SPAKE2, a PAKE
draft-irtf-cfrg-spake2-23
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Watson Ladd
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Benjamin Kaduk
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2021-10-04
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draft-ladd-spake2
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Internet Research Task Force W. Ladd
Internet-Draft Cloudflare
Intended status: Informational B. Kaduk, Ed.
Expires: 30 March 2022 Akamai
September 2021
SPAKE2, a PAKE
draft-irtf-cfrg-spake2-23
Abstract
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.
This document is a product of the Crypto Forum Research Group (CFRG)
in the IRTF.
Status of This Memo
This Internet-Draft is submitted in full conformance with the
provisions of BCP 78 and BCP 79.
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
and may be updated, replaced, or obsoleted by other documents at any
time. It is inappropriate to use Internet-Drafts as reference
material or to cite them other than as "work in progress."
This Internet-Draft will expire on 5 March 2022.
Copyright Notice
Copyright (c) 2021 IETF Trust and the persons identified as the
document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal
Provisions Relating to IETF Documents (https://trustee.ietf.org/
license-info) in effect on the date of publication of this document.
Please review these documents carefully, as they describe your rights
and restrictions with respect to this document.
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2
2. Requirements Notation . . . . . . . . . . . . . . . . . . . . 3
3. Definition of SPAKE2 . . . . . . . . . . . . . . . . . . . . 3
4. Key Schedule and Key Confirmation . . . . . . . . . . . . . . 6
5. Per-User M and N and M=N . . . . . . . . . . . . . . . . . . 7
6. Ciphersuites . . . . . . . . . . . . . . . . . . . . . . . . 8
7. Security Considerations . . . . . . . . . . . . . . . . . . . 10
8. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 10
9. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 11
10. References . . . . . . . . . . . . . . . . . . . . . . . . . 11
10.1. Normative References . . . . . . . . . . . . . . . . . . 11
10.2. Informative References . . . . . . . . . . . . . . . . . 12
Appendix A. Algorithm used for Point Generation . . . . . . . . 13
Appendix B. Test Vectors . . . . . . . . . . . . . . . . . . . . 14
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 17
1. Introduction
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.
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2. Requirements Notation
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
"OPTIONAL" in this document are to be interpreted as described in BCP
14 [RFC2119] [RFC8174] when, and only when, they appear in all
capitals, as shown here.
3. Definition of SPAKE2
3.1. Setup
Let G be a group in which the 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 [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 [I-D.irtf-cfrg-hash-to-curve] 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 [I-D.irtf-cfrg-hash-to-curve], 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
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
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least 32 bytes long. Common choices for H are SHA256 or SHA512
[RFC6234]. Let MHF be a memory-hard hash function designed to slow
down brute-force attackers. Scrypt [RFC7914] is a common example of
this function. The output length of MHF matches that of Hash.
Parameter selection for MHF is out of scope for this document.
Section 6 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 [RFC8265]. The hashing algorithm SHOULD be a MHF so as to
slow down brute-force attackers.
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3.2. Protocol Flow
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 described
later MUST be used. 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. (Section 4 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.
A B
| (setup protocol) |
| |
(compute pA) | pA |
|----------------->|
| pB | (compute pB)
|<-----------------|
| |
| (derive secrets) |
| |
(compute cA) | cA |
|----------------->|
| cB | (compute cB)
| | (check cA)
|<-----------------|
(check cB) | |
3.3. SPAKE2
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.
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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:
TT = len(A) || A
|| len(B) || B
|| len(pA) || pA
|| len(pB) || pB
|| len(K) || K
|| len(w) || w
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 Section 4. 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.
4. Key Schedule and Key Confirmation
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
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.
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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.
TT -> Hash(TT) = Ke || Ka
AAD -> KDF(Ka, nil, "ConfirmationKeys" || AAD) = KcA || KcB
A and B output Ke as the shared secret from the protocol. Ka and its
derived keys are not used for anything except key confirmation.
5. Per-User M and N and M=N
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 [I-D.irtf-cfrg-hash-to-curve] as follows:
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))
In addition M and N may be equal to have a symmetric variant. For
this variant we set
M = hash_to_curve(Hash("M AND N SPAKE2"))
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 [MNVAR]. 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.
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6. Ciphersuites
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 [RFC5869] and HMAC [RFC2104] for G, Hash, KDF,
and MAC functions, respectively.
+==============+==================+================+================+
| G | Hash | KDF | MAC |
+==============+==================+================+================+
| P-256 | SHA256 [RFC6234] | HKDF [RFC5869] | HMAC |
| | | | [RFC2104] |
+--------------+------------------+----------------+----------------+
| P-256 | SHA512 [RFC6234] | HKDF [RFC5869] | HMAC |
| | | | [RFC2104] |
+--------------+------------------+----------------+----------------+
| P-384 | SHA256 [RFC6234] | HKDF [RFC5869] | HMAC |
| | | | [RFC2104] |
+--------------+------------------+----------------+----------------+
| P-384 | SHA512 [RFC6234] | HKDF [RFC5869] | HMAC |
| | | | [RFC2104] |
+--------------+------------------+----------------+----------------+
| P-512 | SHA512 [RFC6234] | HKDF [RFC5869] | HMAC |
| | | | [RFC2104] |
+--------------+------------------+----------------+----------------+
| edwards25519 | SHA256 [RFC6234] | HKDF [RFC5869] | HMAC |
| [RFC7748] | | | [RFC2104] |
+--------------+------------------+----------------+----------------+
| edwards448 | SHA512 [RFC6234] | HKDF [RFC5869] | HMAC |
| [RFC7748] | | | [RFC2104] |
+--------------+------------------+----------------+----------------+
| P-256 | SHA256 [RFC6234] | HKDF [RFC5869] | CMAC-AES-128 |
| | | | [RFC4493] |
+--------------+------------------+----------------+----------------+
| P-256 | SHA512 [RFC6234] | HKDF [RFC5869] | CMAC-AES-128 |
| | | | [RFC4493] |
+--------------+------------------+----------------+----------------+
Table 1: SPAKE2 Ciphersuites
The following points represent permissible point generation seeds for
the groups listed in the Table Table 1, using the algorithm presented
in Appendix A. These bytestrings are compressed points as in [SEC1]
for curves from [SEC1].
For P256:
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M =
02886e2f97ace46e55ba9dd7242579f2993b64e16ef3dcab95afd497333d8fa12f
seed: 1.2.840.10045.3.1.7 point generation seed (M)
N =
03d8bbd6c639c62937b04d997f38c3770719c629d7014d49a24b4f98baa1292b49
seed: 1.2.840.10045.3.1.7 point generation seed (N)
For P384:
M =
030ff0895ae5ebf6187080a82d82b42e2765e3b2f8749c7e05eba366434b363d3dc
36f15314739074d2eb8613fceec2853
seed: 1.3.132.0.34 point generation seed (M)
N =
02c72cf2e390853a1c1c4ad816a62fd15824f56078918f43f922ca21518f9c543bb
252c5490214cf9aa3f0baab4b665c10
seed: 1.3.132.0.34 point generation seed (N)
For P521:
M =
02003f06f38131b2ba2600791e82488e8d20ab889af753a41806c5db18d37d85608
cfae06b82e4a72cd744c719193562a653ea1f119eef9356907edc9b56979962d7aa
seed: 1.3.132.0.35 point generation seed (M)
N =
0200c7924b9ec017f3094562894336a53c50167ba8c5963876880542bc669e494b25
32d76c5b53dfb349fdf69154b9e0048c58a42e8ed04cef052a3bc349d95575cd25
seed: 1.3.132.0.35 point generation seed (N)
For edwards25519:
M =
d048032c6ea0b6d697ddc2e86bda85a33adac920f1bf18e1b0c6d166a5cecdaf
seed: edwards25519 point generation seed (M)
N =
d3bfb518f44f3430f29d0c92af503865a1ed3281dc69b35dd868ba85f886c4ab
seed: edwards25519 point generation seed (N)
For edwards448:
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M =
b6221038a775ecd007a4e4dde39fd76ae91d3cf0cc92be8f0c2fa6d6b66f9a12
942f5a92646109152292464f3e63d354701c7848d9fc3b8880
seed: edwards448 point generation seed (M)
N =
6034c65b66e4cd7a49b0edec3e3c9ccc4588afd8cf324e29f0a84a072531c4db
f97ff9af195ed714a689251f08f8e06e2d1f24a0ffc0146600
seed: edwards448 point generation seed (N)
7. Security Considerations
A security proof of SPAKE2 for prime order groups is found in [REF],
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 may permit
dictionary attacks on the password. 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 [I-D.irtf-cfrg-opaque].
The HMAC keys in this document are shorter than recommended in
[RFC8032]. This is appropriate as the difficulty of the discrete
logarithm problem is comparable with the difficulty of brute forcing
the keys.
8. IANA Considerations
No IANA action is required.
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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
[I-D.irtf-cfrg-hash-to-curve]
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-
Hashing for Message Authentication", RFC 2104,
DOI 10.17487/RFC2104, February 1997,
<https://www.rfc-editor.org/info/rfc2104>.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
<https://www.rfc-editor.org/info/rfc2119>.
[RFC4493] Song, JH., Poovendran, R., Lee, J., and T. Iwata, "The
AES-CMAC Algorithm", RFC 4493, DOI 10.17487/RFC4493, June
2006, <https://www.rfc-editor.org/info/rfc4493>.
[RFC5480] Turner, S., Brown, D., Yiu, K., Housley, R., and T. Polk,
"Elliptic Curve Cryptography Subject Public Key
Information", RFC 5480, DOI 10.17487/RFC5480, March 2009,
<https://www.rfc-editor.org/info/rfc5480>.
[RFC5869] Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand
Key Derivation Function (HKDF)", RFC 5869,
DOI 10.17487/RFC5869, May 2010,
<https://www.rfc-editor.org/info/rfc5869>.
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[RFC6234] Eastlake 3rd, D. and T. Hansen, "US Secure Hash Algorithms
(SHA and SHA-based HMAC and HKDF)", RFC 6234,
DOI 10.17487/RFC6234, May 2011,
<https://www.rfc-editor.org/info/rfc6234>.
[RFC7748] Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves
for Security", RFC 7748, DOI 10.17487/RFC7748, January
2016, <https://www.rfc-editor.org/info/rfc7748>.
[RFC7914] Percival, C. and S. Josefsson, "The scrypt Password-Based
Key Derivation Function", RFC 7914, DOI 10.17487/RFC7914,
August 2016, <https://www.rfc-editor.org/info/rfc7914>.
[RFC8032] Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital
Signature Algorithm (EdDSA)", RFC 8032,
DOI 10.17487/RFC8032, January 2017,
<https://www.rfc-editor.org/info/rfc8032>.
[RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
May 2017, <https://www.rfc-editor.org/info/rfc8174>.
10.2. Informative References
[SEC1] Standards for Efficient Cryptography Group, "SEC 1:
Elliptic Curve Cryptography", May 2009.
[MNVAR] Abdalla, M., Barbosa, M., Bradley, T., Jarecki, S., Katz,
J., and J. Xu, "Universally Composable Relaxed Password
Authentication", August 2020. Appears in Micciancio D.,
Ristenpart T. (eds) Advances in Cryptology -CRYPTO 2020.
Crypto 2020. Lecture notes in Computer Science volume
12170. Springer.
[REF] Abdalla, M. and D. Pointcheval, "Simple Password-Based
Encrypted Key Exchange Protocols.", February 2005.
Appears in A. Menezes, editor. Topics in Cryptography-
CT-RSA 2005, Volume 3376 of Lecture Notes in Computer
Science, pages 191-208, San Francisco, CA, US. Springer-
Verlag, Berlin, Germany.
[TDH] Cash, D., Kiltz, E., and V. Shoup, "The Twin-Diffie
Hellman Problem and Applications", 2008. EUROCRYPT 2008.
Volume 4965 of Lecture notes in Computer Science, pages
127-145. Springer-Verlag, Berlin, Germany.
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[RFC8265] Saint-Andre, P. and A. Melnikov, "Preparation,
Enforcement, and Comparison of Internationalized Strings
Representing Usernames and Passwords", RFC 8265,
DOI 10.17487/RFC8265, October 2017,
<https://www.rfc-editor.org/info/rfc8265>.
[I-D.irtf-cfrg-opaque]
Krawczyk, H., Bourdrez, D., Lewi, K., and C. A. Wood, "The
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
This section describes the algorithm that was used to generate the
points M and N in the table in Section 6.
For each curve in the table below, we construct a string using the
curve OID from [RFC5480] (as an ASCII string) or its name, combined
with the needed constant, e.g., "1.3.132.0.35 point generation seed
(M)" for P-512. This string is turned into a series of blocks by
hashing with SHA256, and hashing that output again to generate the
next 32 bytes, and so on. This pattern is repeated for each group
and value, with the string modified appropriately.
A byte string of length equal to that of an encoded group element is
constructed by concatenating as many blocks as are required, starting
from the first block, and truncating to the desired length. The byte
string is then formatted as required for the group. In the case of
Weierstrass curves, we take the desired length as the length for
representing a compressed point (section 2.3.4 of [SEC1]), and use
the low-order bit of the first byte as the sign bit. In order to
obtain the correct format, the value of the first byte is set to 0x02
or 0x03 (clearing the first six bits and setting the seventh bit),
leaving the sign bit as it was in the byte string constructed by
concatenating hash blocks. For the [RFC8032] curves a different
procedure is used. For edwards448 the 57-byte input has the least-
significant 7 bits of the last byte set to zero, and for edwards25519
the 32-byte input is not modified. For both the [RFC8032] curves the
(modified) input is then interpreted as the representation of the
group element. If this interpretation yields a valid group element
with the correct order (p), the (modified) byte string is the output.
Otherwise, the initial hash block is discarded and a new byte string
constructed from the remaining hash blocks. The procedure of
constructing a byte string of the appropriate length, formatting it
as required for the curve, and checking if it is a valid point of the
correct order, is repeated until a valid element is found.
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The following python snippet generates the above points, assuming an
elliptic curve implementation following the interface of
Edwards25519Point.stdbase() and Edwards448Point.stdbase() in
Appendix A of [RFC8032]:
def iterated_hash(seed, n):
h = seed
for i in range(n):
h = hashlib.sha256(h).digest()
return h
def bighash(seed, start, sz):
n = -(-sz // 32)
hashes = [iterated_hash(seed, i) for i in range(start, start + n)]
return b''.join(hashes)[:sz]
def canon_pointstr(ecname, s):
if ecname == 'edwards25519':
return s
elif ecname == 'edwards448':
return s[:-1] + bytes([s[-1] & 0x80])
else:
return bytes([(s[0] & 1) | 2]) + s[1:]
def gen_point(seed, ecname, ec):
for i in range(1, 1000):
hval = bighash(seed, i, len(ec.encode()))
pointstr = canon_pointstr(ecname, hval)
try:
p = ec.decode(pointstr)
if p != ec.zero_elem() and p * p.l() == ec.zero_elem():
return pointstr, i
except Exception:
pass
Appendix B. Test Vectors
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
[SEC1] A and B identity strings are provided in the protocol
invocation.
B.1. SPAKE2 Test Vectors
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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'
w = 0x0548d8729f730589e579b0475a582c1608138ddf7054b73b5381c7e883e2efae
x = 0x403abbe3b1b4b9ba17e3032849759d723939a27a27b9d921c500edde18ed654b
pA = 0x04a897b769e681c62ac1c2357319a3d363f610839c4477720d24cbe32f5fd85f
44fb92ba966578c1b712be6962498834078262caa5b441ecfa9d4a9485720e918a
y = 0x903023b6598908936ea7c929bd761af6039577a9c3f9581064187c3049d87065
pB = 0x04e0f816fd1c35e22065d5556215c097e799390d16661c386e0ecc84593974a6
1b881a8c82327687d0501862970c64565560cb5671f696048050ca66ca5f8cc7fc
K = 0x048f83ec9f6e4f87cc6f9dc740bdc2769725f923364f01c84148c049a39a735e
bda82eac03e00112fd6a5710682767cff5361f7e819e53d8d3c3a2922e0d837aa6
TT = 0x00000000000000000600000000000000636c69656e74410000000000000004a
897b769e681c62ac1c2357319a3d363f610839c4477720d24cbe32f5fd85f44fb92ba9
66578c1b712be6962498834078262caa5b441ecfa9d4a9485720e918a4100000000000
00004e0f816fd1c35e22065d5556215c097e799390d16661c386e0ecc84593974a61b8
81a8c82327687d0501862970c64565560cb5671f696048050ca66ca5f8cc7fc4100000
000000000048f83ec9f6e4f87cc6f9dc740bdc2769725f923364f01c84148c049a39a7
35ebda82eac03e00112fd6a5710682767cff5361f7e819e53d8d3c3a2922e0d837aa62
0000000000000000548d8729f730589e579b0475a582c1608138ddf7054b73b5381c7e
883e2efae
Hash(TT) = 0x642f05c473c2cd79909f9a841e2f30a70bf89b18180af97353ba198789c2b963
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Ke = 0x642f05c473c2cd79909f9a841e2f30a7
Ka = 0x0bf89b18180af97353ba198789c2b963
KcA = 0xc6be376fc7cd1301fd0a13adf3e7bffd
KcB = 0xb7243f4ae60440a49b3f8cab3c1fba07
A conf = 0x47d29e6666af1b7dd450d571233085d7a9866e4d49d2645e2df975489521232b
B conf = 0x3313c5cefc361d27fb16847a91c2a73b766ffa90a4839122a9b70a2f6bd1d6df
spake2: A='server', B=''
w = 0x626e0cdc7b14c9db3e52a0b1b3a768c98e37852d5db30febe0497b14eae8c254
x = 0x07adb3db6bc623d3399726bfdbfd3d15a58ea776ab8a308b00392621291f9633
pA = 0x04f88fb71c99bfffaea370966b7eb99cd4be0ff1a7d335caac4211c4afd855e2
e15a873b298503ad8ba1d9cbb9a392d2ba309b48bfd7879aefd0f2cea6009763b0
y = 0xb6a4fc8dbb629d4ba51d6f91ed1532cf87adec98f25dd153a75accafafedec16
pB = 0x040c269d6be017dccb15182ac6bfcd9e2a14de019dd587eaf4bdfd353f031101
e7cca177f8eb362a6e83e7d5e729c0732e1b528879c086f39ba0f31a9661bd34db
K = 0x0445ee233b8ecb51ebd6e7da3f307e88a1616bae2166121221fdc0dadb986afa
f3ec8a988dc9c626fa3b99f58a7ca7c9b844bb3e8dd9554aafc5b53813504c1cbe
TT = 0x06000000000000007365727665720000000000000000410000000000000004f
88fb71c99bfffaea370966b7eb99cd4be0ff1a7d335caac4211c4afd855e2e15a873b2
98503ad8ba1d9cbb9a392d2ba309b48bfd7879aefd0f2cea6009763b04100000000000
000040c269d6be017dccb15182ac6bfcd9e2a14de019dd587eaf4bdfd353f031101e7c
ca177f8eb362a6e83e7d5e729c0732e1b528879c086f39ba0f31a9661bd34db4100000
0000000000445ee233b8ecb51ebd6e7da3f307e88a1616bae2166121221fdc0dadb986
afaf3ec8a988dc9c626fa3b99f58a7ca7c9b844bb3e8dd9554aafc5b53813504c1cbe2
000000000000000626e0cdc7b14c9db3e52a0b1b3a768c98e37852d5db30febe0497b1
4eae8c254
Hash(TT) = 0x005184ff460da2ce59062c87733c299c3521297d736598fc0a1127600efa1afb
Ke = 0x005184ff460da2ce59062c87733c299c
Ka = 0x3521297d736598fc0a1127600efa1afb
KcA = 0xf3da53604f0aeecea5a33be7bddf6edf
KcB = 0x9e3f86848736f159bd92b6e107ec6799
A conf = 0xbc9f9bbe99f26d0b2260e6456e05a86196a3307ec6663a18bf6ac825736533b2
B conf = 0xc2370e1bf813b086dff0d834e74425a06e6390f48f5411900276dcccc5a297ec
spake2: A='', B=''
w = 0x7bf46c454b4c1b25799527d896508afd5fc62ef4ec59db1efb49113063d70cca
x = 0x8cef65df64bb2d0f83540c53632de911b5b24b3eab6cc74a97609fd659e95473
pA = 0x04a65b367a3f613cf9f0654b1b28a1e3a8a40387956c8ba6063e8658563890f4
6ca1ef6a676598889fc28de2950ab8120b79a5ef1ea4c9f44bc98f585634b46d66
y = 0xd7a66f64074a84652d8d623a92e20c9675c61cb5b4f6a0063e4648a2fdc02d53
pB = 0x04589f13218822710d98d8b2123a079041052d9941b9cf88c6617ddb2fcc0494
662eea8ba6b64692dc318250030c6af045cb738bc81ba35b043c3dcb46adf6f58d
K = 0x041a3c03d51b452537ca2a1fea6110353c6d5ed483c4f0f86f4492ca3f378d40
a994b4477f93c64d928edbbcd3e85a7c709b7ea73ee97986ce3d1438e135543772
TT = 0x00000000000000000000000000000000410000000000000004a65b367a3f613
cf9f0654b1b28a1e3a8a40387956c8ba6063e8658563890f46ca1ef6a676598889fc28
de2950ab8120b79a5ef1ea4c9f44bc98f585634b46d66410000000000000004589f132
18822710d98d8b2123a079041052d9941b9cf88c6617ddb2fcc0494662eea8ba6b6469
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2dc318250030c6af045cb738bc81ba35b043c3dcb46adf6f58d4100000000000000041
a3c03d51b452537ca2a1fea6110353c6d5ed483c4f0f86f4492ca3f378d40a994b4477
f93c64d928edbbcd3e85a7c709b7ea73ee97986ce3d1438e1355437722000000000000
0007bf46c454b4c1b25799527d896508afd5fc62ef4ec59db1efb49113063d70cca
Hash(TT) = 0xfc6374762ba5cf11f4b2caa08b2cd1b9907ae0e26e8d6234318d91583cd74c86
Ke = 0xfc6374762ba5cf11f4b2caa08b2cd1b9
Ka = 0x907ae0e26e8d6234318d91583cd74c86
KcA = 0x5dbd2f477166b7fb6d61febbd77a5563
KcB = 0x7689b4654407a5faeffdc8f18359d8a3
A conf = 0xdfb4db8d48ae5a675963ea5e6c19d98d4ea028d8e898dad96ea19a80ade95dca
B conf = 0xd0f0609d1613138d354f7e95f19fb556bf52d751947241e8c7118df5ef0ae175
Authors' Addresses
Watson Ladd
Cloudflare
Email: watsonbladd@gmail.com
Benjamin Kaduk (editor)
Akamai Technologies
Email: kaduk@mit.edu
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