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Kaduk, Ed. 5 Expires: March 13, 2021 Akamai 6 September 9, 2020 8 SPAKE2, a PAKE 9 draft-irtf-cfrg-spake2-13 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 with no risk of 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. 19 Status of This Memo 21 This Internet-Draft is submitted in full conformance with the 22 provisions of BCP 78 and BCP 79. 24 Internet-Drafts are working documents of the Internet Engineering 25 Task Force (IETF). Note that other groups may also distribute 26 working documents as Internet-Drafts. The list of current Internet- 27 Drafts is at https://datatracker.ietf.org/drafts/current/. 29 Internet-Drafts are draft documents valid for a maximum of six months 30 and may be updated, replaced, or obsoleted by other documents at any 31 time. It is inappropriate to use Internet-Drafts as reference 32 material or to cite them other than as "work in progress." 34 This Internet-Draft will expire on March 13, 2021. 36 Copyright Notice 38 Copyright (c) 2020 IETF Trust and the persons identified as the 39 document authors. All rights reserved. 41 This document is subject to BCP 78 and the IETF Trust's Legal 42 Provisions Relating to IETF Documents 43 (https://trustee.ietf.org/license-info) in effect on the date of 44 publication of this document. Please review these documents 45 carefully, as they describe your rights and restrictions with respect 46 to this document. Code Components extracted from this document must 47 include Simplified BSD License text as described in Section 4.e of 48 the Trust Legal Provisions and are provided without warranty as 49 described in the Simplified BSD License. 51 Table of Contents 53 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2 54 2. Requirements Notation . . . . . . . . . . . . . . . . . . . . 2 55 3. Definition of SPAKE2 . . . . . . . . . . . . . . . . . . . . 2 56 4. Key Schedule and Key Confirmation . . . . . . . . . . . . . . 5 57 5. Per-User M and N . . . . . . . . . . . . . . . . . . . . . . 6 58 6. Ciphersuites . . . . . . . . . . . . . . . . . . . . . . . . 6 59 7. Security Considerations . . . . . . . . . . . . . . . . . . . 9 60 8. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 9 61 9. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 9 62 10. References . . . . . . . . . . . . . . . . . . . . . . . . . 9 63 Appendix A. Algorithm used for Point Generation . . . . . . . . 11 64 Appendix B. Test Vectors . . . . . . . . . . . . . . . . . . . . 13 65 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 16 67 1. Introduction 69 This document describes SPAKE2, a means for two parties that share a 70 password to derive a strong shared key with no risk of disclosing the 71 password. This password-based key exchange protocol is compatible 72 with any group (requiring only a scheme to map a random input of 73 fixed length per group to a random group element), is computationally 74 efficient, and has a security proof. Predetermined parameters for a 75 selection of commonly used groups are also provided for use by other 76 protocols. 78 2. Requirements Notation 80 The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", 81 "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and 82 "OPTIONAL" in this document are to be interpreted as described in BCP 83 14 [RFC2119] [RFC8174] when, and only when, they appear in all 84 capitals, as shown here. 86 3. Definition of SPAKE2 88 3.1. Setup 90 Let G be a group in which the gap Diffie-Hellman (CDH) problem is 91 hard. Suppose G has order p*h where p is a large prime; h will be 92 called the cofactor. Let I be the unit element in G, e.g., the point 93 at infinity if G is an elliptic curve group. We denote the 94 operations in the group additively. We assume there is a 95 representation of elements of G as byte strings: common choices would 96 be SEC1 [SEC1] uncompressed or compressed for elliptic curve groups 97 or big endian integers of a fixed (per-group) length for prime field 98 DH. We fix two elements M and N in the prime-order subgroup of G as 99 defined in the table in this document for common groups, as well as a 100 generator P of the (large) prime-order subgroup of G. In the case of 101 a composite order group we will work in the quotient group. P is 102 specified in the document defining the group, and so we do not repeat 103 it here. 105 || denotes concatenation of strings. We also let len(S) denote the 106 length of a string in bytes, represented as an eight-byte little- 107 endian number. Finally, let nil represent an empty string, i.e., 108 len(nil) = 0. 110 KDF is a key-derivation function that takes as input a salt, 111 intermediate keying material (IKM), info string, and derived key 112 length L to derive a cryptographic key of length L. MAC is a Message 113 Authentication Code algorithm that takes a secret key and message as 114 input to produce an output. Let Hash be a hash function from 115 arbitrary strings to bit strings of a fixed length. Common choices 116 for H are SHA256 or SHA512 [RFC6234]. Let MHF be a memory-hard hash 117 function designed to slow down brute-force attackers. Scrypt 118 [RFC7914] is a common example of this function. The output length of 119 MHF matches that of Hash. Parameter selection for MHF is out of 120 scope for this document. Section 6 specifies variants of KDF, MAC, 121 and Hash suitable for use with the protocols contained herein. 123 Let A and B be two parties. A and B may also have digital 124 representations of the parties' identities such as Media Access 125 Control addresses or other names (hostnames, usernames, etc). A and 126 B may share Additional Authenticated Data (AAD) of length at most 127 2^16 - 1 bits that is separate from their identities which they may 128 want to include in the protocol execution. One example of AAD is a 129 list of supported protocol versions if SPAKE2(+) were used in a 130 higher-level protocol which negotiates use of a particular PAKE. 131 Including this list would ensure that both parties agree upon the 132 same set of supported protocols and therefore prevent downgrade 133 attacks. We also assume A and B share an integer w; typically w = 134 MHF(pw) mod p, for a user-supplied password pw. Standards such as 135 NIST.SP.800-56Ar3 suggest taking mod p of a hash value that is 64 136 bits longer than that needed to represent p to remove statistical 137 bias introduced by the modulation. Protocols using this 138 specification must define the method used to compute w: it may be 139 necessary to carry out various forms of normalization of the password 140 before hashing [RFC8265]. The hashing algorithm SHOULD be a MHF so 141 as to slow down brute-force attackers. 143 3.2. Protocol Flow 145 SPAKE2 is a one round protocol to establish a shared secret with an 146 additional round for key confirmation. Prior to invocation, A and B 147 are provisioned with information such as the input password needed to 148 run the protocol. During the first round, A sends a public share pA 149 to B, and B responds with its own public share pB. Both A and B then 150 derive a shared secret used to produce encryption and authentication 151 keys. The latter are used during the second round for key 152 confirmation. (Section 4 details the key derivation and confirmation 153 steps.) In particular, A sends a key confirmation message cA to B, 154 and B responds with its own key confirmation message cB. Both 155 parties MUST NOT consider the protocol complete prior to receipt and 156 validation of these key confirmation messages. 158 This sample trace is shown below. 160 A B 161 | (setup protocol) | 162 (compute pA) | pA | 163 |----------------->| 164 | pB | (compute pB) 165 |<-----------------| 166 | | 167 | (derive secrets) | 168 (compute cA) | cA | 169 |----------------->| 170 | cB | (compute cB) 171 |<-----------------| 173 3.3. SPAKE2 175 To begin, A picks x randomly and uniformly from the integers in 176 [0,p), and calculates X=x*P and T=w*M+X, then transmits pA=T to B. 178 B selects y randomly and uniformly from the integers in [0,p), and 179 calculates Y=y*P, S=w*N+Y, then transmits pB=S to A. 181 Both A and B calculate a group element K. A calculates it as 182 h*x*(S-w*N), while B calculates it as h*y*(T-w*M). A knows S because 183 it has received it, and likewise B knows T. The multiplication by h 184 prevents small subgroup confinement attacks by computing a unique 185 value in the quotient group. This is a common mitigation against 186 this kind of attack. 188 K is a shared value, though it MUST NOT be used as a shared secret. 189 Both A and B must derive two shared secrets from the protocol 190 transcript. This prevents man-in-the-middle attackers from inserting 191 themselves into the exchange. The transcript TT is encoded as 192 follows: 194 TT = len(A) || A 195 || len(B) || B 196 || len(S) || S 197 || len(T) || T 198 || len(K) || K 199 || len(w) || w 201 If an identity is absent, it is encoded as a zero-length string. 202 This must only be done for applications in which identities are 203 implicit. Otherwise, the protocol risks Unknown Key Share attacks 204 (discussion of Unknown Key Share attacks in a specific protocol is 205 given in [I-D.ietf-mmusic-sdp-uks]). 207 Upon completion of this protocol, A and B compute shared secrets Ke, 208 KcA, and KcB as specified in Section 4. A MUST send B a key 209 confirmation message so both parties agree upon these shared secrets. 210 This confirmation message F is computed as a MAC over the protocol 211 transcript TT using KcA, as follows: F = MAC(KcA, TT). Similarly, B 212 MUST send A a confirmation message using a MAC computed equivalently 213 except with the use of KcB. Key confirmation verification requires 214 computing F and checking for equality against that which was 215 received. 217 4. Key Schedule and Key Confirmation 219 The protocol transcript TT, as defined in Section Section 3.3, is 220 unique and secret to A and B. Both parties use TT to derive shared 221 symmetric secrets Ke and Ka as Ke || Ka = Hash(TT), with |Ke| = |Ka|. 222 The length of each key is equal to half of the digest output, e.g., 223 128 bits for SHA-256. 225 Both endpoints use Ka to derive subsequent MAC keys for key 226 confirmation messages. Specifically, let KcA and KcB be the MAC keys 227 used by A and B, respectively. A and B compute them as KcA || KcB = 228 KDF(nil, Ka, "ConfirmationKeys" || AAD), where AAD is the associated 229 data each given to each endpoint, or nil if none was provided. The 230 length of each of KcA and KcB is equal to half of the KDF output, 231 e.g., |KcA| = |KcB| = 128 bits for HKDF(SHA256). 233 The resulting key schedule for this protocol, given transcript TT and 234 additional associated data AAD, is as follows. 236 TT -> Hash(TT) = Ka || Ke 237 AAD -> KDF(nil, Ka, "ConfirmationKeys" || AAD) = KcA || KcB 239 A and B output Ke as the shared secret from the protocol. Ka and its 240 derived keys are not used for anything except key confirmation. 242 5. Per-User M and N 244 To avoid concerns that an attacker needs to solve a single ECDH 245 instance to break the authentication of SPAKE2, a variant based on 246 using [I-D.irtf-cfrg-hash-to-curve] is also presented. In this 247 variant, M and N are computed as follows: 249 M = h2c(Hash("M for SPAKE2" || len(A) || A || len(B) || B)) 250 N = h2c(Hash("N for SPAKE2" || len(A) || A || len(B) || B)) 252 In addition M and N may be equal to have a symmetric variant. The 253 security of these variants is examined in [MNVAR]. 255 6. Ciphersuites 257 This section documents SPAKE2 ciphersuite configurations. A 258 ciphersuite indicates a group, cryptographic hash algorithm, and pair 259 of KDF and MAC functions, e.g., SPAKE2-P256-SHA256-HKDF-HMAC. This 260 ciphersuite indicates a SPAKE2 protocol instance over P-256 that uses 261 SHA256 along with HKDF [RFC5869] and HMAC [RFC2104] for G, Hash, KDF, 262 and MAC functions, respectively. 264 +------------------+---------------+-------------+------------------+ 265 | G | Hash | KDF | MAC | 266 +------------------+---------------+-------------+------------------+ 267 | P-256 | SHA256 | HKDF | HMAC [RFC2104] | 268 | | [RFC6234] | [RFC5869] | | 269 | | | | | 270 | P-256 | SHA512 | HKDF | HMAC [RFC2104] | 271 | | [RFC6234] | [RFC5869] | | 272 | | | | | 273 | P-384 | SHA256 | HKDF | HMAC [RFC2104] | 274 | | [RFC6234] | [RFC5869] | | 275 | | | | | 276 | P-384 | SHA512 | HKDF | HMAC [RFC2104] | 277 | | [RFC6234] | [RFC5869] | | 278 | | | | | 279 | P-512 | SHA512 | HKDF | HMAC [RFC2104] | 280 | | [RFC6234] | [RFC5869] | | 281 | | | | | 282 | edwards25519 | SHA256 | HKDF | HMAC [RFC2104] | 283 | [RFC7748] | [RFC6234] | [RFC5869] | | 284 | | | | | 285 | edwards448 | SHA512 | HKDF | HMAC [RFC2104] | 286 | [RFC7748] | [RFC6234] | [RFC5869] | | 287 | | | | | 288 | P-256 | SHA256 | HKDF | CMAC-AES-128 | 289 | | [RFC6234] | [RFC5869] | [RFC4493] | 290 | | | | | 291 | P-256 | SHA512 | HKDF | CMAC-AES-128 | 292 | | [RFC6234] | [RFC5869] | [RFC4493] | 293 +------------------+---------------+-------------+------------------+ 295 Table 1: SPAKE2 Ciphersuites 297 The following points represent permissible point generation seeds for 298 the groups listed in the Table Table 1, using the algorithm presented 299 in Appendix A. These bytestrings are compressed points as in [SEC1] 300 for curves from [SEC1]. 302 For P256: 304 M = 305 02886e2f97ace46e55ba9dd7242579f2993b64e16ef3dcab95afd497333d8fa12f 306 seed: 1.2.840.10045.3.1.7 point generation seed (M) 308 N = 309 03d8bbd6c639c62937b04d997f38c3770719c629d7014d49a24b4f98baa1292b49 310 seed: 1.2.840.10045.3.1.7 point generation seed (N) 311 For P384: 313 M = 314 030ff0895ae5ebf6187080a82d82b42e2765e3b2f8749c7e05eba366434b363d3dc 315 36f15314739074d2eb8613fceec2853 316 seed: 1.3.132.0.34 point generation seed (M) 318 N = 319 02c72cf2e390853a1c1c4ad816a62fd15824f56078918f43f922ca21518f9c543bb 320 252c5490214cf9aa3f0baab4b665c10 321 seed: 1.3.132.0.34 point generation seed (N) 323 For P521: 325 M = 326 02003f06f38131b2ba2600791e82488e8d20ab889af753a41806c5db18d37d85608 327 cfae06b82e4a72cd744c719193562a653ea1f119eef9356907edc9b56979962d7aa 328 seed: 1.3.132.0.35 point generation seed (M) 330 N = 331 0200c7924b9ec017f3094562894336a53c50167ba8c5963876880542bc669e494b25 332 32d76c5b53dfb349fdf69154b9e0048c58a42e8ed04cef052a3bc349d95575cd25 333 seed: 1.3.132.0.35 point generation seed (N) 335 For edwards25519: 337 M = 338 d048032c6ea0b6d697ddc2e86bda85a33adac920f1bf18e1b0c6d166a5cecdaf 339 seed: edwards25519 point generation seed (M) 341 N = 342 d3bfb518f44f3430f29d0c92af503865a1ed3281dc69b35dd868ba85f886c4ab 343 seed: edwards25519 point generation seed (N) 345 For edwards448: 347 M = 348 b6221038a775ecd007a4e4dde39fd76ae91d3cf0cc92be8f0c2fa6d6b66f9a12 349 942f5a92646109152292464f3e63d354701c7848d9fc3b8880 350 seed: edwards448 point generation seed (M) 352 N = 353 6034c65b66e4cd7a49b0edec3e3c9ccc4588afd8cf324e29f0a84a072531c4db 354 f97ff9af195ed714a689251f08f8e06e2d1f24a0ffc0146600 355 seed: edwards448 point generation seed (N) 357 7. Security Considerations 359 A security proof of SPAKE2 for prime order groups is found in [REF], 360 reducing the security of SPAKE2 to the gap Diffie-Hellman assumption. 361 Note that the choice of M and N is critical for the security proof. 362 The generation methods specified in this document is designed to 363 eliminate concerns related to knowing discrete logs of M and N. 365 Elements received from a peer MUST be checked for group membership: 366 failure to properly validate group elements can lead to attacks. It 367 is essential that endpoints verify received points are members of G. 369 The choices of random numbers MUST BE uniform. Randomly generated 370 values (e.g., x and y) MUST NOT be reused; such reuse may permit 371 dictionary attacks on the password. 373 SPAKE2 does not support augmentation. As a result, the server has to 374 store a password equivalent. This is considered a significant 375 drawback in some use cases 377 8. IANA Considerations 379 No IANA action is required. 381 9. Acknowledgments 383 Special thanks to Nathaniel McCallum and Greg Hudson for generation 384 of test vectors. Thanks to Mike Hamburg for advice on how to deal 385 with cofactors. Greg Hudson also suggested the addition of warnings 386 on the reuse of x and y. Thanks to Fedor Brunner, Adam Langley, and 387 the members of the CFRG for comments and advice. Chris Wood 388 contributed substantial text and reformatting to address the 389 excellent review comments from Kenny Paterson. 391 10. References 393 10.1. Normative References 395 [I-D.irtf-cfrg-hash-to-curve] 396 Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R., and 397 C. Wood, "Hashing to Elliptic Curves", draft-irtf-cfrg- 398 hash-to-curve-05 (work in progress), November 2019. 400 [RFC2104] Krawczyk, H., Bellare, M., and R. Canetti, "HMAC: Keyed- 401 Hashing for Message Authentication", RFC 2104, 402 DOI 10.17487/RFC2104, February 1997, 403 . 405 [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate 406 Requirement Levels", BCP 14, RFC 2119, 407 DOI 10.17487/RFC2119, March 1997, 408 . 410 [RFC4493] Song, JH., Poovendran, R., Lee, J., and T. Iwata, "The 411 AES-CMAC Algorithm", RFC 4493, DOI 10.17487/RFC4493, June 412 2006, . 414 [RFC5480] Turner, S., Brown, D., Yiu, K., Housley, R., and T. Polk, 415 "Elliptic Curve Cryptography Subject Public Key 416 Information", RFC 5480, DOI 10.17487/RFC5480, March 2009, 417 . 419 [RFC5869] Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand 420 Key Derivation Function (HKDF)", RFC 5869, 421 DOI 10.17487/RFC5869, May 2010, 422 . 424 [RFC6234] Eastlake 3rd, D. and T. Hansen, "US Secure Hash Algorithms 425 (SHA and SHA-based HMAC and HKDF)", RFC 6234, 426 DOI 10.17487/RFC6234, May 2011, 427 . 429 [RFC7748] Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves 430 for Security", RFC 7748, DOI 10.17487/RFC7748, January 431 2016, . 433 [RFC7914] Percival, C. and S. Josefsson, "The scrypt Password-Based 434 Key Derivation Function", RFC 7914, DOI 10.17487/RFC7914, 435 August 2016, . 437 [RFC8032] Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital 438 Signature Algorithm (EdDSA)", RFC 8032, 439 DOI 10.17487/RFC8032, January 2017, 440 . 442 [RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 443 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, 444 May 2017, . 446 10.2. Informative References 448 [I-D.ietf-mmusic-sdp-uks] 449 Thomson, M. and E. Rescorla, "Unknown Key Share Attacks on 450 uses of TLS with the Session Description Protocol (SDP)", 451 draft-ietf-mmusic-sdp-uks-07 (work in progress), August 452 2019. 454 [MNVAR] Abdalla, M. and M. Barbosa, "Perfect Forward Security of 455 SPAKE2", Oct 2019. 457 IACR eprint 2019/1194 459 [REF] Abdalla, M. and D. Pointcheval, "Simple Password-Based 460 Encrypted Key Exchange Protocols.", Feb 2005. 462 Appears in A. Menezes, editor. Topics in Cryptography- 463 CT-RSA 2005, Volume 3376 of Lecture Notes in Computer 464 Science, pages 191-208, San Francisco, CA, US. Springer- 465 Verlag, Berlin, Germany. 467 [RFC8265] Saint-Andre, P. and A. Melnikov, "Preparation, 468 Enforcement, and Comparison of Internationalized Strings 469 Representing Usernames and Passwords", RFC 8265, 470 DOI 10.17487/RFC8265, October 2017, 471 . 473 [SEC1] Standards for Efficient Cryptography Group, "SEC 1: 474 Elliptic Curve Cryptography", May 2009. 476 [TDH] Cash, D., Kiltz, E., and V. Shoup, "The Twin-Diffie 477 Hellman Problem and Applications", 2008. 479 EUROCRYPT 2008. Volume 4965 of Lecture notes in Computer 480 Science, pages 127-145. Springer-Verlag, Berlin, Germany. 482 Appendix A. Algorithm used for Point Generation 484 This section describes the algorithm that was used to generate the 485 points (M) and (N) in the table in Section 6. 487 For each curve in the table below, we construct a string using the 488 curve OID from [RFC5480] (as an ASCII string) or its name, combined 489 with the needed constant, for instance "1.3.132.0.35 point generation 490 seed (M)" for P-512. This string is turned into a series of blocks 491 by hashing with SHA256, and hashing that output again to generate the 492 next 32 bytes, and so on. This pattern is repeated for each group 493 and value, with the string modified appropriately. 495 A byte string of length equal to that of an encoded group element is 496 constructed by concatenating as many blocks as are required, starting 497 from the first block, and truncating to the desired length. The byte 498 string is then formatted as required for the group. In the case of 499 Weierstrass curves, we take the desired length as the length for 500 representing a compressed point (section 2.3.4 of [SEC1]), and use 501 the low-order bit of the first byte as the sign bit. In order to 502 obtain the correct format, the value of the first byte is set to 0x02 503 or 0x03 (clearing the first six bits and setting the seventh bit), 504 leaving the sign bit as it was in the byte string constructed by 505 concatenating hash blocks. For the [RFC8032] curves a different 506 procedure is used. For edwards448 the 57-byte input has the least- 507 significant 7 bits of the last byte set to zero, and for edwards25519 508 the 32-byte input is not modified. For both the [RFC8032] curves the 509 (modified) input is then interpreted as the representation of the 510 group element. If this interpretation yields a valid group element 511 with the correct order (p), the (modified) byte string is the output. 512 Otherwise, the initial hash block is discarded and a new byte string 513 constructed from the remaining hash blocks. The procedure of 514 constructing a byte string of the appropriate length, formatting it 515 as required for the curve, and checking if it is a valid point of the 516 correct order, is repeated until a valid element is found. 518 The following python snippet generates the above points, assuming an 519 elliptic curve implementation following the interface of 520 Edwards25519Point.stdbase() and Edwards448Point.stdbase() in 521 Appendix A of [RFC8032]: 523 def iterated_hash(seed, n): 524 h = seed 525 for i in range(n): 526 h = hashlib.sha256(h).digest() 527 return h 529 def bighash(seed, start, sz): 530 n = -(-sz // 32) 531 hashes = [iterated_hash(seed, i) for i in range(start, start + n)] 532 return b''.join(hashes)[:sz] 534 def canon_pointstr(ecname, s): 535 if ecname == 'edwards25519': 536 return s 537 elif ecname == 'edwards448': 538 return s[:-1] + bytes([s[-1] & 0x80]) 539 else: 540 return bytes([(s[0] & 1) | 2]) + s[1:] 542 def gen_point(seed, ecname, ec): 543 for i in range(1, 1000): 544 hval = bighash(seed, i, len(ec.encode())) 545 pointstr = canon_pointstr(ecname, hval) 546 try: 547 p = ec.decode(pointstr) 548 if p != ec.zero_elem() and p * p.l() == ec.zero_elem(): 549 return pointstr, i 550 except Exception: 551 pass 553 Appendix B. Test Vectors 555 This section contains test vectors for SPAKE2 using the P256-SHA256- 556 HKDF-HMAC ciphersuite. (Choice of MHF is omitted and values for w 557 and w0,w1 are provided directly.) All points are encoded using the 558 uncompressed format, i.e., with a 0x04 octet prefix, specified in 559 [SEC1] A and B identity strings are provided in the protocol 560 invocation. 562 B.1. SPAKE2 Test Vectors 564 SPAKE2(A='client', B='server') 565 w = 0x7741cf8c80b9bee583abac3d38daa6b807fed38b06580cb75ee85319d25fed 566 e6 567 X = 0x04ac6827b3a9110d1e663bcd4f5de668da34a9f45e464e99067bbea53f1ed4 568 d8abbdd234c05b3a3a8a778ee47f244cca1a79acb7052d5e58530311a9af077ba179 569 T = 0x04e02acfbbfb081fc38b5bab999b5e25a5ffd0b1ac48eae24fcc8e49ac5e0d 570 8a790914419a100e205605f9862daa848e99cea455263f0c6e06bc5a911f3e10a16b 571 Y = 0x0413c45ab093a75c4b2a6e71f957eec3859807858325258b0fa43df5a6efd2 572 63c59b9c1fbfd55bc5e75fd3e7ba8af6799a99b225fe6c30e6c2a2e0ab4962136ba8 573 S = 0x047aad50ba7bd6a5eacbead7689f7146f1a4219fa071cce1755f80280cc6c3 574 a5a73cf469f2a294a0b74a5c07054585ccd447f3f633d8631f3bf43442449e9efeba 575 TT = 0x0600000000000000636c69656e74060000000000000073657276657241000 576 00000000000047aad50ba7bd6a5eacbead7689f7146f1a4219fa071cce1755f80280 577 cc6c3a5a73cf469f2a294a0b74a5c07054585ccd447f3f633d8631f3bf43442449e9 578 efeba410000000000000004e02acfbbfb081fc38b5bab999b5e25a5ffd0b1ac48eae 579 24fcc8e49ac5e0d8a790914419a100e205605f9862daa848e99cea455263f0c6e06b 580 c5a911f3e10a16b410000000000000004d01fc08bbae9b6abe2f4d6893cc9f810433 581 2e19be5f5881c6b9f077e1feff55023da74db65fae320fad8f0dd38e1323f5336f3f 582 53c9c9dec06710f18f556bd2020000000000000007741cf8c80b9bee583abac3d38d 583 aa6b807fed38b06580cb75ee85319d25fede6 584 Ka = 0x2b5e350c58d530c3586f75bf2a155c4b 585 Ke = 0x238509f7adf0dc72500b2d1315737a27 586 KcA = 0xc33d2ef8e37a7e545c14c7fcfdc9db94 587 KcB = 0x18a81cec7eb83416db6615cb3bc03fcb 588 MAC(A) = 0x29e9a63d243f2f0db5532d2eb0dbaa617803b85feb31566d0cb9457e3 589 03bcfa6 590 MAC(B) = 0x487e4cbe98b6287272d043e169a19b6c4682d0481c92f53f1ee03d4b8 591 6c3f43e 593 SPAKE2(A='client', B='') 594 w = 0x7741cf8c80b9bee583abac3d38daa6b807fed38b06580cb75ee85319d25fed 595 e6 596 X = 0x048b5d7b44b02c4c868f4486ec55bd2380ec34cd5fa5dbff1079a79097e305 597 0b34fa91272331729357c86cbb30d371e252dc915aeaa314921b1f09f74816f96a12 598 T = 0x04839f44931b88d12769e601d0ec480b6c9ea95e70ba361ba14bf513e5186a 599 6c302e6f409bd01f1030ad3cdac1e08965217e430ca7f9bce698111ae8a4d0530efd 600 Y = 0x0446419d63037d0bbaca224f89987c776bfea2e0913ccda0790079212f476d 601 6fd1ec997a02821a804f885e4f29b172b27c92251d883efe201cae106c239108c0c7 602 S = 0x042926b2dbcc5d0cb23ca123cc4133242f2998439af5380434a4bd5fd76fbb 603 c030b5563218d0184fa3fd303482a679c9555ccea41098b26b6ee16fe35c792b1fda 604 TT = 0x0600000000000000636c69656e744100000000000000042926b2dbcc5d0cb 605 23ca123cc4133242f2998439af5380434a4bd5fd76fbbc030b5563218d0184fa3fd3 606 03482a679c9555ccea41098b26b6ee16fe35c792b1fda410000000000000004839f4 607 4931b88d12769e601d0ec480b6c9ea95e70ba361ba14bf513e5186a6c302e6f409bd 608 01f1030ad3cdac1e08965217e430ca7f9bce698111ae8a4d0530efd4100000000000 609 000041d9e3c88db68247ab50264a6090e2e524bda3049dbc53c4df708e37bd76913b 610 8cf5954c4d0f835331f185fef4ff1c6115cf0eb8ce27e8224bf5f76c75b182308200 611 00000000000007741cf8c80b9bee583abac3d38daa6b807fed38b06580cb75ee8531 612 9d25fede6 613 Ka = 0xfc8482d5d7623a75ad09721d631d1392 614 Ke = 0x93f618fe24d0d5a54b320f498dbd3ecb 615 KcA = 0x75b20fc4205d6217a22156f918dd03b1 616 KcB = 0x3bf3a5d3876d9d12dc54cab927acd5f7 617 MAC(A) = 0xd4994b751eb832b2836ad674cd615c643053278864a63e263bc2f324b 618 9a04ddd 619 MAC(B) = 0x23cf761999b7603adf5507b50c9bda4eaabe8fa7a9ad0280729dfcd00 620 8b2bf05 622 SPAKE2(A='', B='server') 623 w = 0x7741cf8c80b9bee583abac3d38daa6b807fed38b06580cb75ee85319d25fed 624 e6 625 X = 0x0465e8b4709ba622bc97af5dde3b41757c2114bfc5abb10141245cb01d62ca 626 0d7360e1169cd518f9351bbfa44a66cc5f3bcb60661a04f39b04a3d504046db67884 627 T = 0x0482f64286419ff46362faf781776edf908740b8ff612e0bfe3c90cdc553ba 628 db7f882a4110ee71fa13a693b5ce96ceba5798636555d074648d4521e3b63dc14872 629 Y = 0x041aa11299692627a7cac122d4c14606ff700a8be6a0fb1c42f3762d629893 630 ab3ca51e4a48c798fc8c6b9dcfda1ad33099ed2f73abe6b3500ce383f54011430c26 631 S = 0x04adba3c3b9a74d9769651d09aedb37d22b9471b9e408e2b98fdd4188c12fa 632 c731e9dc87e029f7dee0213660ddf0791f50dd8fd32f7152015be0489125b3831b4b 633 TT = 0x0600000000000000736572766572410000000000000004adba3c3b9a74d97 634 69651d09aedb37d22b9471b9e408e2b98fdd4188c12fac731e9dc87e029f7dee0213 635 660ddf0791f50dd8fd32f7152015be0489125b3831b4b41000000000000000482f64 636 286419ff46362faf781776edf908740b8ff612e0bfe3c90cdc553badb7f882a4110e 637 e71fa13a693b5ce96ceba5798636555d074648d4521e3b63dc148724100000000000 638 00004a406929024a5275372531c85c54fd222f35bfdb1cdf1bd1abe82d5c837744d9 639 3ea2979962eb374d4feda37b178e91711c52edd453178cf69748e0a3d9ef073c2200 640 00000000000007741cf8c80b9bee583abac3d38daa6b807fed38b06580cb75ee8531 641 9d25fede6 642 Ka = 0xcd9c33c6329761919486d0041faccb56 643 Ke = 0xa08125eeed51c61ad93b2ff7d8ec3cd5 644 KcA = 0x60056386cbe06ba199fa6aef81dfb273 645 KcB = 0x5e5a591b4426d47190aecb2fc4527140 646 MAC(A) = 0xf0dcfb4fa874e3fcbadd44b6eb26a64d1d5c6e50034934934551f172d 647 3cdc50e 648 MAC(B) = 0x52e7a505c0b73db656108554a854c3f33bfb01edcc1ee52aa27ceb1cb 649 ef7f47b 651 SPAKE2(A='', B='') 652 w = 0x7741cf8c80b9bee583abac3d38daa6b807fed38b06580cb75ee85319d25fed 653 e6 654 X = 0x04fbeb44d6b772fa390fcced51be7316107e608ddf4ab5dcc9f1b2e24bf667 655 7f3232cdeeb39a61621a9e48028997d449894212eb54b6f12bdbd9baf8f1c909a740 656 T = 0x04887af8439d743215f26d48314835b024b9301ea508eac3a339241672fbba 657 09f63e155b1df5d31ccc63babafc00ffff6e258c692aed84a859fd4960d99fcec777 658 Y = 0x04bb4727c5c5c50ae34d5148ec6797e5ebf93ae51c5c6cfd48579c41436823 659 1ac8769142bf6a0109bd2b86dd901c6054629ce2c6b982326c9cd9a3685c4cf0640d 660 S = 0x04665b5101132528be32f4b4762d6ae80273bbe74e151fc2320da373e146ee 661 cd33038ff8099782f3781160244672cb43b4d9f2007da9b617c1890845440da0ca53 662 TT = 0x410000000000000004665b5101132528be32f4b4762d6ae80273bbe74e151 663 fc2320da373e146eecd33038ff8099782f3781160244672cb43b4d9f2007da9b617c 664 1890845440da0ca53410000000000000004887af8439d743215f26d48314835b024b 665 9301ea508eac3a339241672fbba09f63e155b1df5d31ccc63babafc00ffff6e258c6 666 92aed84a859fd4960d99fcec777410000000000000004aacd2378990cecd338c7cac 667 d132ce633bc424ac5d4ab32f539ccf31f15deef62463253790e139b461c5137944fc 668 6a5ffd895dbe0d3960b01f6d662fc41057a7020000000000000007741cf8c80b9bee 669 583abac3d38daa6b807fed38b06580cb75ee85319d25fede6 670 Ka = 0x16b10f1541c24c630f462f7e0aa57ddf 671 Ke = 0xb7ae8b61938e3dfad8b9ce1d2865533f 672 KcA = 0x3398d6c7de402a9ae89a4594d5576c21 673 KcB = 0x6894ab44d7ba7f3a40a772d1476593d9 674 MAC(A) = 0x12fce7f0aecc1dba393a7e5612e6357becc5e3d07cd41ffd35c6d652f 675 29cde60 676 MAC(B) = 0xac36c6d186c3b824f4cfe099f035cf3aed4162d08886d32fa1806e5bf 677 4015255 679 Authors' Addresses 681 Watson Ladd 682 Cloudflare 684 Email: watsonbladd@gmail.com 686 Benjamin Kaduk (editor) 687 Akamai Technologies 689 Email: kaduk@mit.edu