RFC 9382
SPAKE2, a Password-Authenticated Key Exchange
RFC 9382: SPAKE2, a Password-Authenticated Key Exchange 🎀
okay bestie, let's get into this one 💅 so this doc is an Informational RFC (not a Standards Track thing, just vibes and research 🔬) written by W. Ladd from Akamai 🌸. It came out in September 2023, it's ISSN 2070-1721, and it's a product of the Crypto Forum Research Group (CFRG) inside the Internet Research Task Force (IRTF) 🧚. It doesn't obsolete or update anything, so no drama there ✨.
The whole point is SPAKE2, a protocol where two people who share a password can derive a strong shared key without ever revealing that password 🔑💖. It works with any group, it's computationally efficient, and it has a security proof, so it's giving legit 👑. It lost the CFRG PAKE competition, but because variants are already used in Kerberos and other apps, they still published it 🫶.
Here's the mood: "This document describes SPAKE2..." 🦋
Status of This Memo
This document is not an Internet Standards Track specification 💅. It's published for informational purposes only 🌈. The IRTF shares research and dev results, and those might not be ready for deployment, so chill 🙈. This RFC is just the personal opinion of one or more members of the CFRG, and anything the IRSG approves is not a candidate for any level of Internet Standard (check Section 2 of RFC 7841 if you want the receipts) 📜.
For the current status, errata, and feedback, go to https://www.rfc-editor.org/info/rfc9382 🔗.
Copyright Notice
Copyright (c) 2023 IETF Trust and the persons identified as the document authors. All rights reserved 💜. This doc is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (https://trustee.ietf.org/license-info) as of the publication date, so read those carefully because they describe your rights 🧾.
1. Introduction 🌸
Okay so the intro says SPAKE2 is a way for two parties who share a password to get a strong shared key without disclosing that password 🔐. It's compatible with any group, as long as you have a way to map a random fixed-length input per group to a random group element 🧩. It's efficient, it has a security proof, and the doc also gives predetermined parameters for some commonly used groups so other protocols can just grab them 🎁.
SPAKE2 lost the CFRG PAKE selection competition 😭. But since Kerberos and a bunch of other apps already use variants, publication was still beneficial ✨. Again, this is one or more CFRG members speaking for themselves, not the whole group 🙈.
Lots of those apps showed up before hash-to-elliptic-curve methods existed, or before the competition winners were published. So if you need a symmetric PAKE and you can't hash onto an elliptic curve at execution time, SPAKE2 is your girl 🦋.
2. Requirements Notation 📏
The BCP 14 key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" mean what they're defined to mean in BCP 14 [RFC2119] [RFC8174], but only when they're written in ALL CAPS like that 💅. Lowercase ones are just regular words, no cap 🙅.
3. Definition of SPAKE2 💖
3.1. Protocol Flow 🔁
SPAKE2 is a two-round protocol 🔄. Round one sets up a shared secret between A and B, and round two is for key confirmation ✅. Before anything runs, A and B get provisioned with stuff like the password 🗝️. We assume both sides agree on who's A and who's B: A goes first and uses M, and B goes second and uses N 🎭. If that role assignment isn't possible, a symmetric variant MUST be used, as described later in Section 5 ⚠️. For example, A might be the client when you're using TCP or TLS underneath, and B might be the server, since most protocols have that kind of split 🖥️.
In round one, A sends its public value pA to B, and B replies with its own public value pB 📤📥. Both of them then derive a shared secret that gets used to make encryption and authentication keys 🔐. Those keys are used in round two for key confirmation (Section 4 has the nitty gritty on key derivation and confirmation 🧐). A sends its key confirmation message cA to B, and B replies with cB 💌. A MUST NOT consider the protocol complete until it receives and verifies cB 🚫. Same deal for B: B MUST NOT consider the protocol complete until it receives and verifies cA 🚫. No skipping the check, bestie 🙅♀️.
Here's the sample flow 👇:
A B
| |
| |
(compute pA) | pA |
|---------------------->|
| pB | (compute pB)
|<----------------------|
| |
| (derive secrets) |
| |
(compute cA) | cA |
|---------------------->|
| cB | (compute cB)
| | (check cA)
|<----------------------|
(check cB) | |3.2. Setup 🛠️
Okay this part is the math setup, so get your thinking cap on 🧠✨. Let G be a group where the gap Diffie-Hellman (GDH) problem is hard 💪. Say G has order p*h, where p is a large prime and h is called the cofactor 🔢. I is the unit element in G, like the point at infinity if G is an elliptic curve group 🌌. We use additive notation for the operations ➕. Elements of G are represented as byte strings, and common choices are SEC1 [SEC1] uncompressed or compressed for elliptic curves, or big-endian integers of a fixed per-group length for prime field DH 🧮. Applications MUST specify this encoding, usually by pointing to the doc that defines the group 📚.
We fix two elements, M and N, in the prime-order subgroup of G. Table 1 of this document lists them for common groups, and there's also a generator P for the large prime-order subgroup of G 🎯. For a composite order group, we work in the quotient group 🧩. For the common groups in this doc, P is defined in the group's own document, so we don't repeat it here 🙈.
For elliptic curves that aren't in this doc, the methods in [RFC9380] SHOULD be used to generate M and N 💫. For example, 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 domain separation tag (DST) in this step, as specified in [RFC9380], though it's not required 🎀.
| means concatenation of byte strings 🔗. len(S) is the length of a string in bytes, written as an eight-byte little-endian number 🔢. nil is an empty string, so len(nil) = 0 🫙. Text strings in double quotes are treated as their ASCII encodings throughout the doc 🔤. |
KDF(ikm, salt, info, L) is a key-derivation function. It takes a salt, input keying material (IKM), an info string, and a derived key length L, and gives back a cryptographic key of length L 🗝️. MAC(key, message) is a Message Authentication Code algorithm that takes a secret key and message and produces an output 🛡️. Let Hash be a hash function from arbitrary strings to bit strings of a fixed length, at least 256 bits long 🧱. Common choices are SHA-256 or SHA-512 [RFC6234] 💅. Let MHF be a memory-hard hash function designed to slow down brute-force attackers 🐢. Scrypt [RFC7914] is a common example. MHF's output length matches Hash's 📏. Parameter selection for MHF is out of scope for this document 🙅. Section 6 specifies variants of KDF, MAC, and Hash that work with the protocols in here 🧁.
Let A and B be two parties 👯. A and B may also have digital representations of their identities, like MAC addresses or other names (hostnames, usernames, etc.) 🪪. A and B may share additional authenticated data (AAD) of at most 2^16 - 128 bits, kept separate from their identities, which they may want to include in the run 🎁. One example of AAD is a list of supported protocol versions if SPAKE2 were used in a higher-level protocol that negotiates a particular PAKE 📋. Including that list makes sure both parties agree on the same set of supported protocols, which prevents downgrade attacks 🛑. We also assume A and B share an integer w. Typically, w = MHF(pw) mod p for a user-supplied password pw 🔑. [NIST.SP.800-56Ar3] suggests taking mod p of a hash value that's 64 bits longer than what's needed to represent p, to remove statistical bias from the modulation 📊. Protocols using this spec MUST define how w is computed 📝. Sometimes you need to normalize the password before hashing [RFC8265] 🧼. The hashing algorithm SHOULD be an MHF to slow down brute-force attackers 🐌.
Part 1 of 4 done bestie, more coming soon 💖🎀 Stay tuned for Section 3.3!
3.3. SPAKE2 ✨
okay so A picks x randomly and uniformly from the integers in [0,p) and calculates X=x*P and pA=w*M+X 💅. Then A sends pA over to B.
B picks y randomly and uniformly from the integers in [0,p) and calculates Y=y*P and pB=w*N+Y 🦋. Then B sends pB back to A.
Both A and B work out the group element K. A computes h*x*(pB-w*N) and B computes h*y*(pA-w*M) 🔥. A can do this because it already has pB, and B has pA for the same reason. The multiplication by h is what stops small subgroup confinement attacks, because it gives a unique value in the quotient group ✨.
K is a shared value, but 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 💜. Using the transcript means any messing with the messages gets reflected in the keys, so it's basically a built-in tamper alarm 🚨. The transcript TT is encoded like this:
TT = len(A) || A
|| len(B) || B
|| len(pA) || pA
|| len(pB) || pB
|| len(K) || K
|| len(w) || wHere, w is encoded as a big-endian number padded to the length of p. That representation stops timing attacks that would otherwise reveal the length of w, so len(w) is a constant for a given group. We include it anyway for consistency 🎀.
If an identity is absent, it gets 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 😭.
Once this protocol finishes, A and B compute the shared secrets Ke, KcA, and KcB, as specified in Section 4. A MUST send B a key confirmation message so that both parties agree on these shared secrets 🫶. The confirmation message cA is a MAC over the protocol transcript TT, using KcA: cA = MAC(KcA, TT). Similarly, B MUST send A a confirmation message using a MAC that is computed the same way, except with KcB. To verify key confirmation, you compute cA (or cB, respectively) and check it for equality against what 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 (though it contains some substrings that are not secret) 🔐. Both parties use TT to derive shared symmetric secrets Ke and Ka as Ke || Ka = Hash(TT), with |Ke| = |Ka|. Each key is half the digest output, so for example 128 bits for SHA-256. Keys MUST be at least 128 bits in length 💪.
Both endpoints use Ka to derive the MAC keys for key confirmation messages. Specifically, KcA and KcB are the MAC keys used by A and B. They compute KcA || KcB = KDF(Ka, nil, "ConfirmationKeys" || AAD, L), where AAD is the associated data given to each endpoint, or nil if none was provided. Each of KcA and KcB is half the underlying hash output length, for example |KcA| = |KcB| = 128 bits for HKDF(SHA256), with L=256 bits.
The key schedule for this protocol, given transcript TT and AAD, looks like this:
TT -> Hash(TT) = Ke || Ka
AAD -> KDF(Ka, nil, "ConfirmationKeys" || AAD) = KcA || KcBA 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 Elliptic Curve Diffie-Hellman (ECDH) instance to break the authentication of SPAKE2, it is possible to vary M and N using [RFC9380], like so:
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))There's also a symmetric variant where M=N. 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 whether A or B should use M (or N). That can happen because of asymmetries in the protocol flows, or because you want 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 that don't know the exact identity of the parties before the flow begins 🧚.
6. Ciphersuites ✨
This section documents SPAKE2 ciphersuite configurations. A ciphersuite indicates a group, a cryptographic hash function, and a pair of KDF and MAC functions, for example SPAKE2-P256-SHA256-HKDF-HMAC. This ciphersuite indicates a SPAKE2 protocol instance over P-256 that uses SHA-256, along with HMAC-based Key Derivation Function (HKDF) [RFC5869] and Hashed Message Authentication Code (HMAC) [RFC2104] for G, Hash, KDF, and MAC functions, respectively 🌈. For Ed25519, the compressed encoding is used [RFC8032]. All others use the uncompressed SEC1 encoding.
Here's Table 1 in all its glory 💅:
+==============+==================+================+================+
| 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-521 | SHA512 [RFC6234] | HKDF [RFC5869] | HMAC |
| | | | [RFC2104] |
+--------------+------------------+----------------+----------------+
| edwards25519 | SHA256 [RFC6234] | HKDF [RFC5869] | HMAC |
| [RFC8032] | | | [RFC2104] |
+--------------+------------------+----------------+----------------+
| edwards448 | SHA512 [RFC6234] | HKDF [RFC5869] | HMAC |
| [RFC8032] | | | [RFC2104] |
+--------------+------------------+----------------+----------------+
| P-256 | SHA256 [RFC6234] | HKDF [RFC5869] | CMAC-AES-128 |
| | | | [RFC4493] |
+--------------+------------------+----------------+----------------+
| P-256 | SHA512 [RFC6234] | HKDF [RFC5869] | CMAC-AES-128 |
| | | | [RFC4493] |
+--------------+------------------+----------------+----------------+
Table 1: SPAKE2 CiphersuitesThe following points are the permissible point generation seeds for the groups in Table 1, using the algorithm in Appendix A. These byte strings are compressed points, as in [SEC1], for curves from [SEC1] 🦋.
For P-256:
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 P-384:
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 P-521:
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:
M =
b6221038a775ecd007a4e4dde39fd76ae91d3cf0cc92be8f0c2fa6d6b66f9a12
942f5a92646109152292464f3e63d354701c7848d9fc3b8880
seed: edwards448 point generation seed (M)
N =
6034c65b66e4cd7a49b0edec3e3c9ccc4588afd8cf324e29f0a84a072531c4db
f97ff9af195ed714a689251f08f8e06e2d1f24a0ffc0146600
seed: edwards448 point generation seed (N)7. Security Considerations ✨
Bestie, this section is serious business, so read it carefully 😭🔥
A security proof of SPAKE2 for prime order groups is found in [REF], reducing the security of SPAKE2 to the GDH 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, for example x and y, MUST NOT be reused. Reuse like that violates the security assumptions of the protocol and results in significant insecurity 💀. It is RECOMMENDED to generate these uniform numbers using rejection sampling.
Some implementations of elliptic curve multiplication may leak information about the length of the scalar. These MUST NOT be used. All operations on elliptic curve points must take time independent of the inputs. Hashing of the transcript may take time depending only on the length of the transcript but not the contents 🙈.
SPAKE2 does not support augmentation. As a result, the server has to store a password equivalent. This is considered a significant drawback in some use cases. Applications that need augmented PAKEs should use [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 ✨
Good news, bestie: this document has no IANA actions 🌸.
9. References ✨
The reference list keeps going in the next part 💖.
9.1. Normative References 📚
Okay bestie, these are the must-reads you need to actually implement this. No skipping 💅
[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>.
[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>.
[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>.
[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>.9.2. Informative References 🌸
These are the extra reads. Nice to have, not required.
[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,
J., and J. Xu, "Universally Composable Relaxed Password
Authenticated Key Exchange", in Advances in Cryptology -
CRYPTO 2020, Lecture Notes in Computer Science, Volume
12170, Springer, DOI 10.1007/978-3-030-56784-2_10, August
2020, <https://doi.org/10.1007/978-3-030-56784-2_10>.
[NIST.SP.800-56Ar3]
National Institute of Standards and Technology,
"Recommendation for Pair-Wise Key-Establishment Schemes
Using Discrete Logarithm Cryptography", Revision 3, NIST
Special Publication 800-56A,
DOI 10.6028/NIST.SP.800-56Ar3, April 2018,
<https://doi.org/10.6028/NIST.SP.800-56Ar3>.
[REF] Abdalla, M. and D. Pointcheval, "Simple Password-Based
Encrypted Key Exchange Protocols", Cryptography-CT-RSA
2005, Lecture Notes in Computer Science, Volume 3376,
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,
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>.
[SEC1] Standards for Efficient Cryptography Group, "SEC 1:
Elliptic Curve Cryptography", May 2009.Appendix A. Algorithm Used for Point Generation ✨
Okay so this appendix is the behind-the-scenes story of how the points M and N in Table 1 were actually made 🧚 It's giving math magic.
For each curve in Table 1, they build a string out of either the curve OID from [RFC5480] (as an ASCII string) or the curve's name, then stick on the needed constant. For example, for P-521 you'd get "1.3.132.0.35 point generation seed (M)". That string gets turned into a whole series of blocks: hash it with SHA-256, hash that output again to get the next 32 bytes, and keep going. The same pattern repeats for each group and value, with the string tweaked each time 🔥
To get a byte string the right length for a group element, you stitch together as many blocks as you need, starting from the first one, and then chop it down to the length you want. Next it gets formatted for the specific group. For Weierstrass curves, the length you're aiming for is the length of a compressed point (Section 2.3.4 of [SEC1]), and the low-order bit of the first byte becomes the sign bit. To get the right format, the first byte is set to 0x02 or 0x03 (clearing the first six bits and setting the seventh bit), and the sign bit stays exactly as it was in the byte string built from the hash blocks 💅
The curves in [RFC8032] are a different situation. For edwards448, the 57-byte input has the least-significant 7 bits of the last byte set to zero. For edwards25519, the 32-byte input is left alone, no changes 🦋 For both of those curves, the (modified) input is then read as the representation of a group element. If that reading gives a valid group element with the correct order (p), then the (modified) byte string is the output. If not, the initial hash block gets tossed, and a new byte string is built from the remaining hash blocks. This whole process (build a byte string of the right length, format it for the curve, check that it's a valid point of the correct order) keeps repeating until a valid element shows up. Persistence is everything 👑
Here's the Python snippet that generates the points above. It assumes the elliptic curve implementation follows 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:
passAppendix B. SPAKE2 Test Vectors ✨
okay so this appendix is the receipts 🧾💅. It gives you SPAKE2 test vectors using the P256-SHA256-HKDF-HMAC ciphersuite, so you can check your implementation against known-good numbers. The MHF choice is left out bestie, and the values for w, x, and y are handed to you directly 🔐. Every point is in the uncompressed format, which means it starts with a 0x04 octet prefix, as specified in [SEC1] 🦋. The A and B identity strings come in with the protocol invocation, so check those too 🫶.
Heads up: line breaks were added to the hex below because of line-length limits, so the values are split across lines. Join them back up before you compare! 💜
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) = 0x0e0672dc86f8e45565d338b0540abe6915bdf72e2b35b5c9e5663168e9
60a91b
Ke = 0x0e0672dc86f8e45565d338b0540abe69
Ka = 0x15bdf72e2b35b5c9e5663168e960a91b
KcA = 0x00c12546835755c86d8c0db7851ae86f
KcB = 0xa9fa3406c3b781b93d804485430ca27a
A conf = 0x58ad4aa88e0b60d5061eb6b5dd93e80d9c4f00d127c65b3b35b1b5281f
ee38f0
B conf = 0xd3e2e547f1ae04f2dbdbf0fc4b79f8ecff2dff314b5d32fe9fcef2fb26
dc459b spake2: A='', B='client'
w = 0x0548d8729f730589e579b0475a582c1608138ddf7054b73b5381c7e883e2efae
x = 0x403abbe3b1b4b9ba17e3032849759d723939a27a27b9d921c500edde18ed654b
pA = 0x04a897b769e681c62ac1c2357319a3d363f610839c4477720d24cbe32f5fd8
5f44fb92ba966578c1b712be6962498834078262caa5b441ecfa9d4a9485720e918a
y = 0x903023b6598908936ea7c929bd761af6039577a9c3f9581064187c3049d87065
pB = 0x04e0f816fd1c35e22065d5556215c097e799390d16661c386e0ecc84593974
a61b881a8c82327687d0501862970c64565560cb5671f696048050ca66ca5f8cc7fc
K = 0x048f83ec9f6e4f87cc6f9dc740bdc2769725f923364f01c84148c049a39a735e
bda82eac03e00112fd6a5710682767cff5361f7e819e53d8d3c3a2922e0d837aa6
TT = 0x00000000000000000600000000000000636c69656e74410000000000000004a
897b769e681c62ac1c2357319a3d363f610839c4477720d24cbe32f5fd85f44fb92ba9
66578c1b712be6962498834078262caa5b441ecfa9d4a9485720e918a4100000000000
00004e0f816fd1c35e22065d5556215c097e799390d16661c386e0ecc84593974a61b8
81a8c82327687d0501862970c64565560cb5671f696048050ca66ca5f8cc7fc4100000
000000000048f83ec9f6e4f87cc6f9dc740bdc2769725f923364f01c84148c049a39a7
35ebda82eac03e00112fd6a5710682767cff5361f7e819e53d8d3c3a2922e0d837aa62
0000000000000000548d8729f730589e579b0475a582c1608138ddf7054b73b5381c7e
883e2efae
Hash(TT) = 0x642f05c473c2cd79909f9a841e2f30a70bf89b18180af97353ba198789
c2b963
Ke = 0x642f05c473c2cd79909f9a841e2f30a7
Ka = 0x0bf89b18180af97353ba198789c2b963
KcA = 0xc6be376fc7cd1301fd0a13adf3e7bffd
KcB = 0xb7243f4ae60440a49b3f8cab3c1fba07
A conf = 0x47d29e6666af1b7dd450d571233085d7a9866e4d49d2645e2df9754895
21232b
B conf = 0x3313c5cefc361d27fb16847a91c2a73b766ffa90a4839122a9b70a2f6b
d1d6df 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) = 0x005184ff460da2ce59062c87733c299c3521297d736598fc0a1127600e
fa1afb
Ke = 0x005184ff460da2ce59062c87733c299c
Ka = 0x3521297d736598fc0a1127600efa1afb
KcA = 0xf3da53604f0aeecea5a33be7bddf6edf
KcB = 0x9e3f86848736f159bd92b6e107ec6799
A conf = 0xbc9f9bbe99f26d0b2260e6456e05a86196a3307ec6663a18bf6ac8257365
33b2
B conf = 0xc2370e1bf813b086dff0d834e74425a06e6390f48f5411900276dcccc5a2
97ec 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
e175Acknowledgements
okay the thank-you list 🥹🫶. Huge shoutout to Nathaniel McCallum and Greg Hudson for generating M and N, and to Chris Wood for generating the test vectors 🌸. Mike Hamburg gets a big thank you for the advice on dealing with cofactors 💎. Greg Hudson also suggested adding warnings about reusing x and y ⚠️, which is so important. Thanks to Fedor Brunner, Adam Langley, Liliya Akhmetzyanova, and the CFRG folks for comments and advice 💜. Thanks also to Scott Fluhrer and the Crypto Panel experts involved in the PAKE selection process, found at https://github.com/cfrg/pake-selection, for their super valuable comments ✨. And Chris Wood contributed a ton of text and reformatting to address the excellent review comments from Kenny Paterson 👑. Iconic team behavior 🙌.
Contributors
Benjamin Kaduk Akamai Technologies Email: kaduk@mit.edu
Benjamin is one of the contributors who helped bring this spec together 🦋.
Author's Address
Watson Ladd Akamai Technologies Email: watsonbladd@gmail.com
And that's the whole doc, bestie 💖🎀. Watson Ladd wrote it up, and you now have every single test vector to check your SPAKE2 work against 🧚💅.
📦 where this page comes from
- ietf: rfc9382 2023-09
- 💖 girlified by claude-haiku-5-5 on : a language model's rewrite of the original, under its terms