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Kaduk, Ed. 5 Expires: August 21, 2020 Akamai 6 February 18, 2020 8 SPAKE2, a PAKE 9 draft-irtf-cfrg-spake2-10 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. 18 Status of This Memo 20 This Internet-Draft is submitted in full conformance with the 21 provisions of BCP 78 and BCP 79. 23 Internet-Drafts are working documents of the Internet Engineering 24 Task Force (IETF). Note that other groups may also distribute 25 working documents as Internet-Drafts. The list of current Internet- 26 Drafts is at https://datatracker.ietf.org/drafts/current/. 28 Internet-Drafts are draft documents valid for a maximum of six months 29 and may be updated, replaced, or obsoleted by other documents at any 30 time. It is inappropriate to use Internet-Drafts as reference 31 material or to cite them other than as "work in progress." 33 This Internet-Draft will expire on August 21, 2020. 35 Copyright Notice 37 Copyright (c) 2020 IETF Trust and the persons identified as the 38 document authors. All rights reserved. 40 This document is subject to BCP 78 and the IETF Trust's Legal 41 Provisions Relating to IETF Documents 42 (https://trustee.ietf.org/license-info) in effect on the date of 43 publication of this document. Please review these documents 44 carefully, as they describe your rights and restrictions with respect 45 to this document. Code Components extracted from this document must 46 include Simplified BSD License text as described in Section 4.e of 47 the Trust Legal Provisions and are provided without warranty as 48 described in the Simplified BSD License. 50 Table of Contents 52 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2 53 2. Requirements Notation . . . . . . . . . . . . . . . . . . . . 2 54 3. Definition of SPAKE2 . . . . . . . . . . . . . . . . . . . . 2 55 4. Key Schedule and Key Confirmation . . . . . . . . . . . . . . 5 56 5. Per-user M and N . . . . . . . . . . . . . . . . . . . . . . 6 57 6. Ciphersuites . . . . . . . . . . . . . . . . . . . . . . . . 6 58 7. Security Considerations . . . . . . . . . . . . . . . . . . . 9 59 8. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 9 60 9. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 9 61 10. References . . . . . . . . . . . . . . . . . . . . . . . . . 9 62 Appendix A. Algorithm used for Point Generation . . . . . . . . 11 63 Appendix B. Test Vectors . . . . . . . . . . . . . . . . . . . . 13 64 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 16 66 1. Introduction 68 This document describes SPAKE2, a means for two parties that share a 69 password to derive a strong shared key with no risk of disclosing the 70 password. This password-based key exchange protocol is compatible 71 with any group (requiring only a scheme to map a random input of 72 fixed length per group to a random group element), is computationally 73 efficient, and has a security proof. Predetermined parameters for a 74 selection of commonly used groups are also provided for use by other 75 protocols. 77 2. Requirements Notation 79 The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", 80 "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and 81 "OPTIONAL" in this document are to be interpreted as described in BCP 82 14 [RFC2119] [RFC8174] when, and only when, they appear in all 83 capitals, as shown here. 85 3. Definition of SPAKE2 87 3.1. Setup 89 Let G be a group in which the computational Diffie-Hellman (CDH) 90 problem is hard. Suppose G has order p*h where p is a large prime; h 91 will be called the cofactor. Let I be the unit element in G, e.g., 92 the point at infinity if G is an elliptic curve group. We denote the 93 operations in the group additively. We assume there is a 94 representation of elements of G as byte strings: common choices would 95 be SEC1 [SEC1] uncompressed or compressed for elliptic curve groups 96 or big endian integers of a fixed (per-group) length for prime field 97 DH. We fix two elements M and N in the prime-order subgroup of G as 98 defined in the table in this document for common groups, as well as a 99 generator P of the (large) prime-order subgroup of G. In the case of 100 a composite order group we will work in the quotient group. P is 101 specified in the document defining the group, and so we do not repeat 102 it here. 104 || denotes concatenation of strings. We also let len(S) denote the 105 length of a string in bytes, represented as an eight-byte little- 106 endian number. Finally, let nil represent an empty string, i.e., 107 len(nil) = 0. 109 KDF is a key-derivation function that takes as input a salt, 110 intermediate keying material (IKM), info string, and derived key 111 length L to derive a cryptographic key of length L. MAC is a Message 112 Authentication Code algorithm that takes a secret key and message as 113 input to produce an output. Let Hash be a hash function from 114 arbitrary strings to bit strings of a fixed length. Common choices 115 for H are SHA256 or SHA512 [RFC6234]. Let MHF be a memory-hard hash 116 function designed to slow down brute-force attackers. Scrypt 117 [RFC7914] is a common example of this function. The output length of 118 MHF matches that of Hash. Parameter selection for MHF is out of 119 scope for this document. Section 6 specifies variants of KDF, MAC, 120 and Hash suitable for use with the protocols contained herein. 122 Let A and B be two parties. A and B may also have digital 123 representations of the parties' identities such as Media Access 124 Control addresses or other names (hostnames, usernames, etc). A and 125 B may share Additional Authenticated Data (AAD) of length at most 126 2^16 - 1 bits that is separate from their identities which they may 127 want to include in the protocol execution. One example of AAD is a 128 list of supported protocol versions if SPAKE2(+) were used in a 129 higher-level protocol which negotiates use of a particular PAKE. 130 Including this list would ensure that both parties agree upon the 131 same set of supported protocols and therefore prevent downgrade 132 attacks. We also assume A and B share an integer w; typically w = 133 MHF(pw) mod p, for a user-supplied password pw. Standards such 134 NIST.SP.800-56Ar3 suggest taking mod p of a hash value that is 64 135 bits longer than that needed to represent p to remove statistical 136 bias introduced by the modulation. Protocols using this 137 specification must define the method used to compute w: it may be 138 necessary to carry out various forms of normalization of the password 139 before hashing [RFC8265]. The hashing algorithm SHOULD be a MHF so 140 as to slow down brute-force attackers. 142 3.2. Protocol Flow 144 SPAKE2 is a one round protocols to establish a shared secret with an 145 additional round for key confirmation. Prior to invocation, A and B 146 are provisioned with information such as the input password needed to 147 run the protocol. During the first round, A sends a public share pA 148 to B, and B responds with its own public share pB. Both A and B then 149 derive a shared secret used to produce encryption and authentication 150 keys. The latter are used during the second round for key 151 confirmation. (Section 4 details the key derivation and confirmation 152 steps.) In particular, A sends a key confirmation message cA to B, 153 and B responds with its own key confirmation messgage cB. Both 154 parties MUST NOT consider the protocol complete prior to receipt and 155 validation of these key confirmation messages. 157 This sample trace is shown below. 159 A B 160 | (setup protocol) | 161 (compute pA) | pA | 162 |----------------->| 163 | pB | (compute pB) 164 |<-----------------| 165 | | 166 | (derive secrets) | 167 (compute cA) | cA | 168 |----------------->| 169 | cB | (compute cB) 170 |<-----------------| 172 3.3. SPAKE2 174 To begin, A picks x randomly and uniformly from the integers in 175 [0,p), and calculates X=x*P and T=w*M+X, then transmits pA=T to B. 177 B selects y randomly and uniformly from the integers in [0,p), and 178 calculates Y=y*P, S=w*N+Y, then transmits pB=S to A. 180 Both A and B calculate a group element K. A calculates it as 181 h*x*(S-w*N), while B calculates it as h*y*(T-w*M). A knows S because 182 it has received it, and likewise B knows T. The multiplication by h 183 prevents small subgroup confinement attacks by computing a unique 184 value in the quotient group. (Any text on abstract algebra explains 185 this notion.) 187 K is a shared value, though it MUST NOT be used as a shared secret. 188 Both A and B must derive two shared secrets from K and the protocol 189 transcript. This prevents man-in-the-middle attackers from inserting 190 themselves into the exchange. The transcript TT is encoded as 191 follows: 193 TT = len(A) || A || len(B) || B || len(S) || S 194 || len(T) || T || len(K) || K || len(w) || w 196 If an identity is absent, it is omitted from the transcript entirely. 197 For example, if both A and B are absent, then TT = len(S) || S || 198 len(T) || T || len(K) || K || len(w) || w. Likewise, if only A is 199 absent, TT = len(B) || B || len(S) || S || len(T) || T || len(K) || 200 K || len(w) || w. This must only be done for applications in which 201 identities are implicit. Otherwise, the protocol risks Unknown Key 202 Share attacks (discussion of Unknown Key Share attacks in a specific 203 protocol is given in [I-D.ietf-mmusic-sdp-uks]. 205 Upon completion of this protocol, A and B compute shared secrets Ke, 206 KcA, and KcB as specified in Section 4. A MUST send B a key 207 confirmation message so both parties agree upon these shared secrets. 208 This confirmation message F is computed as a MAC over the protocol 209 transcript TT using KcA, as follows: F = MAC(KcA, TT). Similarly, B 210 MUST send A a confirmation message using a MAC computed equivalently 211 except with the use of KcB. Key confirmation verification requires 212 computing F and checking for equality against that which was 213 received. 215 4. Key Schedule and Key Confirmation 217 The protocol transcript TT, as defined in Section Section 3.3, is 218 unique and secret to A and B. Both parties use TT to derive shared 219 symmetric secrets Ke and Ka as Ke || Ka = Hash(TT). The length of 220 each key is equal to half of the digest output, e.g., |Ke| = |Ka| = 221 128 bits for SHA-256. 223 Both endpoints use Ka to derive subsequent MAC keys for key 224 confirmation messages. Specifically, let KcA and KcB be the MAC keys 225 used by A and B, respectively. A and B compute them as KcA || KcB = 226 KDF(nil, Ka, "ConfirmationKeys" || AAD), where AAD is the associated 227 data each given to each endpoint, or nil if none was provided. The 228 length of each of KcA and KcB is equal to half of the KDF output, 229 e.g., |KcA| = |KcB| = 128 bits for HKDF(SHA256). 231 The resulting key schedule for this protocol, given transcript TT and 232 additional associated data AAD, is as follows. 234 TT -> Hash(TT) = Ka || Ke 235 AAD -> KDF(nil, Ka, "ConfirmationKeys" || AAD) = KcA || KcB 237 A and B output Ke as the shared secret from the protocol. Ka and its 238 derived keys are not used for anything except key confirmation. 240 5. Per-user M and N 242 To avoid concerns that an attacker needs to solve a single ECDH 243 instance to break the authentication of SPAKE2, a variant based on 244 using [I-D.irtf-cfrg-hash-to-curve] is also presented. In this 245 variant M=h2c(HASH(A, B, "M for SPAKE2"), N=h2c(HASH(A, B, "N for 246 SPAKE2")). In addition M and N may be equal to have a symmetric 247 variant. The security of these variants is examined in [MNVAR] 249 6. Ciphersuites 251 This section documents SPAKE2 ciphersuite configurations. A 252 ciphersuite indicates a group, cryptographic hash algorithm, and pair 253 of KDF and MAC functions, e.g., SPAKE2-P256-SHA256-HKDF-HMAC. This 254 ciphersuite indicates a SPAKE2 protocol instance over P-256 that uses 255 SHA256 along with HKDF [RFC5869] and HMAC [RFC2104] for G, Hash, KDF, 256 and MAC functions, respectively. 258 +------------------+---------------+-------------+------------------+ 259 | G | Hash | KDF | MAC | 260 +------------------+---------------+-------------+------------------+ 261 | P-256 | SHA256 | HKDF | HMAC [RFC2104] | 262 | | [RFC6234] | [RFC5869] | | 263 | | | | | 264 | P-256 | SHA512 | HKDF | HMAC [RFC2104] | 265 | | [RFC6234] | [RFC5869] | | 266 | | | | | 267 | P-384 | SHA256 | HKDF | HMAC [RFC2104] | 268 | | [RFC6234] | [RFC5869] | | 269 | | | | | 270 | P-384 | SHA512 | HKDF | HMAC [RFC2104] | 271 | | [RFC6234] | [RFC5869] | | 272 | | | | | 273 | P-512 | SHA512 | HKDF | HMAC [RFC2104] | 274 | | [RFC6234] | [RFC5869] | | 275 | | | | | 276 | edwards25519 | SHA256 | HKDF | HMAC [RFC2104] | 277 | [RFC7748] | [RFC6234] | [RFC5869] | | 278 | | | | | 279 | edwards448 | SHA512 | HKDF | HMAC [RFC2104] | 280 | [RFC7748] | [RFC6234] | [RFC5869] | | 281 | | | | | 282 | P-256 | SHA256 | HKDF | CMAC-AES-128 | 283 | | [RFC6234] | [RFC5869] | [RFC4493] | 284 | | | | | 285 | P-256 | SHA512 | HKDF | CMAC-AES-128 | 286 | | [RFC6234] | [RFC5869] | [RFC4493] | 287 +------------------+---------------+-------------+------------------+ 289 Table 1: SPAKE2(+) Ciphersuites 291 The following points represent permissible point generation seeds for 292 the groups listed in the Table Table 1, using the algorithm presented 293 in Appendix A. These bytestrings are compressed points as in [SEC1] 294 for curves from [SEC1]. 296 For P256: 298 M = 299 02886e2f97ace46e55ba9dd7242579f2993b64e16ef3dcab95afd497333d8fa12f 300 seed: 1.2.840.10045.3.1.7 point generation seed (M) 302 N = 303 03d8bbd6c639c62937b04d997f38c3770719c629d7014d49a24b4f98baa1292b49 304 seed: 1.2.840.10045.3.1.7 point generation seed (N) 305 For P384: 307 M = 308 030ff0895ae5ebf6187080a82d82b42e2765e3b2f8749c7e05eba366434b363d3dc 309 36f15314739074d2eb8613fceec2853 310 seed: 1.3.132.0.34 point generation seed (M) 312 N = 313 02c72cf2e390853a1c1c4ad816a62fd15824f56078918f43f922ca21518f9c543bb 314 252c5490214cf9aa3f0baab4b665c10 315 seed: 1.3.132.0.34 point generation seed (N) 317 For P521: 319 M = 320 02003f06f38131b2ba2600791e82488e8d20ab889af753a41806c5db18d37d85608 321 cfae06b82e4a72cd744c719193562a653ea1f119eef9356907edc9b56979962d7aa 322 seed: 1.3.132.0.35 point generation seed (M) 324 N = 325 0200c7924b9ec017f3094562894336a53c50167ba8c5963876880542bc669e494b25 326 32d76c5b53dfb349fdf69154b9e0048c58a42e8ed04cef052a3bc349d95575cd25 327 seed: 1.3.132.0.35 point generation seed (N) 329 For edwards25519: 331 M = 332 d048032c6ea0b6d697ddc2e86bda85a33adac920f1bf18e1b0c6d166a5cecdaf 333 seed: edwards25519 point generation seed (M) 335 N = 336 d3bfb518f44f3430f29d0c92af503865a1ed3281dc69b35dd868ba85f886c4ab 337 seed: edwards25519 point generation seed (N) 339 For edwards448: 341 M = 342 b6221038a775ecd007a4e4dde39fd76ae91d3cf0cc92be8f0c2fa6d6b66f9a12 343 942f5a92646109152292464f3e63d354701c7848d9fc3b8880 344 seed: edwards448 point generation seed (M) 346 N = 347 6034c65b66e4cd7a49b0edec3e3c9ccc4588afd8cf324e29f0a84a072531c4db 348 f97ff9af195ed714a689251f08f8e06e2d1f24a0ffc0146600 349 seed: edwards448 point generation seed (N) 351 7. Security Considerations 353 A security proof of SPAKE2 for prime order groups is found in [REF]. 354 Note that the choice of M and N is critical for the security proof. 355 The generation method specified in this document is designed to 356 eliminate concerns related to knowing discrete logs of M and N. 358 Elements received from a peer MUST be checked for group membership: 359 failure to properly validate group elements can lead to attacks. It 360 is essential that endpoints verify received points are members of G. 362 The choices of random numbers MUST BE uniform. Randomly generated 363 values (e.g., x and y) MUST NOT be reused; such reuse may permit 364 dictionary attacks on the password. 366 SPAKE2 does not support augmentation. As a result, the server has to 367 store a password equivalent. This is considered a significant 368 drawback in some use cases 370 8. IANA Considerations 372 No IANA action is required. 374 9. Acknowledgments 376 Special thanks to Nathaniel McCallum and Greg Hudson for generation 377 of test vectors. Thanks to Mike Hamburg for advice on how to deal 378 with cofactors. Greg Hudson also suggested the addition of warnings 379 on the reuse of x and y. Thanks to Fedor Brunner, Adam Langley, and 380 the members of the CFRG for comments and advice. Chris Wood 381 contributed substantial text and reformatting to address the 382 excellent review comments from Kenny Paterson. 384 10. References 386 10.1. Normative References 388 [I-D.irtf-cfrg-hash-to-curve] 389 Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R., and 390 C. Wood, "Hashing to Elliptic Curves", draft-irtf-cfrg- 391 hash-to-curve-05 (work in progress), November 2019. 393 [RFC2104] Krawczyk, H., Bellare, M., and R. Canetti, "HMAC: Keyed- 394 Hashing for Message Authentication", RFC 2104, 395 DOI 10.17487/RFC2104, February 1997, 396 . 398 [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate 399 Requirement Levels", BCP 14, RFC 2119, 400 DOI 10.17487/RFC2119, March 1997, 401 . 403 [RFC4493] Song, JH., Poovendran, R., Lee, J., and T. Iwata, "The 404 AES-CMAC Algorithm", RFC 4493, DOI 10.17487/RFC4493, June 405 2006, . 407 [RFC5480] Turner, S., Brown, D., Yiu, K., Housley, R., and T. Polk, 408 "Elliptic Curve Cryptography Subject Public Key 409 Information", RFC 5480, DOI 10.17487/RFC5480, March 2009, 410 . 412 [RFC5869] Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand 413 Key Derivation Function (HKDF)", RFC 5869, 414 DOI 10.17487/RFC5869, May 2010, 415 . 417 [RFC6234] Eastlake 3rd, D. and T. Hansen, "US Secure Hash Algorithms 418 (SHA and SHA-based HMAC and HKDF)", RFC 6234, 419 DOI 10.17487/RFC6234, May 2011, 420 . 422 [RFC7748] Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves 423 for Security", RFC 7748, DOI 10.17487/RFC7748, January 424 2016, . 426 [RFC7914] Percival, C. and S. Josefsson, "The scrypt Password-Based 427 Key Derivation Function", RFC 7914, DOI 10.17487/RFC7914, 428 August 2016, . 430 [RFC8032] Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital 431 Signature Algorithm (EdDSA)", RFC 8032, 432 DOI 10.17487/RFC8032, January 2017, 433 . 435 [RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 436 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, 437 May 2017, . 439 10.2. Informative References 441 [I-D.ietf-mmusic-sdp-uks] 442 Thomson, M. and E. Rescorla, "Unknown Key Share Attacks on 443 uses of TLS with the Session Description Protocol (SDP)", 444 draft-ietf-mmusic-sdp-uks-07 (work in progress), August 445 2019. 447 [MNVAR] Abdalla, M. and M. Barbosa, "Perfect Forward Security of 448 SPAKE2", Oct 2019. 450 IACR eprint 2019/1194 452 [REF] Abdalla, M. and D. Pointcheval, "Simple Password-Based 453 Encrypted Key Exchange Protocols.", Feb 2005. 455 Appears in A. Menezes, editor. Topics in Cryptography- 456 CT-RSA 2005, Volume 3376 of Lecture Notes in Computer 457 Science, pages 191-208, San Francisco, CA, US. Springer- 458 Verlag, Berlin, Germany. 460 [RFC8265] Saint-Andre, P. and A. Melnikov, "Preparation, 461 Enforcement, and Comparison of Internationalized Strings 462 Representing Usernames and Passwords", RFC 8265, 463 DOI 10.17487/RFC8265, October 2017, 464 . 466 [SEC1] Standards for Efficient Cryptography Group, "SEC 1: 467 Elliptic Curve Cryptography", May 2009. 469 [TDH] Cash, D., Kiltz, E., and V. Shoup, "The Twin-Diffie 470 Hellman Problem and Applications", 2008. 472 EUROCRYPT 2008. Volume 4965 of Lecture notes in Computer 473 Science, pages 127-145. Springer-Verlag, Berlin, Germany. 475 Appendix A. Algorithm used for Point Generation 477 This section describes the algorithm that was used to generate the 478 points (M) and (N) in the table in Section 6. 480 For each curve in the table below, we construct a string using the 481 curve OID from [RFC5480] (as an ASCII string) or its name, combined 482 with the needed constant, for instance "1.3.132.0.35 point generation 483 seed (M)" for P-512. This string is turned into a series of blocks 484 by hashing with SHA256, and hashing that output again to generate the 485 next 32 bytes, and so on. This pattern is repeated for each group 486 and value, with the string modified appropriately. 488 A byte string of length equal to that of an encoded group element is 489 constructed by concatenating as many blocks as are required, starting 490 from the first block, and truncating to the desired length. The byte 491 string is then formatted as required for the group. In the case of 492 Weierstrass curves, we take the desired length as the length for 493 representing a compressed point (section 2.3.4 of [SEC1]), and use 494 the low-order bit of the first byte as the sign bit. In order to 495 obtain the correct format, the value of the first byte is set to 0x02 496 or 0x03 (clearing the first six bits and setting the seventh bit), 497 leaving the sign bit as it was in the byte string constructed by 498 concatenating hash blocks. For the [RFC8032] curves a different 499 procedure is used. For edwards448 the 57-byte input has the least- 500 significant 7 bits of the last byte set to zero, and for edwards25519 501 the 32-byte input is not modified. For both the [RFC8032] curves the 502 (modified) input is then interpreted as the representation of the 503 group element. If this interpretation yields a valid group element 504 with the correct order (p), the (modified) byte string is the output. 505 Otherwise, the initial hash block is discarded and a new byte string 506 constructed from the remaining hash blocks. The procedure of 507 constructing a byte string of the appropriate length, formatting it 508 as required for the curve, and checking if it is a valid point of the 509 correct order, is repeated until a valid element is found. 511 The following python snippet generates the above points, assuming an 512 elliptic curve implementation following the interface of 513 Edwards25519Point.stdbase() and Edwards448Point.stdbase() in 514 Appendix A of [RFC8032]: 516 def iterated_hash(seed, n): 517 h = seed 518 for i in range(n): 519 h = hashlib.sha256(h).digest() 520 return h 522 def bighash(seed, start, sz): 523 n = -(-sz // 32) 524 hashes = [iterated_hash(seed, i) for i in range(start, start + n)] 525 return b''.join(hashes)[:sz] 527 def canon_pointstr(ecname, s): 528 if ecname == 'edwards25519': 529 return s 530 elif ecname == 'edwards448': 531 return s[:-1] + bytes([s[-1] & 0x80]) 532 else: 533 return bytes([(s[0] & 1) | 2]) + s[1:] 535 def gen_point(seed, ecname, ec): 536 for i in range(1, 1000): 537 hval = bighash(seed, i, len(ec.encode())) 538 pointstr = canon_pointstr(ecname, hval) 539 try: 540 p = ec.decode(pointstr) 541 if p != ec.zero_elem() and p * p.l() == ec.zero_elem(): 542 return pointstr, i 543 except Exception: 544 pass 546 Appendix B. Test Vectors 548 This section contains test vectors for SPAKE2 using the P256-SHA256- 549 HKDF-HMAC ciphersuite. (Choice of MHF is omitted and values for w 550 and w0,w1 are provided directly.) All points are encoded using the 551 uncompressed format, i.e., with a 0x04 octet prefix, specified in 552 [SEC1] A and B identity strings are provided in the protocol 553 invocation. 555 B.1. SPAKE2 Test Vectors 557 SPAKE2(A='client', B='server') 558 w = 0x7741cf8c80b9bee583abac3d38daa6b807fed38b06580cb75ee85319d25fed 559 e6 560 X = 0x04ac6827b3a9110d1e663bcd4f5de668da34a9f45e464e99067bbea53f1ed4 561 d8abbdd234c05b3a3a8a778ee47f244cca1a79acb7052d5e58530311a9af077ba179 562 T = 0x04e02acfbbfb081fc38b5bab999b5e25a5ffd0b1ac48eae24fcc8e49ac5e0d 563 8a790914419a100e205605f9862daa848e99cea455263f0c6e06bc5a911f3e10a16b 564 Y = 0x0413c45ab093a75c4b2a6e71f957eec3859807858325258b0fa43df5a6efd2 565 63c59b9c1fbfd55bc5e75fd3e7ba8af6799a99b225fe6c30e6c2a2e0ab4962136ba8 566 S = 0x047aad50ba7bd6a5eacbead7689f7146f1a4219fa071cce1755f80280cc6c3 567 a5a73cf469f2a294a0b74a5c07054585ccd447f3f633d8631f3bf43442449e9efeba 568 TT = 0x0600000000000000636c69656e74060000000000000073657276657241000 569 00000000000047aad50ba7bd6a5eacbead7689f7146f1a4219fa071cce1755f80280 570 cc6c3a5a73cf469f2a294a0b74a5c07054585ccd447f3f633d8631f3bf43442449e9 571 efeba410000000000000004e02acfbbfb081fc38b5bab999b5e25a5ffd0b1ac48eae 572 24fcc8e49ac5e0d8a790914419a100e205605f9862daa848e99cea455263f0c6e06b 573 c5a911f3e10a16b410000000000000004d01fc08bbae9b6abe2f4d6893cc9f810433 574 2e19be5f5881c6b9f077e1feff55023da74db65fae320fad8f0dd38e1323f5336f3f 575 53c9c9dec06710f18f556bd2020000000000000007741cf8c80b9bee583abac3d38d 576 aa6b807fed38b06580cb75ee85319d25fede6 577 Ka = 0x2b5e350c58d530c3586f75bf2a155c4b 578 Ke = 0x238509f7adf0dc72500b2d1315737a27 579 KcA = 0xc33d2ef8e37a7e545c14c7fcfdc9db94 580 KcB = 0x18a81cec7eb83416db6615cb3bc03fcb 581 MAC(A) = 0x29e9a63d243f2f0db5532d2eb0dbaa617803b85feb31566d0cb9457e3 582 03bcfa6 583 MAC(B) = 0x487e4cbe98b6287272d043e169a19b6c4682d0481c92f53f1ee03d4b8 584 6c3f43e 586 SPAKE2(A='client', B='') 587 w = 0x7741cf8c80b9bee583abac3d38daa6b807fed38b06580cb75ee85319d25fed 588 e6 589 X = 0x048b5d7b44b02c4c868f4486ec55bd2380ec34cd5fa5dbff1079a79097e305 590 0b34fa91272331729357c86cbb30d371e252dc915aeaa314921b1f09f74816f96a12 591 T = 0x04839f44931b88d12769e601d0ec480b6c9ea95e70ba361ba14bf513e5186a 592 6c302e6f409bd01f1030ad3cdac1e08965217e430ca7f9bce698111ae8a4d0530efd 593 Y = 0x0446419d63037d0bbaca224f89987c776bfea2e0913ccda0790079212f476d 594 6fd1ec997a02821a804f885e4f29b172b27c92251d883efe201cae106c239108c0c7 595 S = 0x042926b2dbcc5d0cb23ca123cc4133242f2998439af5380434a4bd5fd76fbb 596 c030b5563218d0184fa3fd303482a679c9555ccea41098b26b6ee16fe35c792b1fda 597 TT = 0x0600000000000000636c69656e744100000000000000042926b2dbcc5d0cb 598 23ca123cc4133242f2998439af5380434a4bd5fd76fbbc030b5563218d0184fa3fd3 599 03482a679c9555ccea41098b26b6ee16fe35c792b1fda410000000000000004839f4 600 4931b88d12769e601d0ec480b6c9ea95e70ba361ba14bf513e5186a6c302e6f409bd 601 01f1030ad3cdac1e08965217e430ca7f9bce698111ae8a4d0530efd4100000000000 602 000041d9e3c88db68247ab50264a6090e2e524bda3049dbc53c4df708e37bd76913b 603 8cf5954c4d0f835331f185fef4ff1c6115cf0eb8ce27e8224bf5f76c75b182308200 604 00000000000007741cf8c80b9bee583abac3d38daa6b807fed38b06580cb75ee8531 605 9d25fede6 606 Ka = 0xfc8482d5d7623a75ad09721d631d1392 607 Ke = 0x93f618fe24d0d5a54b320f498dbd3ecb 608 KcA = 0x75b20fc4205d6217a22156f918dd03b1 609 KcB = 0x3bf3a5d3876d9d12dc54cab927acd5f7 610 MAC(A) = 0xd4994b751eb832b2836ad674cd615c643053278864a63e263bc2f324b 611 9a04ddd 612 MAC(B) = 0x23cf761999b7603adf5507b50c9bda4eaabe8fa7a9ad0280729dfcd00 613 8b2bf05 615 SPAKE2(A='', B='server') 616 w = 0x7741cf8c80b9bee583abac3d38daa6b807fed38b06580cb75ee85319d25fed 617 e6 618 X = 0x0465e8b4709ba622bc97af5dde3b41757c2114bfc5abb10141245cb01d62ca 619 0d7360e1169cd518f9351bbfa44a66cc5f3bcb60661a04f39b04a3d504046db67884 620 T = 0x0482f64286419ff46362faf781776edf908740b8ff612e0bfe3c90cdc553ba 621 db7f882a4110ee71fa13a693b5ce96ceba5798636555d074648d4521e3b63dc14872 622 Y = 0x041aa11299692627a7cac122d4c14606ff700a8be6a0fb1c42f3762d629893 623 ab3ca51e4a48c798fc8c6b9dcfda1ad33099ed2f73abe6b3500ce383f54011430c26 624 S = 0x04adba3c3b9a74d9769651d09aedb37d22b9471b9e408e2b98fdd4188c12fa 625 c731e9dc87e029f7dee0213660ddf0791f50dd8fd32f7152015be0489125b3831b4b 626 TT = 0x0600000000000000736572766572410000000000000004adba3c3b9a74d97 627 69651d09aedb37d22b9471b9e408e2b98fdd4188c12fac731e9dc87e029f7dee0213 628 660ddf0791f50dd8fd32f7152015be0489125b3831b4b41000000000000000482f64 629 286419ff46362faf781776edf908740b8ff612e0bfe3c90cdc553badb7f882a4110e 630 e71fa13a693b5ce96ceba5798636555d074648d4521e3b63dc148724100000000000 631 00004a406929024a5275372531c85c54fd222f35bfdb1cdf1bd1abe82d5c837744d9 632 3ea2979962eb374d4feda37b178e91711c52edd453178cf69748e0a3d9ef073c2200 633 00000000000007741cf8c80b9bee583abac3d38daa6b807fed38b06580cb75ee8531 634 9d25fede6 635 Ka = 0xcd9c33c6329761919486d0041faccb56 636 Ke = 0xa08125eeed51c61ad93b2ff7d8ec3cd5 637 KcA = 0x60056386cbe06ba199fa6aef81dfb273 638 KcB = 0x5e5a591b4426d47190aecb2fc4527140 639 MAC(A) = 0xf0dcfb4fa874e3fcbadd44b6eb26a64d1d5c6e50034934934551f172d 640 3cdc50e 641 MAC(B) = 0x52e7a505c0b73db656108554a854c3f33bfb01edcc1ee52aa27ceb1cb 642 ef7f47b 644 SPAKE2(A='', B='') 645 w = 0x7741cf8c80b9bee583abac3d38daa6b807fed38b06580cb75ee85319d25fed 646 e6 647 X = 0x04fbeb44d6b772fa390fcced51be7316107e608ddf4ab5dcc9f1b2e24bf667 648 7f3232cdeeb39a61621a9e48028997d449894212eb54b6f12bdbd9baf8f1c909a740 649 T = 0x04887af8439d743215f26d48314835b024b9301ea508eac3a339241672fbba 650 09f63e155b1df5d31ccc63babafc00ffff6e258c692aed84a859fd4960d99fcec777 651 Y = 0x04bb4727c5c5c50ae34d5148ec6797e5ebf93ae51c5c6cfd48579c41436823 652 1ac8769142bf6a0109bd2b86dd901c6054629ce2c6b982326c9cd9a3685c4cf0640d 653 S = 0x04665b5101132528be32f4b4762d6ae80273bbe74e151fc2320da373e146ee 654 cd33038ff8099782f3781160244672cb43b4d9f2007da9b617c1890845440da0ca53 655 TT = 0x410000000000000004665b5101132528be32f4b4762d6ae80273bbe74e151 656 fc2320da373e146eecd33038ff8099782f3781160244672cb43b4d9f2007da9b617c 657 1890845440da0ca53410000000000000004887af8439d743215f26d48314835b024b 658 9301ea508eac3a339241672fbba09f63e155b1df5d31ccc63babafc00ffff6e258c6 659 92aed84a859fd4960d99fcec777410000000000000004aacd2378990cecd338c7cac 660 d132ce633bc424ac5d4ab32f539ccf31f15deef62463253790e139b461c5137944fc 661 6a5ffd895dbe0d3960b01f6d662fc41057a7020000000000000007741cf8c80b9bee 662 583abac3d38daa6b807fed38b06580cb75ee85319d25fede6 663 Ka = 0x16b10f1541c24c630f462f7e0aa57ddf 664 Ke = 0xb7ae8b61938e3dfad8b9ce1d2865533f 665 KcA = 0x3398d6c7de402a9ae89a4594d5576c21 666 KcB = 0x6894ab44d7ba7f3a40a772d1476593d9 667 MAC(A) = 0x12fce7f0aecc1dba393a7e5612e6357becc5e3d07cd41ffd35c6d652f 668 29cde60 669 MAC(B) = 0xac36c6d186c3b824f4cfe099f035cf3aed4162d08886d32fa1806e5bf 670 4015255 672 Authors' Addresses 674 Watson Ladd 675 Cloudflare 677 Email: watsonbladd@gmail.com 679 Benjamin Kaduk (editor) 680 Akamai Technologies 682 Email: kaduk@mit.edu