follows: cA = MAC(KcA, TT). Similarly, B protocol transcript TT, using KcA as follows: cA = MAC(KcA, TT).
MUST send A a confirmation message using a MAC computed equivalently Similarly, B MUST send A a confirmation message using a MAC that is
except with the use of KcB. Key confirmation verification requires computed equivalently, except with the use of KcB. Key confirmation
computing cB and checking for equality against that which was verification requires computing cA (or cB, respectively) and checking
received. for equality against that which was received.
4. Key Schedule and Key Confirmation 4. Key Schedule and Key Confirmation
The protocol transcript TT, as defined in Section 3.3, is unique and The protocol transcript TT, as defined in Section 3.3, is unique and
secret to A and B. Both parties use TT to derive shared symmetric secret to A and B (though it contains some substrings that are not
secrets Ke and Ka as Ke || Ka = Hash(TT), with |Ke| = |Ka|. The secret). Both parties use TT to derive shared symmetric secrets Ke
length of each key is equal to half of the digest output, e.g., 128 and Ka as Ke || Ka = Hash(TT), with |Ke| = |Ka|. The length of each
bits for SHA-256. Keys MUST be at least 128 bits in length. 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 Both endpoints use Ka to derive subsequent MAC keys for key
confirmation messages. Specifically, let KcA and KcB be the MAC keys confirmation messages. Specifically, KcA and KcB are the MAC keys
used by A and B, respectively. A and B compute them as KcA || KcB = used by A and B, respectively. A and B compute them as KcA || KcB =
KDF(Ka, nil, "ConfirmationKeys" || AAD, L), where AAD is the KDF(Ka, nil, "ConfirmationKeys" || AAD, L), where AAD is the
associated data each given to each endpoint, or nil if none was associated data given to each endpoint or AAD is nil if none was
provided. The length of each of KcA and KcB is equal to half of the 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 underlying hash output length, e.g., |KcA| = |KcB| = 128 bits for
HKDF(SHA256), with L=256 bits. HKDF(SHA256), with L=256 bits.
The resulting key schedule for this protocol, given transcript TT and The resulting key schedule for this protocol, given transcript TT and
additional associated data AAD, is as follows. AAD, is as follows.
TT -> Hash(TT) = Ke || Ka TT -> Hash(TT) = Ke || Ka
AAD -> KDF(Ka, nil, "ConfirmationKeys" || AAD) = KcA || KcB AAD -> KDF(Ka, nil, "ConfirmationKeys" || AAD) = KcA || KcB
A and B output Ke as the shared secret from the protocol. Ka and its 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. derived keys are not used for anything except key confirmation.
5. Per-User M and N and M=N 5. Per-User M and N and M=N
To avoid concerns that an attacker needs to solve a single ECDH To avoid concerns that an attacker needs to solve a single Elliptic
instance to break the authentication of SPAKE2, it is possible to Curve Diffie-Hellman (ECDH) instance to break the authentication of
vary M and N using [I-D.irtf-cfrg-hash-to-curve] as follows: SPAKE2, it is possible to vary M and N using [RFC9380] as follows:
M = hash_to_curve(Hash("M SPAKE2" || len(A) || A || len(B) || B)) M = hash_to_curve(Hash("M SPAKE2" || len(A) || A || len(B) || B))
N = hash_to_curve(Hash("N SPAKE2" || len(A) || A || len(B) || B)) N = hash_to_curve(Hash("N SPAKE2" || len(A) || A || len(B) || B))
There is also a symmetric variant where M=N. For this variant we set There is also a symmetric variant where M=N. For this variant, we
set:
M = hash_to_curve(Hash("M AND N SPAKE2")) M = hash_to_curve(Hash("M AND N SPAKE2"))
This variant MUST be used when it is not possible to determine which This variant MUST be used when it is not possible to determine
of A and B should use M or N, due to asymmetries in the protocol whether A or B should use M (or N), due to asymmetries in the
flows or the desire to use only a single shared secret with nil protocol flows or the desire to use only a single shared secret with
identities for authentication. The security of these variants is nil identities for authentication. The security of these variants is
examined in [MNVAR]. The variant with per-user M and N may not be examined in [MNVAR]. The variant with per-user M and N may not be
suitable for protocols that require the initial messages to 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 generated by each party at the same time and that do not know the
identity of the parties before the flow begins. exact identity of the parties before the flow begins.
6. Ciphersuites 6. Ciphersuites
This section documents SPAKE2 ciphersuite configurations. A This section documents SPAKE2 ciphersuite configurations. A
ciphersuite indicates a group, cryptographic hash function, and pair ciphersuite indicates a group, cryptographic hash function, and pair
of KDF and MAC functions, e.g., SPAKE2-P256-SHA256-HKDF-HMAC. This of KDF and MAC functions, e.g., SPAKE2-P256-SHA256-HKDF-HMAC. This
ciphersuite indicates a SPAKE2 protocol instance over P-256 that uses ciphersuite indicates a SPAKE2 protocol instance over P-256 that uses
SHA256 along with HKDF [RFC5869] and HMAC [RFC2104] for G, Hash, KDF, SHA-256, along with HMAC-based Key Derivation Function (HKDF)
and MAC functions, respectively. For Ed25519 the compressed encoding [RFC5869] and Hashed Message Authentication Code (HMAC) [RFC2104] for
is used [RFC8032], all others use the uncompressed SEC1 encoding. G, Hash, KDF, and MAC functions, respectively. For Ed25519, the
compressed encoding is used [RFC8032]; all others use the
uncompressed SEC1 encoding.
+==============+==================+================+================+ +==============+==================+================+================+
| G | Hash | KDF | MAC | | G | Hash | KDF | MAC |
+==============+==================+================+================+ +==============+==================+================+================+
| P-256 | SHA256 [RFC6234] | HKDF [RFC5869] | HMAC | | P-256 | SHA256 [RFC6234] | HKDF [RFC5869] | HMAC |
| | | | [RFC2104] | | | | | [RFC2104] |
+--------------+------------------+----------------+----------------+ +--------------+------------------+----------------+----------------+
| P-256 | SHA512 [RFC6234] | HKDF [RFC5869] | HMAC | | P-256 | SHA512 [RFC6234] | HKDF [RFC5869] | HMAC |
| | | | [RFC2104] | | | | | [RFC2104] |
+--------------+------------------+----------------+----------------+ +--------------+------------------+----------------+----------------+
skipping to change at page 8, line 40 skipping to change at line 360
| P-256 | SHA256 [RFC6234] | HKDF [RFC5869] | CMAC-AES-128 | | P-256 | SHA256 [RFC6234] | HKDF [RFC5869] | CMAC-AES-128 |
| | | | [RFC4493] | | | | | [RFC4493] |
+--------------+------------------+----------------+----------------+ +--------------+------------------+----------------+----------------+
| P-256 | SHA512 [RFC6234] | HKDF [RFC5869] | CMAC-AES-128 | | P-256 | SHA512 [RFC6234] | HKDF [RFC5869] | CMAC-AES-128 |
| | | | [RFC4493] | | | | | [RFC4493] |
+--------------+------------------+----------------+----------------+ +--------------+------------------+----------------+----------------+
Table 1: SPAKE2 Ciphersuites Table 1: SPAKE2 Ciphersuites
The following points represent permissible point generation seeds for The following points represent permissible point generation seeds for
the groups listed in the Table Table 1, using the algorithm presented the groups listed in Table 1, using the algorithm presented in
in Appendix A. These bytestrings are compressed points as in [SEC1] Appendix A. These byte strings are compressed points, as in [SEC1],
for curves from [SEC1]. for curves from [SEC1].
For P256: For P-256:
M = M =
02886e2f97ace46e55ba9dd7242579f2993b64e16ef3dcab95afd497333d8fa12f 02886e2f97ace46e55ba9dd7242579f2993b64e16ef3dcab95afd497333d8fa12f
seed: 1.2.840.10045.3.1.7 point generation seed (M) seed: 1.2.840.10045.3.1.7 point generation seed (M)
N = N =
03d8bbd6c639c62937b04d997f38c3770719c629d7014d49a24b4f98baa1292b49 03d8bbd6c639c62937b04d997f38c3770719c629d7014d49a24b4f98baa1292b49
seed: 1.2.840.10045.3.1.7 point generation seed (N) seed: 1.2.840.10045.3.1.7 point generation seed (N)
For P384: For P-384:
M = M =
030ff0895ae5ebf6187080a82d82b42e2765e3b2f8749c7e05eba366434b363d3dc 030ff0895ae5ebf6187080a82d82b42e2765e3b2f8749c7e05eba366434b363d3dc
36f15314739074d2eb8613fceec2853 36f15314739074d2eb8613fceec2853
seed: 1.3.132.0.34 point generation seed (M) seed: 1.3.132.0.34 point generation seed (M)
N = N =
02c72cf2e390853a1c1c4ad816a62fd15824f56078918f43f922ca21518f9c543bb 02c72cf2e390853a1c1c4ad816a62fd15824f56078918f43f922ca21518f9c543bb
252c5490214cf9aa3f0baab4b665c10 252c5490214cf9aa3f0baab4b665c10
seed: 1.3.132.0.34 point generation seed (N) seed: 1.3.132.0.34 point generation seed (N)
For P521: For P-521:
M = M =
02003f06f38131b2ba2600791e82488e8d20ab889af753a41806c5db18d37d85608 02003f06f38131b2ba2600791e82488e8d20ab889af753a41806c5db18d37d85608
cfae06b82e4a72cd744c719193562a653ea1f119eef9356907edc9b56979962d7aa cfae06b82e4a72cd744c719193562a653ea1f119eef9356907edc9b56979962d7aa
seed: 1.3.132.0.35 point generation seed (M) seed: 1.3.132.0.35 point generation seed (M)
N = N =
0200c7924b9ec017f3094562894336a53c50167ba8c5963876880542bc669e494b25 0200c7924b9ec017f3094562894336a53c50167ba8c5963876880542bc669e494b25
32d76c5b53dfb349fdf69154b9e0048c58a42e8ed04cef052a3bc349d95575cd25 32d76c5b53dfb349fdf69154b9e0048c58a42e8ed04cef052a3bc349d95575cd25
seed: 1.3.132.0.35 point generation seed (N) seed: 1.3.132.0.35 point generation seed (N)
skipping to change at page 10, line 18 skipping to change at line 423
seed: edwards448 point generation seed (M) seed: edwards448 point generation seed (M)
N = N =
6034c65b66e4cd7a49b0edec3e3c9ccc4588afd8cf324e29f0a84a072531c4db 6034c65b66e4cd7a49b0edec3e3c9ccc4588afd8cf324e29f0a84a072531c4db
f97ff9af195ed714a689251f08f8e06e2d1f24a0ffc0146600 f97ff9af195ed714a689251f08f8e06e2d1f24a0ffc0146600
seed: edwards448 point generation seed (N) seed: edwards448 point generation seed (N)
7. Security Considerations 7. Security Considerations
A security proof of SPAKE2 for prime order groups is found in [REF], A security proof of SPAKE2 for prime order groups is found in [REF],
reducing the security of SPAKE2 to the gap Diffie-Hellman assumption. reducing the security of SPAKE2 to the GDH assumption. Note that the
Note that the choice of M and N is critical for the security proof. choice of M and N is critical for the security proof. The generation
The generation methods specified in this document are designed to methods specified in this document are designed to eliminate concerns
eliminate concerns related to knowing discrete logs of M and N. related to knowing discrete logs of M and N.
Elements received from a peer MUST be checked for group membership: Elements received from a peer MUST be checked for group membership.
failure to properly deserialize and validate group elements can lead Failure to properly deserialize and validate group elements can lead
to attacks. An endpoint MUST abort the protocol if any received to attacks. An endpoint MUST abort the protocol if any received
public value is not a member of G. public value is not a member of G.
The choices of random numbers MUST BE uniform. Randomly generated The choices of random numbers MUST be uniform. Randomly generated
values, e.g., x and y, MUST NOT be reused; such reuse violates the values, e.g., x and y, MUST NOT be reused; such reuse violates the
security assumptions of the protocol and results in significant security assumptions of the protocol and results in significant
insecurity. It is RECOMMENDED to generate these uniform numbers insecurity. It is RECOMMENDED to generate these uniform numbers
using rejection sampling. using rejection sampling.
Some implementations of elliptic curve multiplication may leak Some implementations of elliptic curve multiplication may leak
information about the length of the scalar. These MUST NOT be used. information about the length of the scalar. These MUST NOT be used.
All operations on elliptic curve points must take time independent of All operations on elliptic curve points must take time independent of
the inputs. Hashing of the transcript may take time depending only the inputs. Hashing of the transcript may take time depending only
on the length of the transcript, but not the contents. on the length of the transcript but not the contents.
SPAKE2 does not support augmentation. As a result, the server has to SPAKE2 does not support augmentation. As a result, the server has to
store a password equivalent. This is considered a significant store a password equivalent. This is considered a significant
drawback in some use cases. Applications that need augmented PAKEs drawback in some use cases. Applications that need augmented PAKEs
should use [I-D.irtf-cfrg-opaque]. should use [CFRG-OPAQUE].
The HMAC keys in this document are shorter than recommended in The HMAC keys in this document are shorter than recommended in
[RFC8032]. This is appropriate as the difficulty of the discrete [RFC8032]. This is appropriate, as the difficulty of the discrete
logarithm problem is comparable with the difficulty of brute forcing logarithm problem is comparable with the difficulty of brute forcing
the keys. the keys.
8. IANA Considerations 8. IANA Considerations
No IANA action is required. This document has no IANA actions.
9. Acknowledgments
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.
10. References
10.1. Normative References 9. References
[I-D.irtf-cfrg-hash-to-curve] 9.1. Normative References
Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R., and
C. Wood, "Hashing to Elliptic Curves", Work in Progress,
Internet-Draft, draft-irtf-cfrg-hash-to-curve-05, 2
November 2019, <http://www.ietf.org/internet-drafts/draft-
irtf-cfrg-hash-to-curve-05.txt>.
[RFC2104] Krawczyk, H., Bellare, M., and R. Canetti, "HMAC: Keyed- [RFC2104] Krawczyk, H., Bellare, M., and R. Canetti, "HMAC: Keyed-
Hashing for Message Authentication", RFC 2104, Hashing for Message Authentication", RFC 2104,
DOI 10.17487/RFC2104, February 1997, DOI 10.17487/RFC2104, February 1997,
<https://www.rfc-editor.org/info/rfc2104>. <https://www.rfc-editor.org/info/rfc2104>.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119, Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997, DOI 10.17487/RFC2119, March 1997,
<https://www.rfc-editor.org/info/rfc2119>. <https://www.rfc-editor.org/info/rfc2119>.
skipping to change at page 12, line 28 skipping to change at line 505
[RFC8032] Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital [RFC8032] Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital
Signature Algorithm (EdDSA)", RFC 8032, Signature Algorithm (EdDSA)", RFC 8032,
DOI 10.17487/RFC8032, January 2017, DOI 10.17487/RFC8032, January 2017,
<https://www.rfc-editor.org/info/rfc8032>. <https://www.rfc-editor.org/info/rfc8032>.
[RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC [RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
May 2017, <https://www.rfc-editor.org/info/rfc8174>. May 2017, <https://www.rfc-editor.org/info/rfc8174>.
10.2. Informative References [RFC9380] Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R. S.,
and C. A. Wood, "Hashing to Elliptic Curves", RFC 9380,
DOI 10.17487/RFC9380, August 2023,
<https://www.rfc-editor.org/info/rfc9380>.
[SEC1] Standards for Efficient Cryptography Group, "SEC 1: 9.2. Informative References
Elliptic Curve Cryptography", May 2009.
[CFRG-OPAQUE]
Bourdrez, D., Krawczyk, H., Lewi, K., and C. A. Wood, "The
OPAQUE Asymmetric PAKE Protocol", Work in Progress,
Internet-Draft, draft-irtf-cfrg-opaque-11, 8 June 2023,
<https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-
opaque-11>.
[MNVAR] Abdalla, M., Barbosa, M., Bradley, T., Jarecki, S., Katz, [MNVAR] Abdalla, M., Barbosa, M., Bradley, T., Jarecki, S., Katz,
J., and J. Xu, "Universally Composable Relaxed Password J., and J. Xu, "Universally Composable Relaxed Password
Authentication", August 2020. Appears in Micciancio D., Authenticated Key Exchange", in Advances in Cryptology -
Ristenpart T. (eds) Advances in Cryptology -CRYPTO 2020. CRYPTO 2020, Lecture Notes in Computer Science, Volume
Crypto 2020. Lecture notes in Computer Science volume 12170, Springer, DOI 10.1007/978-3-030-56784-2_10, August
12170. Springer. 2020, <https://doi.org/10.1007/978-3-030-56784-2_10>.
[REF] Abdalla, M. and D. Pointcheval, "Simple Password-Based [NIST.SP.800-56Ar3]
Encrypted Key Exchange Protocols.", February 2005. National Institute of Standards and Technology,
Appears in A. Menezes, editor. Topics in Cryptography- "Recommendation for Pair-Wise Key-Establishment Schemes
CT-RSA 2005, Volume 3376 of Lecture Notes in Computer Using Discrete Logarithm Cryptography", Revision 3, NIST
Science, pages 191-208, San Francisco, CA, US. Springer- Special Publication 800-56A,
Verlag, Berlin, Germany. DOI 10.6028/NIST.SP.800-56Ar3, April 2018,
<https://doi.org/10.6028/NIST.SP.800-56Ar3>.
[TDH] Cash, D., Kiltz, E., and V. Shoup, "The Twin-Diffie [REF] Abdalla, M. and D. Pointcheval, "Simple Password-Based
Hellman Problem and Applications", 2008. EUROCRYPT 2008. Encrypted Key Exchange Protocols", Cryptography-CT-RSA
Volume 4965 of Lecture notes in Computer Science, pages 2005, Lecture Notes in Computer Science, Volume 3376,
127-145. Springer-Verlag, Berlin, Germany. pages 191-208, Springer, DOI 10.1007/978-3-540-30574-3_14,
February 2005,
<https://doi.org/10.1007/978-3-540-30574-3_14>.
[RFC8265] Saint-Andre, P. and A. Melnikov, "Preparation, [RFC8265] Saint-Andre, P. and A. Melnikov, "Preparation,
Enforcement, and Comparison of Internationalized Strings Enforcement, and Comparison of Internationalized Strings
Representing Usernames and Passwords", RFC 8265, Representing Usernames and Passwords", RFC 8265,
DOI 10.17487/RFC8265, October 2017, DOI 10.17487/RFC8265, October 2017,
<https://www.rfc-editor.org/info/rfc8265>. <https://www.rfc-editor.org/info/rfc8265>.
[I-D.irtf-cfrg-opaque] [SEC1] Standards for Efficient Cryptography Group, "SEC 1:
Krawczyk, H., Bourdrez, D., Lewi, K., and C. A. Wood, "The Elliptic Curve Cryptography", May 2009.
OPAQUE Asymmetric PAKE Protocol", Work in Progress,
Internet-Draft, draft-irtf-cfrg-opaque-06, 12 July 2021,
<https://www.ietf.org/archive/id/draft-irtf-cfrg-opaque-
06.txt>.
Appendix A. Algorithm used for Point Generation Appendix A. Algorithm Used for Point Generation
This section describes the algorithm that was used to generate the This section describes the algorithm that was used to generate points
points M and N in the table in Section 6. M and N in Table 1.
For each curve in the table below, we construct a string using the For each curve in Table 1, we construct a string using the curve OID
curve OID from [RFC5480] (as an ASCII string) or its name, combined from [RFC5480] (as an ASCII string) or its name, combined with the
with the needed constant, e.g., "1.3.132.0.35 point generation seed needed constant, e.g., "1.3.132.0.35 point generation seed (M)" for
(M)" for P-521. This string is turned into a series of blocks by P-521. This string is turned into a series of blocks by hashing with
hashing with SHA256, and hashing that output again to generate the SHA-256 and hashing that output again to generate the next 32 bytes
next 32 bytes, and so on. This pattern is repeated for each group and so on. This pattern is repeated for each group and value, with
and value, with the string modified appropriately. the string modified appropriately.
A byte string of length equal to that of an encoded group element is A byte string of a length equal to that of an encoded group element
constructed by concatenating as many blocks as are required, starting is constructed by concatenating as many blocks as are required,
from the first block, and truncating to the desired length. The byte starting from the first block and truncating to the desired length.
string is then formatted as required for the group. In the case of The byte string is then formatted as required for the group. In the
Weierstrass curves, we take the desired length as the length for case of Weierstrass curves, we take the desired length as the length
representing a compressed point (section 2.3.4 of [SEC1]), and use 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 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 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), 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 leaving the sign bit as it was in the byte string constructed by
concatenating hash blocks. For the [RFC8032] curves a different concatenating hash blocks. For the curves in [RFC8032], a different
procedure is used. For edwards448 the 57-byte input has the least- 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 significant 7 bits of the last byte set to zero, and for
the 32-byte input is not modified. For both the [RFC8032] curves the edwards25519, the 32-byte input is not modified. For both the curves
(modified) input is then interpreted as the representation of the in [RFC8032], the (modified) input is then interpreted as the
group element. If this interpretation yields a valid group element representation of the group element. If this interpretation yields a
with the correct order (p), the (modified) byte string is the output. valid group element with the correct order (p), the (modified) byte
Otherwise, the initial hash block is discarded and a new byte string string is the output. Otherwise, the initial hash block is discarded
constructed from the remaining hash blocks. The procedure of and a new byte string is constructed from the remaining hash blocks.
constructing a byte string of the appropriate length, formatting it The procedure of constructing a byte string of the appropriate
as required for the curve, and checking if it is a valid point of the length, formatting it as required for the curve, and checking if it
correct order, is repeated until a valid element is found. 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 The following Python snippet generates the above points, assuming an
elliptic curve implementation following the interface of elliptic curve implementation follows the interface of
Edwards25519Point.stdbase() and Edwards448Point.stdbase() in Edwards25519Point.stdbase() and Edwards448Point.stdbase() in
Appendix A of [RFC8032]: Appendix A of [RFC8032]:
def iterated_hash(seed, n): def iterated_hash(seed, n):
h = seed h = seed
for i in range(n): for i in range(n):
h = hashlib.sha256(h).digest() h = hashlib.sha256(h).digest()
return h return h
def bighash(seed, start, sz): def bighash(seed, start, sz):
n = -(-sz // 32) n = -(-sz // 32)
hashes = [iterated_hash(seed, i) for i in range(start, start + n)] hashes = [iterated_hash(seed, i)
return b''.join(hashes)[:sz] for i in range(start, start + n)]
return b''.join(hashes)[:sz]
def canon_pointstr(ecname, s): def canon_pointstr(ecname, s):
if ecname == 'edwards25519': if ecname == 'edwards25519':
return s return s
elif ecname == 'edwards448': elif ecname == 'edwards448':
return s[:-1] + bytes([s[-1] & 0x80]) return s[:-1] + bytes([s[-1] & 0x80])
else: else:
return bytes([(s[0] & 1) | 2]) + s[1:] return bytes([(s[0] & 1) | 2]) + s[1:]
def gen_point(seed, ecname, ec): def gen_point(seed, ecname, ec):
for i in range(1, 1000): for i in range(1, 1000):
hval = bighash(seed, i, len(ec.encode())) hval = bighash(seed, i, len(ec.encode()))
pointstr = canon_pointstr(ecname, hval) pointstr = canon_pointstr(ecname, hval)
try: try:
p = ec.decode(pointstr) p = ec.decode(pointstr)
if p != ec.zero_elem() and p * p.l() == ec.zero_elem(): if p != ec.zero_elem() and p * p.l() == ec.zero_elem():
return pointstr, i return pointstr, i
except Exception: except Exception:
pass pass
Appendix B. Test Vectors Appendix B. SPAKE2 Test Vectors
This section contains test vectors for SPAKE2 using the P256-SHA256- This section contains test vectors for SPAKE2, using the P256-SHA256-
HKDF-HMAC ciphersuite. (Choice of MHF is omitted and values for w, HKDF-HMAC ciphersuite. (The choice of MHF is omitted, and the values
x, and y are provided directly.) All points are encoded using the for w, x, and y are provided directly.) All points are encoded using
uncompressed format, i.e., with a 0x04 octet prefix, specified in the uncompressed format, i.e., with a 0x04 octet prefix, specified in
[SEC1] A and B identity strings are provided in the protocol [SEC1]. A and B identity strings are provided in the protocol
invocation. invocation.
B.1. SPAKE2 Test Vectors Line breaks have been added due to line-length limitations.
spake2: A='server', B='client'
w = 0x2ee57912099d31560b3a44b1184b9b4866e904c49d12ac5042c97dca461b1a5f
x = 0x43dd0fd7215bdcb482879fca3220c6a968e66d70b1356cac18bb26c84a78d729
pA = 0x04a56fa807caaa53a4d28dbb9853b9815c61a411118a6fe516a8798434751470
f9010153ac33d0d5f2047ffdb1a3e42c9b4e6be662766e1eeb4116988ede5f912c
y = 0xdcb60106f276b02606d8ef0a328c02e4b629f84f89786af5befb0bc75b6e66be
pB = 0x0406557e482bd03097ad0cbaa5df82115460d951e3451962f1eaf4367a420676
d09857ccbc522686c83d1852abfa8ed6e4a1155cf8f1543ceca528afb591a1e0b7
K = 0x0412af7e89717850671913e6b469ace67bd90a4df8ce45c2af19010175e37eed
69f75897996d539356e2fa6a406d528501f907e04d97515fbe83db277b715d3325
TT = 0x06000000000000007365727665720600000000000000636c69656e744100000
00000000004a56fa807caaa53a4d28dbb9853b9815c61a411118a6fe516a8798434751
470f9010153ac33d0d5f2047ffdb1a3e42c9b4e6be662766e1eeb4116988ede5f912c4
1000000000000000406557e482bd03097ad0cbaa5df82115460d951e3451962f1eaf43
67a420676d09857ccbc522686c83d1852abfa8ed6e4a1155cf8f1543ceca528afb591a
1e0b741000000000000000412af7e89717850671913e6b469ace67bd90a4df8ce45c2a
f19010175e37eed69f75897996d539356e2fa6a406d528501f907e04d97515fbe83db2
77b715d332520000000000000002ee57912099d31560b3a44b1184b9b4866e904c49d1
2ac5042c97dca461b1a5f
Hash(TT) = 0x0e0672dc86f8e45565d338b0540abe6915bdf72e2b35b5c9e5663168e960a91b
Ke = 0x0e0672dc86f8e45565d338b0540abe69
Ka = 0x15bdf72e2b35b5c9e5663168e960a91b
KcA = 0x00c12546835755c86d8c0db7851ae86f
KcB = 0xa9fa3406c3b781b93d804485430ca27a
A conf = 0x58ad4aa88e0b60d5061eb6b5dd93e80d9c4f00d127c65b3b35b1b5281fee38f0
B conf = 0xd3e2e547f1ae04f2dbdbf0fc4b79f8ecff2dff314b5d32fe9fcef2fb26dc459b
spake2: A='', B='client' spake2: A='server', B='client'
w = 0x0548d8729f730589e579b0475a582c1608138ddf7054b73b5381c7e883e2efae w = 0x2ee57912099d31560b3a44b1184b9b4866e904c49d12ac5042c97dca461b1a5f
x = 0x403abbe3b1b4b9ba17e3032849759d723939a27a27b9d921c500edde18ed654b x = 0x43dd0fd7215bdcb482879fca3220c6a968e66d70b1356cac18bb26c84a78d729
pA = 0x04a897b769e681c62ac1c2357319a3d363f610839c4477720d24cbe32f5fd85f pA = 0x04a56fa807caaa53a4d28dbb9853b9815c61a411118a6fe516a8798434751470
44fb92ba966578c1b712be6962498834078262caa5b441ecfa9d4a9485720e918a f9010153ac33d0d5f2047ffdb1a3e42c9b4e6be662766e1eeb4116988ede5f912c
y = 0x903023b6598908936ea7c929bd761af6039577a9c3f9581064187c3049d87065 y = 0xdcb60106f276b02606d8ef0a328c02e4b629f84f89786af5befb0bc75b6e66be
pB = 0x04e0f816fd1c35e22065d5556215c097e799390d16661c386e0ecc84593974a6 pB = 0x0406557e482bd03097ad0cbaa5df82115460d951e3451962f1eaf4367a420676
1b881a8c82327687d0501862970c64565560cb5671f696048050ca66ca5f8cc7fc d09857ccbc522686c83d1852abfa8ed6e4a1155cf8f1543ceca528afb591a1e0b7
K = 0x048f83ec9f6e4f87cc6f9dc740bdc2769725f923364f01c84148c049a39a735e K = 0x0412af7e89717850671913e6b469ace67bd90a4df8ce45c2af19010175e37eed
bda82eac03e00112fd6a5710682767cff5361f7e819e53d8d3c3a2922e0d837aa6 69f75897996d539356e2fa6a406d528501f907e04d97515fbe83db277b715d3325
TT = 0x00000000000000000600000000000000636c69656e74410000000000000004a TT = 0x06000000000000007365727665720600000000000000636c69656e744100000
897b769e681c62ac1c2357319a3d363f610839c4477720d24cbe32f5fd85f44fb92ba9 00000000004a56fa807caaa53a4d28dbb9853b9815c61a411118a6fe516a8798434751
66578c1b712be6962498834078262caa5b441ecfa9d4a9485720e918a4100000000000 470f9010153ac33d0d5f2047ffdb1a3e42c9b4e6be662766e1eeb4116988ede5f912c4
00004e0f816fd1c35e22065d5556215c097e799390d16661c386e0ecc84593974a61b8 1000000000000000406557e482bd03097ad0cbaa5df82115460d951e3451962f1eaf43
81a8c82327687d0501862970c64565560cb5671f696048050ca66ca5f8cc7fc4100000 67a420676d09857ccbc522686c83d1852abfa8ed6e4a1155cf8f1543ceca528afb591a
000000000048f83ec9f6e4f87cc6f9dc740bdc2769725f923364f01c84148c049a39a7 1e0b741000000000000000412af7e89717850671913e6b469ace67bd90a4df8ce45c2a
35ebda82eac03e00112fd6a5710682767cff5361f7e819e53d8d3c3a2922e0d837aa62 f19010175e37eed69f75897996d539356e2fa6a406d528501f907e04d97515fbe83db2
0000000000000000548d8729f730589e579b0475a582c1608138ddf7054b73b5381c7e 77b715d332520000000000000002ee57912099d31560b3a44b1184b9b4866e904c49d1
883e2efae 2ac5042c97dca461b1a5f
Hash(TT) = 0x642f05c473c2cd79909f9a841e2f30a70bf89b18180af97353ba198789c2b963 HASH(TT) = 0x0e0672dc86f8e45565d338b0540abe6915bdf72e2b35b5c9e5663168e9
Ke = 0x642f05c473c2cd79909f9a841e2f30a7 60a91b
Ka = 0x0bf89b18180af97353ba198789c2b963 Ke = 0x0e0672dc86f8e45565d338b0540abe69
KcA = 0xc6be376fc7cd1301fd0a13adf3e7bffd Ka = 0x15bdf72e2b35b5c9e5663168e960a91b
KcB = 0xb7243f4ae60440a49b3f8cab3c1fba07 KcA = 0x00c12546835755c86d8c0db7851ae86f
A conf = 0x47d29e6666af1b7dd450d571233085d7a9866e4d49d2645e2df975489521232b KcB = 0xa9fa3406c3b781b93d804485430ca27a
B conf = 0x3313c5cefc361d27fb16847a91c2a73b766ffa90a4839122a9b70a2f6bd1d6df A conf = 0x58ad4aa88e0b60d5061eb6b5dd93e80d9c4f00d127c65b3b35b1b5281f
ee38f0
B conf = 0xd3e2e547f1ae04f2dbdbf0fc4b79f8ecff2dff314b5d32fe9fcef2fb26
dc459b
spake2: A='server', B='' spake2: A='', B='client'
w = 0x626e0cdc7b14c9db3e52a0b1b3a768c98e37852d5db30febe0497b14eae8c254 w = 0x0548d8729f730589e579b0475a582c1608138ddf7054b73b5381c7e883e2efae
x = 0x07adb3db6bc623d3399726bfdbfd3d15a58ea776ab8a308b00392621291f9633 x = 0x403abbe3b1b4b9ba17e3032849759d723939a27a27b9d921c500edde18ed654b
pA = 0x04f88fb71c99bfffaea370966b7eb99cd4be0ff1a7d335caac4211c4afd855e2 pA = 0x04a897b769e681c62ac1c2357319a3d363f610839c4477720d24cbe32f5fd8
e15a873b298503ad8ba1d9cbb9a392d2ba309b48bfd7879aefd0f2cea6009763b0 5f44fb92ba966578c1b712be6962498834078262caa5b441ecfa9d4a9485720e918a
y = 0xb6a4fc8dbb629d4ba51d6f91ed1532cf87adec98f25dd153a75accafafedec16 y = 0x903023b6598908936ea7c929bd761af6039577a9c3f9581064187c3049d87065
pB = 0x040c269d6be017dccb15182ac6bfcd9e2a14de019dd587eaf4bdfd353f031101 pB = 0x04e0f816fd1c35e22065d5556215c097e799390d16661c386e0ecc84593974
e7cca177f8eb362a6e83e7d5e729c0732e1b528879c086f39ba0f31a9661bd34db a61b881a8c82327687d0501862970c64565560cb5671f696048050ca66ca5f8cc7fc
K = 0x0445ee233b8ecb51ebd6e7da3f307e88a1616bae2166121221fdc0dadb986afa K = 0x048f83ec9f6e4f87cc6f9dc740bdc2769725f923364f01c84148c049a39a735e
f3ec8a988dc9c626fa3b99f58a7ca7c9b844bb3e8dd9554aafc5b53813504c1cbe bda82eac03e00112fd6a5710682767cff5361f7e819e53d8d3c3a2922e0d837aa6
TT = 0x06000000000000007365727665720000000000000000410000000000000004f TT = 0x00000000000000000600000000000000636c69656e74410000000000000004a
88fb71c99bfffaea370966b7eb99cd4be0ff1a7d335caac4211c4afd855e2e15a873b2 897b769e681c62ac1c2357319a3d363f610839c4477720d24cbe32f5fd85f44fb92ba9
98503ad8ba1d9cbb9a392d2ba309b48bfd7879aefd0f2cea6009763b04100000000000 66578c1b712be6962498834078262caa5b441ecfa9d4a9485720e918a4100000000000
000040c269d6be017dccb15182ac6bfcd9e2a14de019dd587eaf4bdfd353f031101e7c 00004e0f816fd1c35e22065d5556215c097e799390d16661c386e0ecc84593974a61b8
ca177f8eb362a6e83e7d5e729c0732e1b528879c086f39ba0f31a9661bd34db4100000 81a8c82327687d0501862970c64565560cb5671f696048050ca66ca5f8cc7fc4100000
0000000000445ee233b8ecb51ebd6e7da3f307e88a1616bae2166121221fdc0dadb986 000000000048f83ec9f6e4f87cc6f9dc740bdc2769725f923364f01c84148c049a39a7
afaf3ec8a988dc9c626fa3b99f58a7ca7c9b844bb3e8dd9554aafc5b53813504c1cbe2 35ebda82eac03e00112fd6a5710682767cff5361f7e819e53d8d3c3a2922e0d837aa62
000000000000000626e0cdc7b14c9db3e52a0b1b3a768c98e37852d5db30febe0497b1 0000000000000000548d8729f730589e579b0475a582c1608138ddf7054b73b5381c7e
4eae8c254 883e2efae
Hash(TT) = 0x005184ff460da2ce59062c87733c299c3521297d736598fc0a1127600efa1afb Hash(TT) = 0x642f05c473c2cd79909f9a841e2f30a70bf89b18180af97353ba198789
Ke = 0x005184ff460da2ce59062c87733c299c c2b963
Ka = 0x3521297d736598fc0a1127600efa1afb Ke = 0x642f05c473c2cd79909f9a841e2f30a7
KcA = 0xf3da53604f0aeecea5a33be7bddf6edf Ka = 0x0bf89b18180af97353ba198789c2b963
KcB = 0x9e3f86848736f159bd92b6e107ec6799 KcA = 0xc6be376fc7cd1301fd0a13adf3e7bffd
A conf = 0xbc9f9bbe99f26d0b2260e6456e05a86196a3307ec6663a18bf6ac825736533b2 KcB = 0xb7243f4ae60440a49b3f8cab3c1fba07
B conf = 0xc2370e1bf813b086dff0d834e74425a06e6390f48f5411900276dcccc5a297ec A conf = 0x47d29e6666af1b7dd450d571233085d7a9866e4d49d2645e2df9754895
21232b
B conf = 0x3313c5cefc361d27fb16847a91c2a73b766ffa90a4839122a9b70a2f6b
d1d6df
spake2: A='', B='' spake2: A='server', B=''
w = 0x7bf46c454b4c1b25799527d896508afd5fc62ef4ec59db1efb49113063d70cca w = 0x626e0cdc7b14c9db3e52a0b1b3a768c98e37852d5db30febe0497b14eae8c254
x = 0x8cef65df64bb2d0f83540c53632de911b5b24b3eab6cc74a97609fd659e95473 x = 0x07adb3db6bc623d3399726bfdbfd3d15a58ea776ab8a308b00392621291f9633
pA = 0x04a65b367a3f613cf9f0654b1b28a1e3a8a40387956c8ba6063e8658563890f4 pA = 0x04f88fb71c99bfffaea370966b7eb99cd4be0ff1a7d335caac4211c4afd855e2
6ca1ef6a676598889fc28de2950ab8120b79a5ef1ea4c9f44bc98f585634b46d66 e15a873b298503ad8ba1d9cbb9a392d2ba309b48bfd7879aefd0f2cea6009763b0
y = 0xd7a66f64074a84652d8d623a92e20c9675c61cb5b4f6a0063e4648a2fdc02d53 y = 0xb6a4fc8dbb629d4ba51d6f91ed1532cf87adec98f25dd153a75accafafedec16
pB = 0x04589f13218822710d98d8b2123a079041052d9941b9cf88c6617ddb2fcc0494 pB = 0x040c269d6be017dccb15182ac6bfcd9e2a14de019dd587eaf4bdfd353f031101
662eea8ba6b64692dc318250030c6af045cb738bc81ba35b043c3dcb46adf6f58d e7cca177f8eb362a6e83e7d5e729c0732e1b528879c086f39ba0f31a9661bd34db
K = 0x041a3c03d51b452537ca2a1fea6110353c6d5ed483c4f0f86f4492ca3f378d40 K = 0x0445ee233b8ecb51ebd6e7da3f307e88a1616bae2166121221fdc0dadb986afa
a994b4477f93c64d928edbbcd3e85a7c709b7ea73ee97986ce3d1438e135543772 f3ec8a988dc9c626fa3b99f58a7ca7c9b844bb3e8dd9554aafc5b53813504c1cbe
TT = 0x00000000000000000000000000000000410000000000000004a65b367a3f613 TT = 0x06000000000000007365727665720000000000000000410000000000000004f
cf9f0654b1b28a1e3a8a40387956c8ba6063e8658563890f46ca1ef6a676598889fc28 88fb71c99bfffaea370966b7eb99cd4be0ff1a7d335caac4211c4afd855e2e15a873b2
de2950ab8120b79a5ef1ea4c9f44bc98f585634b46d66410000000000000004589f132 98503ad8ba1d9cbb9a392d2ba309b48bfd7879aefd0f2cea6009763b04100000000000
18822710d98d8b2123a079041052d9941b9cf88c6617ddb2fcc0494662eea8ba6b6469 000040c269d6be017dccb15182ac6bfcd9e2a14de019dd587eaf4bdfd353f031101e7c
2dc318250030c6af045cb738bc81ba35b043c3dcb46adf6f58d4100000000000000041 ca177f8eb362a6e83e7d5e729c0732e1b528879c086f39ba0f31a9661bd34db4100000
a3c03d51b452537ca2a1fea6110353c6d5ed483c4f0f86f4492ca3f378d40a994b4477 0000000000445ee233b8ecb51ebd6e7da3f307e88a1616bae2166121221fdc0dadb986
f93c64d928edbbcd3e85a7c709b7ea73ee97986ce3d1438e1355437722000000000000 afaf3ec8a988dc9c626fa3b99f58a7ca7c9b844bb3e8dd9554aafc5b53813504c1cbe2
0007bf46c454b4c1b25799527d896508afd5fc62ef4ec59db1efb49113063d70cca 000000000000000626e0cdc7b14c9db3e52a0b1b3a768c98e37852d5db30febe0497b1
Hash(TT) = 0xfc6374762ba5cf11f4b2caa08b2cd1b9907ae0e26e8d6234318d91583cd74c86 4eae8c254
Ke = 0xfc6374762ba5cf11f4b2caa08b2cd1b9 Hash(TT) = 0x005184ff460da2ce59062c87733c299c3521297d736598fc0a1127600e
Ka = 0x907ae0e26e8d6234318d91583cd74c86 fa1afb
KcA = 0x5dbd2f477166b7fb6d61febbd77a5563 Ke = 0x005184ff460da2ce59062c87733c299c
KcB = 0x7689b4654407a5faeffdc8f18359d8a3 Ka = 0x3521297d736598fc0a1127600efa1afb
A conf = 0xdfb4db8d48ae5a675963ea5e6c19d98d4ea028d8e898dad96ea19a80ade95dca KcA = 0xf3da53604f0aeecea5a33be7bddf6edf
B conf = 0xd0f0609d1613138d354f7e95f19fb556bf52d751947241e8c7118df5ef0ae175 KcB = 0x9e3f86848736f159bd92b6e107ec6799
A conf = 0xbc9f9bbe99f26d0b2260e6456e05a86196a3307ec6663a18bf6ac8257365
33b2
B conf = 0xc2370e1bf813b086dff0d834e74425a06e6390f48f5411900276dcccc5a2
97ec
Authors' Addresses spake2: A='', B=''
w = 0x7bf46c454b4c1b25799527d896508afd5fc62ef4ec59db1efb49113063d70cca
x = 0x8cef65df64bb2d0f83540c53632de911b5b24b3eab6cc74a97609fd659e95473
pA = 0x04a65b367a3f613cf9f0654b1b28a1e3a8a40387956c8ba6063e8658563890f4
6ca1ef6a676598889fc28de2950ab8120b79a5ef1ea4c9f44bc98f585634b46d66
y = 0xd7a66f64074a84652d8d623a92e20c9675c61cb5b4f6a0063e4648a2fdc02d53
pB = 0x04589f13218822710d98d8b2123a079041052d9941b9cf88c6617ddb2fcc0494
662eea8ba6b64692dc318250030c6af045cb738bc81ba35b043c3dcb46adf6f58d
K = 0x041a3c03d51b452537ca2a1fea6110353c6d5ed483c4f0f86f4492ca3f378d40
a994b4477f93c64d928edbbcd3e85a7c709b7ea73ee97986ce3d1438e135543772
TT = 0x00000000000000000000000000000000410000000000000004a65b367a3f613
cf9f0654b1b28a1e3a8a40387956c8ba6063e8658563890f46ca1ef6a676598889fc28
de2950ab8120b79a5ef1ea4c9f44bc98f585634b46d66410000000000000004589f132
18822710d98d8b2123a079041052d9941b9cf88c6617ddb2fcc0494662eea8ba6b6469
2dc318250030c6af045cb738bc81ba35b043c3dcb46adf6f58d4100000000000000041
a3c03d51b452537ca2a1fea6110353c6d5ed483c4f0f86f4492ca3f378d40a994b4477
f93c64d928edbbcd3e85a7c709b7ea73ee97986ce3d1438e1355437722000000000000
0007bf46c454b4c1b25799527d896508afd5fc62ef4ec59db1efb49113063d70cca
Hash(TT) = 0xfc6374762ba5cf11f4b2caa08b2cd1b9907ae0e26e8d6234318d91583c
d74c86
Ke = 0xfc6374762ba5cf11f4b2caa08b2cd1b9
Ka = 0x907ae0e26e8d6234318d91583cd74c86
KcA = 0x5dbd2f477166b7fb6d61febbd77a5563
KcB = 0x7689b4654407a5faeffdc8f18359d8a3
A conf = 0xdfb4db8d48ae5a675963ea5e6c19d98d4ea028d8e898dad96ea19a80ade9
5dca
B conf = 0xd0f0609d1613138d354f7e95f19fb556bf52d751947241e8c7118df5ef0a
e175
Watson Ladd Acknowledgements
Sealance
Email: watsonbladd@gmail.com Special thanks to Nathaniel McCallum and Greg Hudson for generating M
and N and Chris Wood for generating 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.
Benjamin Kaduk (editor) Contributors
Akamai Technologies
Benjamin Kaduk
Akamai Technologies
Email: kaduk@mit.edu Email: kaduk@mit.edu
Author's Address
Watson Ladd
Akamai Technologies
Email: watsonbladd@gmail.com
 End of changes. 84 change blocks. 
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