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Kaduk, Ed. 5 Expires: 12 August 2022 Akamai 6 February 2022 8 SPAKE2, a PAKE 9 draft-irtf-cfrg-spake2-26 11 Abstract 13 This document describes SPAKE2 which is a protocol for two parties 14 that share a password to derive a strong shared key without 15 disclosing the password. This method is compatible with any group, 16 is computationally efficient, and SPAKE2 has a security proof. This 17 document predated the CFRG PAKE competition and it was not selected, 18 however, given existing use of variants in Kerberos and other 19 applications it was felt publication was beneficial. Applications 20 that need a symmetric PAKE (password authenticated key exchange) and 21 where hashing onto an elliptic curve at execution time is not 22 possible can use SPAKE2. This document is a product of the Crypto 23 Forum Research Group (CFRG) in the IRTF. 25 Status of This Memo 27 This Internet-Draft is submitted in full conformance with the 28 provisions of BCP 78 and BCP 79. 30 Internet-Drafts are working documents of the Internet Engineering 31 Task Force (IETF). Note that other groups may also distribute 32 working documents as Internet-Drafts. The list of current Internet- 33 Drafts is at https://datatracker.ietf.org/drafts/current/. 35 Internet-Drafts are draft documents valid for a maximum of six months 36 and may be updated, replaced, or obsoleted by other documents at any 37 time. It is inappropriate to use Internet-Drafts as reference 38 material or to cite them other than as "work in progress." 40 This Internet-Draft will expire on 5 August 2022. 42 Copyright Notice 44 Copyright (c) 2022 IETF Trust and the persons identified as the 45 document authors. All rights reserved. 47 This document is subject to BCP 78 and the IETF Trust's Legal 48 Provisions Relating to IETF Documents (https://trustee.ietf.org/ 49 license-info) in effect on the date of publication of this document. 50 Please review these documents carefully, as they describe your rights 51 and restrictions with respect to this document. 53 Table of Contents 55 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2 56 2. Requirements Notation . . . . . . . . . . . . . . . . . . . . 3 57 3. Definition of SPAKE2 . . . . . . . . . . . . . . . . . . . . 3 58 4. Key Schedule and Key Confirmation . . . . . . . . . . . . . . 6 59 5. Per-User M and N and M=N . . . . . . . . . . . . . . . . . . 7 60 6. Ciphersuites . . . . . . . . . . . . . . . . . . . . . . . . 7 61 7. Security Considerations . . . . . . . . . . . . . . . . . . . 10 62 8. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 11 63 9. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 11 64 10. References . . . . . . . . . . . . . . . . . . . . . . . . . 11 65 10.1. Normative References . . . . . . . . . . . . . . . . . . 11 66 10.2. Informative References . . . . . . . . . . . . . . . . . 12 67 Appendix A. Algorithm used for Point Generation . . . . . . . . 13 68 Appendix B. Test Vectors . . . . . . . . . . . . . . . . . . . . 14 69 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 17 71 1. Introduction 73 This document describes SPAKE2, a means for two parties that share a 74 password to derive a strong shared key without disclosing the 75 password. This password-based key exchange protocol is compatible 76 with any group (requiring only a scheme to map a random input of 77 fixed length per group to a random group element), is computationally 78 efficient, and has a security proof. Predetermined parameters for a 79 selection of commonly used groups are also provided for use by other 80 protocols. 82 SPAKE2 was not selected as the result of the CFRG PAKE selection 83 competition. However, given existing use of variants in Kerberos and 84 other applications it was felt publication was beneficial. This RFC 85 represents the individual opinion(s) of one or more members of the 86 Crypto Forum Research Group of the Internet Research Task Force 87 (IRTF). 89 Many of these applications predated methods to hash to elliptic 90 curves being available or predated the publication of the PAKEs that 91 were chosen as an outcome of the PAKE selection competition. In 92 cases where a symmetric PAKE is needed, and hashing onto an elliptic 93 curve at protocol execution time is not available, SPAKE2 is useful. 95 2. Requirements Notation 97 The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", 98 "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and 99 "OPTIONAL" in this document are to be interpreted as described in BCP 100 14 [RFC2119] [RFC8174] when, and only when, they appear in all 101 capitals, as shown here. 103 3. Definition of SPAKE2 105 3.1. Protocol Flow 107 SPAKE2 is a two round protocol, wherein the first round establishes a 108 shared secret between A and B, and the second round serves as key 109 confirmation. Prior to invocation, A and B are provisioned with 110 information such as the input password needed to run the protocol. 111 We assume that the roles of A and B are agreed upon by both sides: A 112 goes first and uses M, and B goes second and uses N. If this 113 assignment of roles is not possible a symmetric variant MUST be used 114 as described later Section 5. For instance A may be the client when 115 using TCP or TLS as an underlying protocol and B the server. Most 116 protocols have such a distinction. During the first round, A sends a 117 public value pA to B, and B responds with its own public value pB. 118 Both A and B then derive a shared secret used to produce encryption 119 and authentication keys. The latter are used during the second round 120 for key confirmation. (Section 4 details the key derivation and 121 confirmation steps.) In particular, A sends a key confirmation 122 message cA to B, and B responds with its own key confirmation message 123 cB. A MUST NOT consider the protocol complete until it receives and 124 verifies cB. Likewise, B MUST NOT consider the protocol complete 125 until it receives and verifies cA. 127 This sample flow is shown below. 129 A B 130 | (setup protocol) | 131 | | 132 (compute pA) | pA | 133 |----------------->| 134 | pB | (compute pB) 135 |<-----------------| 136 | | 137 | (derive secrets) | 138 | | 139 (compute cA) | cA | 140 |----------------->| 141 | cB | (compute cB) 142 | | (check cA) 143 |<-----------------| 144 (check cB) | | 146 3.2. Setup 148 Let G be a group in which the gap Diffie-Hellman (GDH) problem is 149 hard. Suppose G has order p*h where p is a large prime; h will be 150 called the cofactor. Let I be the unit element in G, e.g., the point 151 at infinity if G is an elliptic curve group. We denote the 152 operations in the group additively. We assume there is a 153 representation of elements of G as byte strings: common choices would 154 be SEC1 [SEC1] uncompressed or compressed for elliptic curve groups 155 or big endian integers of a fixed (per-group) length for prime field 156 DH. Applications MUST specify this encoding, typically by referring 157 to the document defining the group. We fix two elements M and N in 158 the prime-order subgroup of G as defined in the table in this 159 document for common groups, as well as a generator P of the (large) 160 prime-order subgroup of G. In the case of a composite order group we 161 will work in the quotient group. For common groups used in this 162 document, P is specified in the document defining the group, and so 163 we do not repeat it here. 165 For elliptic curves other than the ones in this document the methods 166 of [I-D.irtf-cfrg-hash-to-curve] SHOULD be used to generate M and N, 167 e.g. via M = hash_to_curve("M SPAKE2 seed OID x") and N = 168 hash_to_curve("N SPAKE2 seed OID x"), where x is an OID for the 169 curve. Applications MAY include a DST tag in this step, as specified 170 in [I-D.irtf-cfrg-hash-to-curve], though this is not required. 172 || denotes concatenation of byte strings. We also let len(S) denote 173 the length of a string in bytes, represented as an eight-byte little- 174 endian number. Finally, let nil represent an empty string, i.e., 175 len(nil) = 0. Text strings in double quotes are treated as their 176 ASCII encodings throughout this document. 178 KDF(ikm, salt, info, L) is a key-derivation function that takes as 179 input a salt, intermediate keying material (IKM), info string, and 180 derived key length L to derive a cryptographic key of length L. 181 MAC(key, message) is a Message Authentication Code algorithm that 182 takes a secret key and message as input to produce an output. Let 183 Hash be a hash function from arbitrary strings to bit strings of a 184 fixed length, at least 256 bits long. Common choices for Hash are 185 SHA256 or SHA512 [RFC6234]. Let MHF be a memory-hard hash function 186 designed to slow down brute-force attackers. Scrypt [RFC7914] is a 187 common example of this function. The output length of MHF matches 188 that of Hash. Parameter selection for MHF is out of scope for this 189 document. Section 6 specifies variants of KDF, MAC, and Hash 190 suitable for use with the protocols contained herein. 192 Let A and B be two parties. A and B may also have digital 193 representations of the parties' identities such as Media Access 194 Control addresses or other names (hostnames, usernames, etc). A and 195 B may share Additional Authenticated Data (AAD) of length at most 196 2^16 - 128 bits that is separate from their identities which they may 197 want to include in the protocol execution. One example of AAD is a 198 list of supported protocol versions if SPAKE2 were used in a higher- 199 level protocol which negotiates use of a particular PAKE. Including 200 this list would ensure that both parties agree upon the same set of 201 supported protocols and therefore prevent downgrade attacks. We also 202 assume A and B share an integer w; typically w = MHF(pw) mod p, for a 203 user-supplied password pw. Standards such as NIST.SP.800-56Ar3 204 suggest taking mod p of a hash value that is 64 bits longer than that 205 needed to represent p to remove statistical bias introduced by the 206 modulation. Protocols using this specification MUST define the 207 method used to compute w. In some cases, it may be necessary to 208 carry out various forms of normalization of the password before 209 hashing [RFC8265]. The hashing algorithm SHOULD be a MHF so as to 210 slow down brute-force attackers. 212 3.3. SPAKE2 214 To begin, A picks x randomly and uniformly from the integers in 215 [0,p), and calculates X=x*P and pA=w*M+X, then transmits pA to B. 217 B selects y randomly and uniformly from the integers in [0,p), and 218 calculates Y=y*P, pB=w*N+Y, then transmits pB to A. 220 Both A and B calculate a group element K. A calculates it as 221 h*x*(pB-w*N), while B calculates it as h*y*(pA-w*M). A knows pB 222 because it has received it, and likewise B knows pA. The 223 multiplication by h prevents small subgroup confinement attacks by 224 computing a unique value in the quotient group. 226 K is a shared value, though it MUST NOT be used or output as a shared 227 secret from the protocol. Both A and B must derive two additional 228 shared secrets from the protocol transcript, which includes K. This 229 prevents man-in-the-middle attackers from inserting themselves into 230 the exchange. The transcript TT is encoded as follows: 232 TT = len(A) || A 233 || len(B) || B 234 || len(pA) || pA 235 || len(pB) || pB 236 || len(K) || K 237 || len(w) || w 239 Here w is encoded as a big endian number padded to the length of p. 240 This representation prevents timing attacks that otherwise would 241 reveal the length of w. len(w) is thus a constant. We include it for 242 consistency. 244 If an identity is absent, it is encoded as a zero-length string. 245 This MUST only be done for applications in which identities are 246 implicit. Otherwise, the protocol risks unknown key share attacks, 247 where both sides of a connection disagree over who is authenticated. 249 Upon completion of this protocol, A and B compute shared secrets Ke, 250 KcA, and KcB as specified in Section 4. A MUST send B a key 251 confirmation message so both parties agree upon these shared secrets. 252 This confirmation message cA is computed as a MAC over the protocol 253 transcript TT using KcA, as follows: cA = MAC(KcA, TT). Similarly, B 254 MUST send A a confirmation message using a MAC computed equivalently 255 except with the use of KcB. Key confirmation verification requires 256 computing cB and checking for equality against that which was 257 received. 259 4. Key Schedule and Key Confirmation 261 The protocol transcript TT, as defined in Section 3.3, is unique and 262 secret to A and B. Both parties use TT to derive shared symmetric 263 secrets Ke and Ka as Ke || Ka = Hash(TT), with |Ke| = |Ka|. The 264 length of each key is equal to half of the digest output, e.g., 128 265 bits for SHA-256. Keys MUST be at least 128 bits in length. 267 Both endpoints use Ka to derive subsequent MAC keys for key 268 confirmation messages. Specifically, let KcA and KcB be the MAC keys 269 used by A and B, respectively. A and B compute them as KcA || KcB = 270 KDF(Ka, nil, "ConfirmationKeys" || AAD, L), where AAD is the 271 associated data each given to each endpoint, or nil if none was 272 provided. The length of each of KcA and KcB is equal to half of the 273 underlying hash output length, e.g., |KcA| = |KcB| = 128 bits for 274 HKDF(SHA256), with L=256 bits. 276 The resulting key schedule for this protocol, given transcript TT and 277 additional associated data AAD, is as follows. 279 TT -> Hash(TT) = Ke || Ka 280 AAD -> KDF(Ka, nil, "ConfirmationKeys" || AAD) = KcA || KcB 282 A and B output Ke as the shared secret from the protocol. Ka and its 283 derived keys are not used for anything except key confirmation. 285 5. Per-User M and N and M=N 287 To avoid concerns that an attacker needs to solve a single ECDH 288 instance to break the authentication of SPAKE2, it is possible to 289 vary M and N using [I-D.irtf-cfrg-hash-to-curve] as follows: 291 M = hash_to_curve(Hash("M SPAKE2" || len(A) || A || len(B) || B)) 292 N = hash_to_curve(Hash("N SPAKE2" || len(A) || A || len(B) || B)) 294 There is also a symmetric variant where M=N. For this variant we set 296 M = hash_to_curve(Hash("M AND N SPAKE2")) 298 This variant MUST be used when it is not possible to determine which 299 of A and B should use M or N, due to asymmetries in the protocol 300 flows or the desire to use only a single shared secret with nil 301 identities for authentication. The security of these variants is 302 examined in [MNVAR]. The variant with per-user M and N may not be 303 suitable for protocols that require the initial messages to be 304 generated by each party at the same time and do not know the exact 305 identity of the parties before the flow begins. 307 6. Ciphersuites 309 This section documents SPAKE2 ciphersuite configurations. A 310 ciphersuite indicates a group, cryptographic hash function, and pair 311 of KDF and MAC functions, e.g., SPAKE2-P256-SHA256-HKDF-HMAC. This 312 ciphersuite indicates a SPAKE2 protocol instance over P-256 that uses 313 SHA256 along with HKDF [RFC5869] and HMAC [RFC2104] for G, Hash, KDF, 314 and MAC functions, respectively. For Ed25519 the compressed encoding 315 is used [RFC8032], all others use the uncompressed SEC1 encoding. 317 +==============+==================+================+================+ 318 | G | Hash | KDF | MAC | 319 +==============+==================+================+================+ 320 | P-256 | SHA256 [RFC6234] | HKDF [RFC5869] | HMAC | 321 | | | | [RFC2104] | 322 +--------------+------------------+----------------+----------------+ 323 | P-256 | SHA512 [RFC6234] | HKDF [RFC5869] | HMAC | 324 | | | | [RFC2104] | 325 +--------------+------------------+----------------+----------------+ 326 | P-384 | SHA256 [RFC6234] | HKDF [RFC5869] | HMAC | 327 | | | | [RFC2104] | 328 +--------------+------------------+----------------+----------------+ 329 | P-384 | SHA512 [RFC6234] | HKDF [RFC5869] | HMAC | 330 | | | | [RFC2104] | 331 +--------------+------------------+----------------+----------------+ 332 | P-521 | SHA512 [RFC6234] | HKDF [RFC5869] | HMAC | 333 | | | | [RFC2104] | 334 +--------------+------------------+----------------+----------------+ 335 | edwards25519 | SHA256 [RFC6234] | HKDF [RFC5869] | HMAC | 336 | [RFC8032] | | | [RFC2104] | 337 +--------------+------------------+----------------+----------------+ 338 | edwards448 | SHA512 [RFC6234] | HKDF [RFC5869] | HMAC | 339 | [RFC8032] | | | [RFC2104] | 340 +--------------+------------------+----------------+----------------+ 341 | P-256 | SHA256 [RFC6234] | HKDF [RFC5869] | CMAC-AES-128 | 342 | | | | [RFC4493] | 343 +--------------+------------------+----------------+----------------+ 344 | P-256 | SHA512 [RFC6234] | HKDF [RFC5869] | CMAC-AES-128 | 345 | | | | [RFC4493] | 346 +--------------+------------------+----------------+----------------+ 348 Table 1: SPAKE2 Ciphersuites 350 The following points represent permissible point generation seeds for 351 the groups listed in the Table Table 1, using the algorithm presented 352 in Appendix A. These bytestrings are compressed points as in [SEC1] 353 for curves from [SEC1]. 355 For P256: 357 M = 358 02886e2f97ace46e55ba9dd7242579f2993b64e16ef3dcab95afd497333d8fa12f 359 seed: 1.2.840.10045.3.1.7 point generation seed (M) 361 N = 362 03d8bbd6c639c62937b04d997f38c3770719c629d7014d49a24b4f98baa1292b49 363 seed: 1.2.840.10045.3.1.7 point generation seed (N) 365 For P384: 367 M = 368 030ff0895ae5ebf6187080a82d82b42e2765e3b2f8749c7e05eba366434b363d3dc 369 36f15314739074d2eb8613fceec2853 370 seed: 1.3.132.0.34 point generation seed (M) 372 N = 373 02c72cf2e390853a1c1c4ad816a62fd15824f56078918f43f922ca21518f9c543bb 374 252c5490214cf9aa3f0baab4b665c10 375 seed: 1.3.132.0.34 point generation seed (N) 377 For P521: 379 M = 380 02003f06f38131b2ba2600791e82488e8d20ab889af753a41806c5db18d37d85608 381 cfae06b82e4a72cd744c719193562a653ea1f119eef9356907edc9b56979962d7aa 382 seed: 1.3.132.0.35 point generation seed (M) 384 N = 385 0200c7924b9ec017f3094562894336a53c50167ba8c5963876880542bc669e494b25 386 32d76c5b53dfb349fdf69154b9e0048c58a42e8ed04cef052a3bc349d95575cd25 387 seed: 1.3.132.0.35 point generation seed (N) 389 For edwards25519: 391 M = 392 d048032c6ea0b6d697ddc2e86bda85a33adac920f1bf18e1b0c6d166a5cecdaf 393 seed: edwards25519 point generation seed (M) 395 N = 396 d3bfb518f44f3430f29d0c92af503865a1ed3281dc69b35dd868ba85f886c4ab 397 seed: edwards25519 point generation seed (N) 399 For edwards448: 401 M = 402 b6221038a775ecd007a4e4dde39fd76ae91d3cf0cc92be8f0c2fa6d6b66f9a12 403 942f5a92646109152292464f3e63d354701c7848d9fc3b8880 404 seed: edwards448 point generation seed (M) 406 N = 407 6034c65b66e4cd7a49b0edec3e3c9ccc4588afd8cf324e29f0a84a072531c4db 408 f97ff9af195ed714a689251f08f8e06e2d1f24a0ffc0146600 409 seed: edwards448 point generation seed (N) 411 7. Security Considerations 413 A security proof of SPAKE2 for prime order groups is found in [REF], 414 reducing the security of SPAKE2 to the gap Diffie-Hellman assumption. 415 Note that the choice of M and N is critical for the security proof. 416 The generation methods specified in this document are designed to 417 eliminate concerns related to knowing discrete logs of M and N. 419 Elements received from a peer MUST be checked for group membership: 420 failure to properly deserialize and validate group elements can lead 421 to attacks. An endpoint MUST abort the protocol if any received 422 public value is not a member of G. 424 The choices of random numbers MUST BE uniform. Randomly generated 425 values, e.g., x and y, MUST NOT be reused; such reuse violates the 426 security assumptions of the protocol and results in significant 427 insecurity. It is RECOMMENDED to generate these uniform numbers 428 using rejection sampling. 430 Some implementations of elliptic curve multiplication may leak 431 information about the length of the scalar. These MUST NOT be used. 432 All operations on elliptic curve points must take time independent of 433 the inputs. Hashing of the transcript may take time depending only 434 on the length of the transcript, but not the contents. 436 SPAKE2 does not support augmentation. As a result, the server has to 437 store a password equivalent. This is considered a significant 438 drawback in some use cases. Applications that need augmented PAKEs 439 should use [I-D.irtf-cfrg-opaque]. 441 The HMAC keys in this document are shorter than recommended in 442 [RFC8032]. This is appropriate as the difficulty of the discrete 443 logarithm problem is comparable with the difficulty of brute forcing 444 the keys. 446 8. IANA Considerations 448 No IANA action is required. 450 9. Acknowledgments 452 Special thanks to Nathaniel McCallum and Greg Hudson for generation 453 of M and N, and Chris Wood for test vectors. Thanks to Mike Hamburg 454 for advice on how to deal with cofactors. Greg Hudson also suggested 455 the addition of warnings on the reuse of x and y. Thanks to Fedor 456 Brunner, Adam Langley, Liliya Akhmetzyanova, and the members of the 457 CFRG for comments and advice. Thanks to Scott Fluhrer and those 458 Crypto Panel experts involved in the PAKE selection process 459 (https://github.com/cfrg/pake-selection) who have provided valuable 460 comments. Chris Wood contributed substantial text and reformatting 461 to address the excellent review comments from Kenny Paterson. 463 10. References 465 10.1. Normative References 467 [I-D.irtf-cfrg-hash-to-curve] 468 Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R., and 469 C. Wood, "Hashing to Elliptic Curves", Work in Progress, 470 Internet-Draft, draft-irtf-cfrg-hash-to-curve-05, 2 471 November 2019, . 474 [RFC2104] Krawczyk, H., Bellare, M., and R. Canetti, "HMAC: Keyed- 475 Hashing for Message Authentication", RFC 2104, 476 DOI 10.17487/RFC2104, February 1997, 477 . 479 [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate 480 Requirement Levels", BCP 14, RFC 2119, 481 DOI 10.17487/RFC2119, March 1997, 482 . 484 [RFC4493] Song, JH., Poovendran, R., Lee, J., and T. Iwata, "The 485 AES-CMAC Algorithm", RFC 4493, DOI 10.17487/RFC4493, June 486 2006, . 488 [RFC5480] Turner, S., Brown, D., Yiu, K., Housley, R., and T. Polk, 489 "Elliptic Curve Cryptography Subject Public Key 490 Information", RFC 5480, DOI 10.17487/RFC5480, March 2009, 491 . 493 [RFC5869] Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand 494 Key Derivation Function (HKDF)", RFC 5869, 495 DOI 10.17487/RFC5869, May 2010, 496 . 498 [RFC6234] Eastlake 3rd, D. and T. Hansen, "US Secure Hash Algorithms 499 (SHA and SHA-based HMAC and HKDF)", RFC 6234, 500 DOI 10.17487/RFC6234, May 2011, 501 . 503 [RFC7914] Percival, C. and S. Josefsson, "The scrypt Password-Based 504 Key Derivation Function", RFC 7914, DOI 10.17487/RFC7914, 505 August 2016, . 507 [RFC8032] Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital 508 Signature Algorithm (EdDSA)", RFC 8032, 509 DOI 10.17487/RFC8032, January 2017, 510 . 512 [RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 513 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, 514 May 2017, . 516 10.2. Informative References 518 [SEC1] Standards for Efficient Cryptography Group, "SEC 1: 519 Elliptic Curve Cryptography", May 2009. 521 [MNVAR] Abdalla, M., Barbosa, M., Bradley, T., Jarecki, S., Katz, 522 J., and J. Xu, "Universally Composable Relaxed Password 523 Authentication", August 2020. Appears in Micciancio D., 524 Ristenpart T. (eds) Advances in Cryptology -CRYPTO 2020. 525 Crypto 2020. Lecture notes in Computer Science volume 526 12170. Springer. 528 [REF] Abdalla, M. and D. Pointcheval, "Simple Password-Based 529 Encrypted Key Exchange Protocols.", February 2005. 530 Appears in A. Menezes, editor. Topics in Cryptography- 531 CT-RSA 2005, Volume 3376 of Lecture Notes in Computer 532 Science, pages 191-208, San Francisco, CA, US. Springer- 533 Verlag, Berlin, Germany. 535 [TDH] Cash, D., Kiltz, E., and V. Shoup, "The Twin-Diffie 536 Hellman Problem and Applications", 2008. EUROCRYPT 2008. 537 Volume 4965 of Lecture notes in Computer Science, pages 538 127-145. Springer-Verlag, Berlin, Germany. 540 [RFC8265] Saint-Andre, P. and A. Melnikov, "Preparation, 541 Enforcement, and Comparison of Internationalized Strings 542 Representing Usernames and Passwords", RFC 8265, 543 DOI 10.17487/RFC8265, October 2017, 544 . 546 [I-D.irtf-cfrg-opaque] 547 Krawczyk, H., Bourdrez, D., Lewi, K., and C. A. Wood, "The 548 OPAQUE Asymmetric PAKE Protocol", Work in Progress, 549 Internet-Draft, draft-irtf-cfrg-opaque-06, 12 July 2021, 550 . 553 Appendix A. Algorithm used for Point Generation 555 This section describes the algorithm that was used to generate the 556 points M and N in the table in Section 6. 558 For each curve in the table below, we construct a string using the 559 curve OID from [RFC5480] (as an ASCII string) or its name, combined 560 with the needed constant, e.g., "1.3.132.0.35 point generation seed 561 (M)" for P-521. This string is turned into a series of blocks by 562 hashing with SHA256, and hashing that output again to generate the 563 next 32 bytes, and so on. This pattern is repeated for each group 564 and value, with the string modified appropriately. 566 A byte string of length equal to that of an encoded group element is 567 constructed by concatenating as many blocks as are required, starting 568 from the first block, and truncating to the desired length. The byte 569 string is then formatted as required for the group. In the case of 570 Weierstrass curves, we take the desired length as the length for 571 representing a compressed point (section 2.3.4 of [SEC1]), and use 572 the low-order bit of the first byte as the sign bit. In order to 573 obtain the correct format, the value of the first byte is set to 0x02 574 or 0x03 (clearing the first six bits and setting the seventh bit), 575 leaving the sign bit as it was in the byte string constructed by 576 concatenating hash blocks. For the [RFC8032] curves a different 577 procedure is used. For edwards448 the 57-byte input has the least- 578 significant 7 bits of the last byte set to zero, and for edwards25519 579 the 32-byte input is not modified. For both the [RFC8032] curves the 580 (modified) input is then interpreted as the representation of the 581 group element. If this interpretation yields a valid group element 582 with the correct order (p), the (modified) byte string is the output. 583 Otherwise, the initial hash block is discarded and a new byte string 584 constructed from the remaining hash blocks. The procedure of 585 constructing a byte string of the appropriate length, formatting it 586 as required for the curve, and checking if it is a valid point of the 587 correct order, is repeated until a valid element is found. 589 The following python snippet generates the above points, assuming an 590 elliptic curve implementation following the interface of 591 Edwards25519Point.stdbase() and Edwards448Point.stdbase() in 592 Appendix A of [RFC8032]: 594 def iterated_hash(seed, n): 595 h = seed 596 for i in range(n): 597 h = hashlib.sha256(h).digest() 598 return h 600 def bighash(seed, start, sz): 601 n = -(-sz // 32) 602 hashes = [iterated_hash(seed, i) for i in range(start, start + n)] 603 return b''.join(hashes)[:sz] 605 def canon_pointstr(ecname, s): 606 if ecname == 'edwards25519': 607 return s 608 elif ecname == 'edwards448': 609 return s[:-1] + bytes([s[-1] & 0x80]) 610 else: 611 return bytes([(s[0] & 1) | 2]) + s[1:] 613 def gen_point(seed, ecname, ec): 614 for i in range(1, 1000): 615 hval = bighash(seed, i, len(ec.encode())) 616 pointstr = canon_pointstr(ecname, hval) 617 try: 618 p = ec.decode(pointstr) 619 if p != ec.zero_elem() and p * p.l() == ec.zero_elem(): 620 return pointstr, i 621 except Exception: 622 pass 624 Appendix B. Test Vectors 626 This section contains test vectors for SPAKE2 using the P256-SHA256- 627 HKDF-HMAC ciphersuite. (Choice of MHF is omitted and values for w, 628 x, and y are provided directly.) All points are encoded using the 629 uncompressed format, i.e., with a 0x04 octet prefix, specified in 630 [SEC1] A and B identity strings are provided in the protocol 631 invocation. 633 B.1. SPAKE2 Test Vectors 634 spake2: A='server', B='client' 635 w = 0x2ee57912099d31560b3a44b1184b9b4866e904c49d12ac5042c97dca461b1a5f 636 x = 0x43dd0fd7215bdcb482879fca3220c6a968e66d70b1356cac18bb26c84a78d729 637 pA = 0x04a56fa807caaa53a4d28dbb9853b9815c61a411118a6fe516a8798434751470 638 f9010153ac33d0d5f2047ffdb1a3e42c9b4e6be662766e1eeb4116988ede5f912c 639 y = 0xdcb60106f276b02606d8ef0a328c02e4b629f84f89786af5befb0bc75b6e66be 640 pB = 0x0406557e482bd03097ad0cbaa5df82115460d951e3451962f1eaf4367a420676 641 d09857ccbc522686c83d1852abfa8ed6e4a1155cf8f1543ceca528afb591a1e0b7 642 K = 0x0412af7e89717850671913e6b469ace67bd90a4df8ce45c2af19010175e37eed 643 69f75897996d539356e2fa6a406d528501f907e04d97515fbe83db277b715d3325 644 TT = 0x06000000000000007365727665720600000000000000636c69656e744100000 645 00000000004a56fa807caaa53a4d28dbb9853b9815c61a411118a6fe516a8798434751 646 470f9010153ac33d0d5f2047ffdb1a3e42c9b4e6be662766e1eeb4116988ede5f912c4 647 1000000000000000406557e482bd03097ad0cbaa5df82115460d951e3451962f1eaf43 648 67a420676d09857ccbc522686c83d1852abfa8ed6e4a1155cf8f1543ceca528afb591a 649 1e0b741000000000000000412af7e89717850671913e6b469ace67bd90a4df8ce45c2a 650 f19010175e37eed69f75897996d539356e2fa6a406d528501f907e04d97515fbe83db2 651 77b715d332520000000000000002ee57912099d31560b3a44b1184b9b4866e904c49d1 652 2ac5042c97dca461b1a5f 653 Hash(TT) = 0x0e0672dc86f8e45565d338b0540abe6915bdf72e2b35b5c9e5663168e960a91b 654 Ke = 0x0e0672dc86f8e45565d338b0540abe69 655 Ka = 0x15bdf72e2b35b5c9e5663168e960a91b 656 KcA = 0x00c12546835755c86d8c0db7851ae86f 657 KcB = 0xa9fa3406c3b781b93d804485430ca27a 658 A conf = 0x58ad4aa88e0b60d5061eb6b5dd93e80d9c4f00d127c65b3b35b1b5281fee38f0 659 B conf = 0xd3e2e547f1ae04f2dbdbf0fc4b79f8ecff2dff314b5d32fe9fcef2fb26dc459b 661 spake2: A='', B='client' 662 w = 0x0548d8729f730589e579b0475a582c1608138ddf7054b73b5381c7e883e2efae 663 x = 0x403abbe3b1b4b9ba17e3032849759d723939a27a27b9d921c500edde18ed654b 664 pA = 0x04a897b769e681c62ac1c2357319a3d363f610839c4477720d24cbe32f5fd85f 665 44fb92ba966578c1b712be6962498834078262caa5b441ecfa9d4a9485720e918a 666 y = 0x903023b6598908936ea7c929bd761af6039577a9c3f9581064187c3049d87065 667 pB = 0x04e0f816fd1c35e22065d5556215c097e799390d16661c386e0ecc84593974a6 668 1b881a8c82327687d0501862970c64565560cb5671f696048050ca66ca5f8cc7fc 669 K = 0x048f83ec9f6e4f87cc6f9dc740bdc2769725f923364f01c84148c049a39a735e 670 bda82eac03e00112fd6a5710682767cff5361f7e819e53d8d3c3a2922e0d837aa6 671 TT = 0x00000000000000000600000000000000636c69656e74410000000000000004a 672 897b769e681c62ac1c2357319a3d363f610839c4477720d24cbe32f5fd85f44fb92ba9 673 66578c1b712be6962498834078262caa5b441ecfa9d4a9485720e918a4100000000000 674 00004e0f816fd1c35e22065d5556215c097e799390d16661c386e0ecc84593974a61b8 675 81a8c82327687d0501862970c64565560cb5671f696048050ca66ca5f8cc7fc4100000 676 000000000048f83ec9f6e4f87cc6f9dc740bdc2769725f923364f01c84148c049a39a7 677 35ebda82eac03e00112fd6a5710682767cff5361f7e819e53d8d3c3a2922e0d837aa62 678 0000000000000000548d8729f730589e579b0475a582c1608138ddf7054b73b5381c7e 679 883e2efae 680 Hash(TT) = 0x642f05c473c2cd79909f9a841e2f30a70bf89b18180af97353ba198789c2b963 681 Ke = 0x642f05c473c2cd79909f9a841e2f30a7 682 Ka = 0x0bf89b18180af97353ba198789c2b963 683 KcA = 0xc6be376fc7cd1301fd0a13adf3e7bffd 684 KcB = 0xb7243f4ae60440a49b3f8cab3c1fba07 685 A conf = 0x47d29e6666af1b7dd450d571233085d7a9866e4d49d2645e2df975489521232b 686 B conf = 0x3313c5cefc361d27fb16847a91c2a73b766ffa90a4839122a9b70a2f6bd1d6df 688 spake2: A='server', B='' 689 w = 0x626e0cdc7b14c9db3e52a0b1b3a768c98e37852d5db30febe0497b14eae8c254 690 x = 0x07adb3db6bc623d3399726bfdbfd3d15a58ea776ab8a308b00392621291f9633 691 pA = 0x04f88fb71c99bfffaea370966b7eb99cd4be0ff1a7d335caac4211c4afd855e2 692 e15a873b298503ad8ba1d9cbb9a392d2ba309b48bfd7879aefd0f2cea6009763b0 693 y = 0xb6a4fc8dbb629d4ba51d6f91ed1532cf87adec98f25dd153a75accafafedec16 694 pB = 0x040c269d6be017dccb15182ac6bfcd9e2a14de019dd587eaf4bdfd353f031101 695 e7cca177f8eb362a6e83e7d5e729c0732e1b528879c086f39ba0f31a9661bd34db 696 K = 0x0445ee233b8ecb51ebd6e7da3f307e88a1616bae2166121221fdc0dadb986afa 697 f3ec8a988dc9c626fa3b99f58a7ca7c9b844bb3e8dd9554aafc5b53813504c1cbe 698 TT = 0x06000000000000007365727665720000000000000000410000000000000004f 699 88fb71c99bfffaea370966b7eb99cd4be0ff1a7d335caac4211c4afd855e2e15a873b2 700 98503ad8ba1d9cbb9a392d2ba309b48bfd7879aefd0f2cea6009763b04100000000000 701 000040c269d6be017dccb15182ac6bfcd9e2a14de019dd587eaf4bdfd353f031101e7c 702 ca177f8eb362a6e83e7d5e729c0732e1b528879c086f39ba0f31a9661bd34db4100000 703 0000000000445ee233b8ecb51ebd6e7da3f307e88a1616bae2166121221fdc0dadb986 704 afaf3ec8a988dc9c626fa3b99f58a7ca7c9b844bb3e8dd9554aafc5b53813504c1cbe2 705 000000000000000626e0cdc7b14c9db3e52a0b1b3a768c98e37852d5db30febe0497b1 706 4eae8c254 707 Hash(TT) = 0x005184ff460da2ce59062c87733c299c3521297d736598fc0a1127600efa1afb 708 Ke = 0x005184ff460da2ce59062c87733c299c 709 Ka = 0x3521297d736598fc0a1127600efa1afb 710 KcA = 0xf3da53604f0aeecea5a33be7bddf6edf 711 KcB = 0x9e3f86848736f159bd92b6e107ec6799 712 A conf = 0xbc9f9bbe99f26d0b2260e6456e05a86196a3307ec6663a18bf6ac825736533b2 713 B conf = 0xc2370e1bf813b086dff0d834e74425a06e6390f48f5411900276dcccc5a297ec 715 spake2: A='', B='' 716 w = 0x7bf46c454b4c1b25799527d896508afd5fc62ef4ec59db1efb49113063d70cca 717 x = 0x8cef65df64bb2d0f83540c53632de911b5b24b3eab6cc74a97609fd659e95473 718 pA = 0x04a65b367a3f613cf9f0654b1b28a1e3a8a40387956c8ba6063e8658563890f4 719 6ca1ef6a676598889fc28de2950ab8120b79a5ef1ea4c9f44bc98f585634b46d66 720 y = 0xd7a66f64074a84652d8d623a92e20c9675c61cb5b4f6a0063e4648a2fdc02d53 721 pB = 0x04589f13218822710d98d8b2123a079041052d9941b9cf88c6617ddb2fcc0494 722 662eea8ba6b64692dc318250030c6af045cb738bc81ba35b043c3dcb46adf6f58d 723 K = 0x041a3c03d51b452537ca2a1fea6110353c6d5ed483c4f0f86f4492ca3f378d40 724 a994b4477f93c64d928edbbcd3e85a7c709b7ea73ee97986ce3d1438e135543772 725 TT = 0x00000000000000000000000000000000410000000000000004a65b367a3f613 726 cf9f0654b1b28a1e3a8a40387956c8ba6063e8658563890f46ca1ef6a676598889fc28 727 de2950ab8120b79a5ef1ea4c9f44bc98f585634b46d66410000000000000004589f132 728 18822710d98d8b2123a079041052d9941b9cf88c6617ddb2fcc0494662eea8ba6b6469 729 2dc318250030c6af045cb738bc81ba35b043c3dcb46adf6f58d4100000000000000041 730 a3c03d51b452537ca2a1fea6110353c6d5ed483c4f0f86f4492ca3f378d40a994b4477 731 f93c64d928edbbcd3e85a7c709b7ea73ee97986ce3d1438e1355437722000000000000 732 0007bf46c454b4c1b25799527d896508afd5fc62ef4ec59db1efb49113063d70cca 733 Hash(TT) = 0xfc6374762ba5cf11f4b2caa08b2cd1b9907ae0e26e8d6234318d91583cd74c86 734 Ke = 0xfc6374762ba5cf11f4b2caa08b2cd1b9 735 Ka = 0x907ae0e26e8d6234318d91583cd74c86 736 KcA = 0x5dbd2f477166b7fb6d61febbd77a5563 737 KcB = 0x7689b4654407a5faeffdc8f18359d8a3 738 A conf = 0xdfb4db8d48ae5a675963ea5e6c19d98d4ea028d8e898dad96ea19a80ade95dca 739 B conf = 0xd0f0609d1613138d354f7e95f19fb556bf52d751947241e8c7118df5ef0ae175 741 Authors' Addresses 743 Watson Ladd 744 Sealance 746 Email: watsonbladd@gmail.com 748 Benjamin Kaduk (editor) 749 Akamai Technologies 751 Email: kaduk@mit.edu