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Checking references for intended status: Proposed Standard ---------------------------------------------------------------------------- (See RFCs 3967 and 4897 for information about using normative references to lower-maturity documents in RFCs) == Missing Reference: 'DH76' is mentioned on line 42, but not defined -- Looks like a reference, but probably isn't: '0' on line 128 -- Looks like a reference, but probably isn't: '2' on line 129 == Missing Reference: 'SEED' is mentioned on line 340, but not defined == Unused Reference: 'FIPS-46-1' is defined on line 464, but no explicit reference was found in the text == Unused Reference: 'FIPS-81' is defined on line 468, but no explicit reference was found in the text == Outdated reference: A later version (-13) exists of draft-ietf-smime-cms-07 -- Possible downref: Non-RFC (?) normative reference: ref. 'FIPS-46-1' -- Possible downref: Non-RFC (?) normative reference: ref. 'FIPS-81' -- Possible downref: Non-RFC (?) normative reference: ref. 'FIPS-180' -- Possible downref: Non-RFC (?) normative reference: ref. 'FIPS-186' -- Possible downref: Non-RFC (?) normative reference: ref. 'P1363' ** Obsolete normative reference: RFC 2459 (ref. 'PKIX') (Obsoleted by RFC 3280) -- Possible downref: Non-RFC (?) normative reference: ref. 'LAW98' -- Possible downref: Non-RFC (?) normative reference: ref. 'LL97' -- Possible downref: Non-RFC (?) normative reference: ref. 'X942' Summary: 10 errors (**), 0 flaws (~~), 8 warnings (==), 12 comments (--). Run idnits with the --verbose option for more detailed information about the items above. -------------------------------------------------------------------------------- 2 E. Rescorla 3 INTERNET-DRAFT RTFM Inc. 4 March 1999 (Expires September 1999) 6 Diffie-Hellman Key Agreement Method 8 Status of this Memo 10 This document is an Internet-Draft and is in full conformance with 11 all provisions of Section 10 of RFC2026. Internet-Drafts are working 12 documents of the Internet Engineering Task Force (IETF), its areas, 13 and its working groups. Note that other groups may also distribute 14 working documents as Internet-Drafts. 16 Internet-Drafts are draft documents valid for a maximum of six months 17 and may be updated, replaced, or obsoleted by other documents at any 18 time. It is inappropriate to use Internet-Drafts as reference mate- 19 rial or to cite them other than as ``work in progress.'' 21 To learn the current status of any Internet-Draft, please check the 22 ``1id-abstracts.txt'' listing contained in the Internet-Drafts Shadow 23 Directories on ftp.is.co.za (Africa), nic.nordu.net (Europe), 24 munnari.oz.au (Pacific Rim), ftp.ietf.org (US East Coast), or 25 ftp.isi.edu (US West Coast). 27 Abstract 29 This document standardizes one particular Diffie-Hellman variant, 30 based on the ANSI X9.42 draft, developed by the ANSI X9F1 working 31 group. Diffie-Hellman is a key agreement algorithm used by two par- 32 ties to agree on a shared secret. An algorithm for converting the 33 shared secret into an arbitrary amount of keying material is pro- 34 vided. The resulting keying material is used as a symmetric encryp- 35 tion key. The Diffie-Hellman variant described requires the recipi- 36 ent to have a certificate, but the originator may have a static key 37 pair (with the public key placed in a certificate) or an ephemeral 38 key pair. 40 1. Introduction 42 In [DH76] Diffie and Hellman describe a means for two parties to 43 agree upon a shared secret in such a way that the secret will be 44 unavailable to eavesdroppers. This secret may then be converted into 45 cryptographic keying material for other (symmetric) algorithms. A 46 large number of minor variants of this process exist. This document 47 describes one such variant, based on the ANSI X9.42 specification. 49 1.1. Discussion of this Draft 51 This draft is being discussed on the "ietf-smime" mailing list. To 52 join the list, send a message to with 53 the single word "subscribe" in the body of the message. Also, there 54 is a Web site for the mailing list at . 57 1.2. Requirements Terminology 59 Keywords "MUST", "MUST NOT", "REQUIRED", "SHOULD", "SHOULD NOT" and 60 "MAY" that appear in this document are to be interpreted as described 61 in [RFC2119]. 63 2. Overview Of Method 65 Diffie-Hellman key agreement requires that both the sender and recip- 66 ient of a message have key pairs. By combining one's private key and 67 the other party's public key, both parties can compute the same 68 shared secret number. This number can then be converted into crypto- 69 graphic keying material. That keying material is typically used as a 70 key-encryption key (KEK) to encrypt (wrap) a content-encryption key 71 (CEK) which is in turn used to encrypt the message data. 73 2.1. Key Agreement 75 The first stage of the key agreement process is to compute a shared 76 secret number, called ZZ. When the same originator and recipient 77 public/private key pairs are used, the same ZZ value will result. 78 The ZZ value is then converted into a shared symmetric cryptographic 79 key. When the originator employs a static private/public key pair, 80 the introduction of a public random value ensures that the resulting 81 symmetric key will be different for each key agreement. 83 2.1.1. Generation of ZZ 85 X9.42 defines that the shared secret ZZ is generated as follows: 87 ZZ = g ^ (xb * xa) mod p 89 Note that the individual parties actually perform the computations: 91 ZZ = (yb ^ xa) mod p = (ya ^ xb) mod p 93 where ^ denotes exponentiation 94 ya is party a's public key; ya = g ^ xa mod p 95 yb is party b's public key; yb = g ^ xb mod p 96 xa is party a's private key 97 xb is party b's private key 98 p is a large prime 99 q is a large prime 100 g = h^{(p-1)/q} mod p, where 101 h is any integer with 1 < h < p-1 such that h{(p-1)/q} mod p > 1 102 (g has order q mod p; i.e. g^q mod p = 1 if g!=1) 103 j a large integer such that p=qj + 1 104 (See Section 2.2 for criteria for keys and parameters) 106 In [CMS], the recipient's key is identified by the CMS RecipientIden- 107 tifier, which points to the recipient's certificate. The sender's 108 public key is identified using the OriginatorIdentifierOrKey field, 109 either by reference to the sender's certificate or by inline inclu- 110 sion of a public key. 112 2.1.2. Generation of Keying Material 114 X9.42 provides an algorithm for generating an essentially arbitrary 115 amount of keying material from ZZ. Our algorithm is derived from that 116 algorithm by mandating some optional fields and omitting others. 118 KM = H ( ZZ || OtherInfo) 120 H is the message digest function SHA-1 [FIPS-180] 121 ZZ is the shared secret value computed in Section 2.1.1. Leading zeros MUST 122 be preserved, so that ZZ occupies as many octets as p. For 123 instance, if p is 1024 bits, ZZ should be 128 bytes long. 124 OtherInfo is the DER encoding of the following structure: 126 OtherInfo ::= SEQUENCE { 127 keyInfo KeySpecificInfo, 128 partyAInfo [0] OCTET STRING OPTIONAL, 129 suppPubInfo [2] OCTET STRING 130 } 132 KeySpecificInfo ::= SEQUENCE { 133 algorithm OBJECT IDENTIFIER, 134 counter OCTET STRING SIZE (4..4) } 136 Note that these ASN.1 definitions use EXPLICIT tagging. (In ASN.1, 137 EXPLICIT tagging is implicit unless IMPLICIT is explicitly specified.) 139 algorithm is the ASN.1 algorithm OID of the CEK wrapping algorithm with 140 which this KEK will be used. Note that this is NOT an 141 AlgorithmIdentifier, but simply the OBJECT IDENTIFIER. No parameters 142 are used. 143 counter is a 32 bit number, represented in network byte order. Its 144 initial value is 1 for any ZZ, i.e. the byte sequence 00 00 00 01 145 (hex), and it is incremented by one every time the above key 146 generation function is run for a given KEK. 147 partyAInfo is a random string provided by the sender. In CMS, it is 148 provided as a parameter in the UserKeyingMaterial field (encoded as 149 an OCTET STRING). If provided, partyAInfo MUST contain 512 bits. 150 suppPubInfo is the length of the generated KEK, in bits, represented 151 as a 32 bit number in network byte order. E.g. for 3DES it 152 would be the byte sequence 00 00 00 C0. 154 To generate a KEK, one generates one or more KM blocks (incrementing 155 counter appropriately) until enough material has been generated. The 156 KM blocks are concatenated left to right I.e. KM(counter=1) || 157 KM(counter=2)... 159 Note that the only source of secret entropy in this computation is 160 ZZ. Even if a string longer than ZZ is generated, the effective key 161 space of the KEK is limited by the size of ZZ, in addition to any 162 security level considerations imposed by the parameters p and q.How- 163 ever, if partyAInfo is different for each message, a different KEK 164 will be generated for each message. Note that partyAInfo MUST be used 165 in Static-Static mode, but MAY appear in Ephemeral-Static mode. 167 2.1.3. KEK Computation 169 Each key encryption algorithm requires a specific size key (n). The 170 KEK is generated by mapping the left n-most bytes of KM onto the key. 171 For 3DES, which requires 192 bits of keying material, the algorithm 172 must be run twice, once with a counter value of 1 (to generate K1', 173 K2', and the first 32 bits of K3') and once with a counter value of 2 174 (to generate the last 32 bits of K3). K1',K2' and K3' are then parity 175 adjusted to generate the 3 DES keys K1,K2 and K3. For RC2-128, which 176 requires 128 bits of keying material, the algorithm is run once, with 177 a counter value of 1, and the left-most 128 bits are directly con- 178 verted to an RC2 key. Similarly, for RC2-40, which requires 40 bits 179 of keying material, the algorithm is run once, with a counter value 180 of 1, and the leftmost 40 bits are used as the key. 182 2.1.4. Keylengths for common algorithms 184 Some common key encryption algorithms have KEKs of the following 185 lengths. 187 3-key 3DES 192 bits 188 RC2-128 128 bits 189 RC2-40 40 bits 191 RC2 effective key lengths are equal to RC2 real key lengths. 193 2.1.5. Public Key Validation 195 The following algorithm MAY be used to validate a received public key 196 y. 198 1. Verify that y lies within the interval [2,p-1]. If it does not, 199 the key is invalid. 200 2. Compute y^q mod p. If the result == 1, the key is valid. 201 Otherwise the key is invalid. 203 The primary purpose of public key validation is to prevent a small 204 subgroup attack [LAW98] on the sender's key pair. If Ephemeral-Static 205 mode is used, this check may not be necessary. See also [P1363] for 206 more information on Public Key validation. 208 Note that this procedure may be subject to pending patents. 210 2.1.6. Example 1 212 ZZ is the 20 bytes 00 01 02 03 04 05 06 07 08 09 213 0a 0b 0c 0d 0e 0f 10 11 12 13 215 The key wrap algorithm is 3DES-EDE wrap. 217 No partyAInfo is used. 219 Consequently, the input to the first invocation of SHA-1 is: 221 00 01 02 03 04 05 06 07 08 09 0a 0b 0c 0d 0e 0f 10 11 12 13 ; ZZ 222 30 1d 223 30 13 224 06 0b 2a 86 48 86 f7 0d 01 09 10 03 06 ; 3DES wrap OID 225 04 04 226 00 00 00 01 ; Counter 227 a2 06 228 04 04 229 00 00 00 c0 ; key length 230 And the output is the 20 bytes: 232 a0 96 61 39 23 76 f7 04 4d 90 52 a3 97 88 32 46 b6 7f 5f 1e 234 The input to the second invocation of SHA-1 is: 236 00 01 02 03 04 05 06 07 08 09 0a 0b 0c 0d 0e 0f 10 11 12 13 ; ZZ 237 30 1d 238 30 13 239 06 0b 2a 86 48 86 f7 0d 01 09 10 03 06 ; 3DES wrap OID 240 04 04 241 00 00 00 02 ; Counter 242 a2 06 243 04 04 244 00 00 00 c0 ; key length 246 And the output is the 20 bytes: 248 f6 3e b5 fb 5f 56 d9 b6 a8 34 03 91 c2 d3 45 34 93 2e 11 30 250 Consequently, 251 K1'=a0 96 61 39 23 76 f7 04 252 K2'=4d 90 52 a3 97 88 32 46 253 K3'=b6 7f 5f 1e f6 3e b5 fb 255 Note: These keys are not parity adjusted 257 2.1.7. Example 2 259 ZZ is the 20 bytes 00 01 02 03 04 05 06 07 08 09 260 0a 0b 0c 0d 0e 0f 10 11 12 13 262 The key wrap algorithm is RC2-128 key wrap, so we need 128 bits (16 263 bytes) of keying material. 265 The partyAInfo used is the 64 bytes 267 01 23 45 67 89 ab cd ef fe dc ba 98 76 54 32 01 268 01 23 45 67 89 ab cd ef fe dc ba 98 76 54 32 01 269 01 23 45 67 89 ab cd ef fe dc ba 98 76 54 32 01 270 01 23 45 67 89 ab cd ef fe dc ba 98 76 54 32 01 272 Consequently, the input to SHA-1 is: 274 00 01 02 03 04 05 06 07 08 09 0a 0b 0c 0d 0e 0f 10 11 12 13 ; ZZ 275 30 61 276 30 13 277 06 0b 2a 86 48 86 f7 0d 01 09 10 03 07 ; RC2 wrap OID 278 04 04 279 00 00 00 01 ; Counter 280 a0 42 281 04 40 282 01 23 45 67 89 ab cd ef fe dc ba 98 76 54 32 01 ; partyAInfo 283 01 23 45 67 89 ab cd ef fe dc ba 98 76 54 32 01 284 01 23 45 67 89 ab cd ef fe dc ba 98 76 54 32 01 285 01 23 45 67 89 ab cd ef fe dc ba 98 76 54 32 01 286 a2 06 287 04 04 288 00 00 00 80 ; key length 290 And the output is the 20 bytes: 292 48 95 0c 46 e0 53 00 75 40 3c ce 72 88 96 04 e0 3e 7b 5d e9 294 Consequently, 295 K=48 95 0c 46 e0 53 00 75 40 3c ce 72 88 96 04 e0 297 2.2. Key and Parameter Requirements 299 X9.42 requires that the group parameters be of the form p=jq + 1 300 where q is a large prime of length m and j>=2. An algorithm for gen- 301 erating primes of this form (derived from the algorithms in FIPS PUB 302 186-1[FIPS-186] and [X942]can be found in appendix A. 304 X9.42 requires that the private key x be in the interval [2, (q - 305 2)]. x should be randomly generated in this interval. y is then com- 306 puted by calculating g^x mod p. To comply with this draft, m MUST be 307 >=160 bits in length, (consequently, q MUST be at least 160 bits 308 long). When symmetric ciphers stronger than DES are to be used, a 309 larger m may be advisable. p must be a minimum of 512 bits long. 311 2.2.1. Group Parameter Generation 313 Agents SHOULD generate domain parameters (g,p,q) using the following 314 algorithm, derived from [FIPS-186] and [X942]. When this algorithm is 315 used, the correctness of the generation procedure can be verified by 316 a third party by the algorithm of 2.2.2. 318 2.2.1.1. Generation of p, q 320 This algorithm generates a p, q pair where q is of length m and 321 p is of length L. 323 Let L - 1 = n*m + b where both b and n are integers and 324 0 <= b < 160. 326 1. Choose an arbitrary sequence of at least m bits and call it SEED. 327 Let g be the length of SEED in bits. 329 2. Set U = 0 331 3. Set m' = m / 160, where / represents integer division, 332 consequently, if m = 200, m' = 1. 334 4. for i = 0 to m' - 1 336 U = U + SHA[SEED + i] XOR SHA[(SEED + m' + i) mod 2^160] * 2^(160 * i) 338 Note that for m=160, this reduces to the algorithm of [FIPS-186] 340 U = SHA[SEED] XOR SHA[(SEED+1) mod 2^160 ]. 342 5. Form q from U by setting the most significant bit (the 2^(m-1) bit) 343 and the least significant bit to 1. In terms of boolean operations, 344 q = U OR 2^(m-1) OR 1. Note that 2^(m-1) < q < 2^m 346 6. Use a robust primality algorithm to test whether q is prime. 348 7. If q is not prime then go to 1. 350 8. Let counter = 0 and offset = 2 352 9. For k = 0 to n let 354 Vk = SHA[(SEED + offset + k) mod 2^160 ]. 356 10. Let W be the integer 358 W = V0 + V1*2^160 + ... + Vn-1*2(n-1)*160 + (Vn mod 2^b) 359 * 2n*160 360 and let 361 X = W + 2^(L-1) 363 Note that 0 <= W < 2^(L-1) and hence 2^(L-1) 365 11. Let c = X mod (2 * q) and set p = X - (c-1). Note that p is congruent 366 to 1 mod (2 * q). 368 12. If p < 2^(L -1) then go to step 15. 370 13. Perform a robust primality test on p. 372 14. If p passes the test performed in step 13 go to step 17. 374 15. Let counter = counter + 1 and offset = offset + n + 1. 376 16. If counter >= 4096 go to step 1. Otherwise go to step 9. 378 17. Save the value of SEED and the value of counter for use 379 in certifying the proper generation of p and q. 381 Note: A robust primality test is one where the probability of 382 a non-prime number passing the test is at most 2^-80. [FIPS-186] 383 provides a suitable algorithm, as does [X9.42]. 385 2.2.1.2. Generation of g 387 This section gives an algorithm (derived from [FIPS-186]) for gener- 388 ating g. 389 1. Let j = (p - 1)/q. 391 2. Set h = any integer, where 1 < h < p - 1 and h differs 392 from any value previously tried. 394 3. Set g = h^j mod p 396 4. If g = 1 go to step 2 398 2.2.2. Group Parameter Validation 400 The ASN.1 for DH keys in [PKIX] includes elements j and validation- 401 Parms which MAY be used by recipients of a key to verify that the 402 group parameters were correctly generated. Two checks are possible: 404 1. Verify that p=qj + 1. This demonstrates that the parameters meet 405 the X9.42 parameter criteria. 406 2. Verify that when the p,q generation procedure of [FIPS-186] 407 Appendix 2 is followed with seed 'seed', that p is found when 408 'counter' = pgenCounter. 409 This demonstrates that the parameters were randomly chosen and 410 do not have a special form. 412 Whether agents provide validation information in their certificates 413 is a local matter between the agents and their CA. 415 2.3. Ephemeral-Static Mode 417 In Ephemeral-Static mode, the recipient has a static (and certified) 418 key pair, but the sender generates a new key pair for each message 419 and sends it using the originatorKey production. If the sender's key 420 is freshly generated for each message, the shared secret ZZ will be 421 similarly different for each message and partyAInfo MAY be omitted, 422 since it serves merely to decouple multiple KEKs generated by the 423 same set of pairwise keys. If, however, the same ephemeral sender key 424 is used for multiple messages (e.g. it is cached as a performance 425 optimization) then a separate partyAInfo MUST be used for each mes- 426 sage. All implementations of this standard MUST implement Ephemeral- 427 Static mode. 429 In order to resist small subgroup attacks, the recipient SHOULD per- 430 form the check described in 2.1.5. If an opponent cannot determine 431 success or failure of a decryption operation by the recipient, the 432 recipient MAY choose to omit this check. See also [LL97] for a method 433 of generating keys which are not subject to small subgroup attack. 435 2.4. Static-Static Mode 437 In Static-Static mode, both the sender and the recipient have a 438 static (and certified) key pair. Since the sender's and recipient's 439 keys are therefore the same for each message, ZZ will be the same for 440 each message. Thus, partyAInfo MUST be used (and different for each 441 message) in order to ensure that different messages use different 442 KEKs. Implementations MAY implement Static-Static mode. 444 In order to prevent small subgroup attacks, both originator and 445 recipient SHOULD either perform the validation step described in Sec- 446 tion 2.1.5 or verify that the CA has properly verified the validity 447 of the key. See also [LL97] for a method of generating keys which 448 are not subject to small subgroup attack. 450 Acknowledgements 452 The Key Agreement method described in this document is based on work 453 done by the ANSI X9F1 working group. The author wishes to extend his 454 thanks for their assistance. 456 The author also wishes to thank Stephen Henson, Paul Hoffman, Russ 457 Housley, Burt Kaliski, Brian Korver, John Linn, Jim Schaad, Mark 458 Schertler, Peter Yee, and Robert Zuccherato for their expert advice 459 and review. 461 References 462 [CMS] Housley, R., "Cryptographic Message Syntax", draft-ietf-smime-cms-07.txt. 464 [FIPS-46-1] Federal Information Processing Standards Publication (FIPS PUB) 465 46-1, Data Encryption Standard, Reaffirmed 1988 January 22 466 (supersedes FIPS PUB 46, 1977 January 15). 468 [FIPS-81] Federal Information Processing Standards Publication (FIPS PUB) 469 81, DES Modes of Operation, 1980 December 2. 471 [FIPS-180] Federal Information Processing Standards Publication (FIPS PUB) 472 180-1, "Secure Hash Standard", 1995 April 17. 474 [FIPS-186] Federal Information Processing Standards Publication (FIPS PUB) 475 186, "Digital Signature Standard", 1994 May 19. 477 [P1363] "Standard Specifications for Public Key Cryptography", IEEE P1363 478 working group draft, 1998, Annex D. 480 [PKIX] Housley, R., Ford, W., Polk, W., Solo, D., "Internet X.509 Public 481 Key Infrastructure Certificate and CRL Profile", RFC-2459. January 1999. 483 [LAW98] L. Law, A. Menezes, M. Qu, J. Solinas and S. Vanstone, 484 "An efficient protocol for authenticated key agreement", 485 Technical report CORR 98-05, University of Waterloo, 1998. 487 [LL97] C.H. Lim and P.J. Lee, "A key recovery attack on discrete log-based 488 schemes using a prime order subgroup", B.S. Kaliski, Jr., editor, 489 Advances in Cryptology - Crypto '97, Lecture Notes in Computer Science, 490 vol. 1295, 1997, Springer-Verlag, pp. 249-263. 492 [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate Requirement 493 Levels." RFC 2119. March 1997. 495 [X942] "Agreement Of Symmetric Keys Using Diffie-Hellman and MQV Algorithms", 496 ANSI draft, 1998. 498 Security Considerations 500 All the security in this system is provided by the secrecy of the 501 private keying material. If either sender or recipient private keys 502 are disclosed, all messages sent or received using that key are com- 503 promised. Similarly, loss of the private key results in an inability 504 to read messages sent using that key. 506 Static Diffie-Hellman keys are vulnerable to a small subgroup attack 507 [LAW98]. In practice, this issue arises for both sides in Static- 508 Static mode and for the receiver during Ephemeral-Static mode. Sec- 509 tions 2.3 and 2.4 describe appropriate practices to protect against 510 this attack. Alternatively, it is possible to generate keys in such a 511 fashion that they are resistant to this attack. See [LL97] 513 The security level provided by these methods depends on several fac- 514 tors. It depends on the length of the symmetric key (typically, a 2^l 515 security level if the length is l bits); the size of the prime q (a 516 2^{m/2} security level); and the size of the prime p (where the secu- 517 rity level grows as a subexponential function of the size in bits). 518 A good design principle is to have a balanced system, where all three 519 security levels are approximately the same. If many keys are derived 520 from a given pair of primes p and q, it may be prudent to have higher 521 levels for the primes. In any case, the overall security is limited 522 by the lowest of the three levels. 524 Author's Address 526 Eric Rescorla 527 RTFM Inc. 528 30 Newell Road, #16 529 East Palo Alto, CA 94303 530 Table of Contents 532 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1 533 1.1. Discussion of this Draft . . . . . . . . . . . . . . . . . . . 2 534 1.2. Requirements Terminology . . . . . . . . . . . . . . . . . . . 2 535 2. Overview Of Method . . . . . . . . . . . . . . . . . . . . . . . 2 536 2.1. Key Agreement . . . . . . . . . . . . . . . . . . . . . . . . . 2 537 2.1.1. Generation of ZZ . . . . . . . . . . . . . . . . . . . . . . 2 538 2.1.2. Generation of Keying Material . . . . . . . . . . . . . . . . 3 539 2.1.3. KEK Computation . . . . . . . . . . . . . . . . . . . . . . . 4 540 2.1.4. Keylengths for common algorithms . . . . . . . . . . . . . . 4 541 2.1.5. Public Key Validation . . . . . . . . . . . . . . . . . . . . 5 542 2.1.6. Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 5 543 2.1.7. Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 6 544 2.2. Key and Parameter Requirements . . . . . . . . . . . . . . . . 7 545 2.2.1. Group Parameter Generation . . . . . . . . . . . . . . . . . 7 546 2.2.1.1. Generation of p, q . . . . . . . . . . . . . . . . . . . . 7 547 2.2.1.2. Generation of g . . . . . . . . . . . . . . . . . . . . . . 9 548 2.2.2. Group Parameter Validation . . . . . . . . . . . . . . . . . 9 549 2.3. Ephemeral-Static Mode . . . . . . . . . . . . . . . . . . . . . 9 550 2.4. Static-Static Mode . . . . . . . . . . . . . . . . . . . . . . 10 551 2.4. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . 10 552 2.4. References . . . . . . . . . . . . . . . . . . . . . . . . . . 10 553 Security Considerations . . . . . . . . . . . . . . . . . . . . . . 11 554 Author's Address . . . . . . . . . . . . . . . . . . . . . . . . . . 12