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Checking references for intended status: Informational ---------------------------------------------------------------------------- -- Looks like a reference, but probably isn't: '1' on line 750 ** Obsolete normative reference: RFC 4492 (Obsoleted by RFC 8422) ** Obsolete normative reference: RFC 5226 (Obsoleted by RFC 8126) ** Obsolete normative reference: RFC 5246 (Obsoleted by RFC 8446) Summary: 3 errors (**), 0 flaws (~~), 1 warning (==), 7 comments (--). Run idnits with the --verbose option for more detailed information about the items above. -------------------------------------------------------------------------------- 2 Internet Engineering Task Force D. Gillmor 3 Internet-Draft ACLU 4 Updates: 4492, 5246, 4346, 2246 (if May 12, 2015 5 approved) 6 Intended status: Informational 7 Expires: November 13, 2015 9 Negotiated Finite Field Diffie-Hellman Ephemeral Parameters for TLS 10 draft-ietf-tls-negotiated-ff-dhe-09 12 Abstract 14 Traditional finite-field-based Diffie-Hellman (DH) key exchange 15 during the TLS handshake suffers from a number of security, 16 interoperability, and efficiency shortcomings. These shortcomings 17 arise from lack of clarity about which DH group parameters TLS 18 servers should offer and clients should accept. This document offers 19 a solution to these shortcomings for compatible peers by using a 20 section of the TLS "EC Named Curve Registry" to establish common 21 finite-field DH parameters with known structure and a mechanism for 22 peers to negotiate support for these groups. 24 Status of This Memo 26 This Internet-Draft is submitted in full conformance with the 27 provisions of BCP 78 and BCP 79. 29 Internet-Drafts are working documents of the Internet Engineering 30 Task Force (IETF). Note that other groups may also distribute 31 working documents as Internet-Drafts. The list of current Internet- 32 Drafts is at http://datatracker.ietf.org/drafts/current/. 34 Internet-Drafts are draft documents valid for a maximum of six months 35 and may be updated, replaced, or obsoleted by other documents at any 36 time. It is inappropriate to use Internet-Drafts as reference 37 material or to cite them other than as "work in progress." 39 This Internet-Draft will expire on November 13, 2015. 41 Copyright Notice 43 Copyright (c) 2015 IETF Trust and the persons identified as the 44 document authors. All rights reserved. 46 This document is subject to BCP 78 and the IETF Trust's Legal 47 Provisions Relating to IETF Documents 48 (http://trustee.ietf.org/license-info) in effect on the date of 49 publication of this document. Please review these documents 50 carefully, as they describe your rights and restrictions with respect 51 to this document. Code Components extracted from this document must 52 include Simplified BSD License text as described in Section 4.e of 53 the Trust Legal Provisions and are provided without warranty as 54 described in the Simplified BSD License. 56 Table of Contents 58 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3 59 1.1. Requirements Language . . . . . . . . . . . . . . . . . . 3 60 1.2. Vocabulary . . . . . . . . . . . . . . . . . . . . . . . 4 61 2. Named Group Overview . . . . . . . . . . . . . . . . . . . . 4 62 3. Client Behavior . . . . . . . . . . . . . . . . . . . . . . . 5 63 3.1. Client Local Policy on Custom Groups . . . . . . . . . . 6 64 4. Server Behavior . . . . . . . . . . . . . . . . . . . . . . . 6 65 5. Optimizations . . . . . . . . . . . . . . . . . . . . . . . . 7 66 5.1. Checking the Peer's Public Key . . . . . . . . . . . . . 7 67 5.2. Short Exponents . . . . . . . . . . . . . . . . . . . . . 8 68 5.3. Table Acceleration . . . . . . . . . . . . . . . . . . . 8 69 6. Operational Considerations . . . . . . . . . . . . . . . . . 8 70 6.1. Preference Ordering . . . . . . . . . . . . . . . . . . . 8 71 7. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 9 72 8. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 9 73 9. Security Considerations . . . . . . . . . . . . . . . . . . . 10 74 9.1. Negotiation resistance to active attacks . . . . . . . . 10 75 9.2. Group strength considerations . . . . . . . . . . . . . . 11 76 9.3. Finite-Field DHE only . . . . . . . . . . . . . . . . . . 12 77 9.4. Deprecating weak groups . . . . . . . . . . . . . . . . . 12 78 9.5. Choice of groups . . . . . . . . . . . . . . . . . . . . 12 79 9.6. Timing attacks . . . . . . . . . . . . . . . . . . . . . 13 80 9.7. Replay attacks from non-negotiated FFDHE . . . . . . . . 13 81 9.8. Forward Secrecy . . . . . . . . . . . . . . . . . . . . . 13 82 9.9. False Start . . . . . . . . . . . . . . . . . . . . . . . 14 83 10. Privacy Considerations . . . . . . . . . . . . . . . . . . . 14 84 10.1. Client fingerprinting . . . . . . . . . . . . . . . . . 14 85 11. References . . . . . . . . . . . . . . . . . . . . . . . . . 14 86 11.1. Normative References . . . . . . . . . . . . . . . . . . 14 87 11.2. Informative References . . . . . . . . . . . . . . . . . 15 88 11.3. URIs . . . . . . . . . . . . . . . . . . . . . . . . . . 16 89 Appendix A. Named Group Registry . . . . . . . . . . . . . . . . 16 90 A.1. ffdhe2048 . . . . . . . . . . . . . . . . . . . . . . . . 17 91 A.2. ffdhe3072 . . . . . . . . . . . . . . . . . . . . . . . . 18 92 A.3. ffdhe4096 . . . . . . . . . . . . . . . . . . . . . . . . 19 93 A.4. ffdhe6144 . . . . . . . . . . . . . . . . . . . . . . . . 21 94 A.5. ffdhe8192 . . . . . . . . . . . . . . . . . . . . . . . . 23 95 Author's Address . . . . . . . . . . . . . . . . . . . . . . . . 26 97 1. Introduction 99 Traditional TLS [RFC5246] offers a Diffie-Hellman ephemeral (DHE) key 100 exchange mode which provides Forward Secrecy for the connection. The 101 client offers a ciphersuite in the ClientHello that includes DHE, and 102 the server offers the client group parameters generator g and modulus 103 p. If the client does not consider the group strong enough (e.g., if 104 p is too small, or if p is not prime, or there are small subgroups 105 that cannot be easily avoided), or if it is unable to process the 106 group for other reasons, the client has no recourse but to terminate 107 the connection. 109 Conversely, when a TLS server receives a suggestion for a DHE 110 ciphersuite from a client, it has no way of knowing what kinds of DH 111 groups the client is capable of handling, or what the client's 112 security requirements are for this key exchange session. For 113 example, some widely-distributed TLS clients are not capable of DH 114 groups where p > 1024 bits. Other TLS clients may by policy wish to 115 use DHE only if the server can offer a stronger group (and are 116 willing to use a non-PFS key-exchange mechanism otherwise). The 117 server has no way of knowing which type of client is connecting, but 118 must select DH parameters with insufficient knowledge. 120 Additionally, the DH parameters selected by the server may have a 121 known structure which renders them secure against a small subgroup 122 attack, but a client receiving an arbitrary p and g has no efficient 123 way to verify that the structure of a new group is reasonable for 124 use. 126 This modification to TLS solves these problems by using a section of 127 the "EC Named Curves" registry to select common DH groups with known 128 structure and defining the use of the "elliptic_curves(10)" extension 129 (described here as "Supported Groups" extension) for clients 130 advertising support for DHE with these groups. This document also 131 provides guidance for compatible peers to take advantage of the 132 additional security, availability, and efficiency offered. 134 The use of this mechanism by one compatible peer when interacting 135 with a non-compatible peer should have no detrimental effects. 137 1.1. Requirements Language 139 The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", 140 "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this 141 document are to be interpreted as described in [RFC2119]. The term 142 "PRIVATE USE" is to be interpreted as described in [RFC5226]. 144 1.2. Vocabulary 146 The terms "DHE" or "FFDHE" are used in this document to refer to the 147 finite-field-based Diffie-Hellman ephemeral key exchange mechanism in 148 TLS. TLS also supports elliptic-curve-based Diffie-Hellman (ECDHE) 149 ephemeral key exchanges [RFC4492], but this document does not 150 document their use. A registry previously used only by ECHDE-capable 151 implementations is expanded in this document to cover FFDHE groups as 152 well. "FFDHE ciphersuites" is used in this document to refer 153 exclusively to ciphersuites with FFDHE key exchange mechanisms, but 154 note that these suites are typically labeled with a TLS_DHE_ prefix. 156 2. Named Group Overview 158 We use previously-unallocated codepoints within the extension 159 currently known as "elliptic_curves" (section 5.1.1. of [RFC4492]) to 160 indicate known finite field groups. The extension's semantics are 161 expanded from "Supported Elliptic Curves" to "Supported Groups". The 162 semantics of the extension's data type (enum NamedCurve) is also 163 expanded from "named curve" to "named group". 165 Additionally, we explicitly relax the requirement about when the 166 Supported Groups extension can be sent. This extension MAY be sent 167 by the client when either FFDHE or ECDHE ciphersuites are listed. 169 Codepoints in the NamedCurve registry with a high byte of 0x01 (that 170 is, between 256 and 511 inclusive) are set aside for FFDHE groups, 171 though only a small number of them are initially defined and we do 172 not expect many other FFDHE groups to be added to this range. No 173 codepoints outside of this range will be allocated to FFDHE groups. 174 The new code points for the NamedCurve registry are: 176 enum { 177 // other already defined elliptic curves (see RFC 4492) 178 ffdhe2048(256), ffdhe3072(257), ffdhe4096(258), 179 ffdhe6144(259), ffdhe8192(260), 180 // 181 } NamedCurve; 183 These additions to the Named Curve registry are described in detail 184 in Appendix A. They are all safe primes derived from the base of the 185 natural logarithm ("e"), with the high and low 64 bits set to 1 for 186 efficient Montgomery or Barrett reduction. 188 The use of the base of the natural logarithm here is as a "nothing- 189 up-my-sleeve" number. The goal is to guarantee that the bits in the 190 middle of the modulus are effectively random, while avoiding any 191 suspicion that the primes have secretly been selected to be weak 192 according to some secret criteria. [RFC3526] used pi for this value. 193 See Section 9.5 for reasons that this draft does not reuse pi. 195 3. Client Behavior 197 A TLS client that is capable of using strong finite field Diffie- 198 Hellman groups can advertise its capabilities and its preferences for 199 stronger key exchange by using this mechanism. 201 The compatible client that wants to be able to negotiate strong FFDHE 202 sends a "Supported Groups" extension (identified by type 203 elliptic_curves(10) in [RFC4492]) in the ClientHello, and include a 204 list of known FFDHE groups in the extension data, ordered from most 205 preferred to least preferred. If the client also supports and wants 206 to offer ECDHE key exchange, it MUST use a single "Supported Groups" 207 extension to include all supported groups (both ECDHE and FFDHE 208 groups). The ordering SHOULD be based on client preference, but see 209 Section 6.1 for more nuance. 211 A client that offers a "Supported Groups" extension containing an 212 FFDHE group SHOULD also include at least one FFDHE ciphersuite in the 213 Client Hello. 215 A client that offers a group MUST be able and willing to perform a DH 216 key exchange using that group. 218 A client that offers one or more FFDHE groups in the "Supported 219 Groups" extension and an FFDHE ciphersuite, and receives an FFDHE 220 ciphersuite from the server SHOULD take the following steps upon 221 receiving the ServerKeyExchange: 223 For non-anonymous ciphersuites where the offered Certificate is 224 valid and appropriate for the peer, validate the signature over 225 the ServerDHParams. If not valid, terminate the connection. 227 If the signature over ServerDHParams is valid, compare the 228 selected dh_p and dh_g with the FFDHE groups offered by the 229 client. If none of the offered groups match, the server is not 230 compatible with this draft. The client MAY decide to continue the 231 connection if the selected group is acceptable under local policy, 232 or it MAY decide to terminate the connection with a fatal 233 insufficient_security(71) alert. 235 If the client continues (either because the server offered a 236 matching group, or because local policy permits the offered custom 237 group), the client MUST verify that dh_Ys is in the range 1 < 238 dh_Ys < dh_p - 1. If dh_Ys is not in this range, the client MUST 239 terminate the connection with a fatal handshake_failure(40) alert. 241 If dh_Ys is in range, then the client SHOULD continue with the 242 connection as usual. 244 3.1. Client Local Policy on Custom Groups 246 Compatible clients that are willing to accept FFDHE ciphersuites from 247 non-compatible servers may have local policy about what custom FFDHE 248 groups they are willing to accept. This local policy presents a risk 249 to clients, who may accept weakly-protected communications from 250 misconfigured servers. 252 This draft cannot enumerate all possible safe local policy (the 253 safest may be to simply reject all custom groups), but compatible 254 clients that accept some custom groups from the server MUST do at 255 least cursory checks on group size, and may take other properties 256 into consideration as well. 258 A compatible client that accepts FFDHE ciphersuites using custom 259 groups from non-compatible servers MUST reject any group with |dh_p| 260 < 768 bits, and SHOULD reject any group with |dh_p| < 1024 bits. 262 A compatible client that rejects a non-compatible server's custom 263 group may decide to retry the connection while omitting all FFDHE 264 ciphersuites from the ClientHello. A client SHOULD only use this 265 approach if it successfully verified the server's expected signature 266 over the ServerDHParams, to avoid being forced by an active attacker 267 into a non-preferred ciphersuite. 269 4. Server Behavior 271 If a compatible TLS server receives a Supported Groups extension from 272 a client that includes any FFDHE group (i.e. any codepoint between 273 256 and 511 inclusive, even if unknown to the server), and if none of 274 the client-proposed FFDHE groups are known and acceptable to the 275 server, then the server MUST NOT select an FFDHE ciphersuite. In 276 this case, the server SHOULD select an acceptable non-FFDHE 277 ciphersuite from the client's offered list. If the extension is 278 present with FFDHE groups, none of the client's offered groups are 279 acceptable by the server, and none of the client's proposed non-FFDHE 280 ciphersuites are acceptable to the server, the server MUST end the 281 connection with a fatal TLS alert of type insufficient_security(71). 283 If at least one FFDHE ciphersuite is present in the client 284 ciphersuite list, and the Supported Groups extension is either absent 285 from the ClientHello entirely or contains no FFDHE groups (i.e. no 286 codepoints between 256 and 511 inclusive), then the server knows that 287 the client is not compatible with this document. In this scenario, a 288 server MAY select a non-FFDHE ciphersuite, or MAY select an FFDHE 289 ciphersuite and offer an FFDHE group of its choice to the client as 290 part of a traditional ServerKeyExchange. 292 A compatible TLS server that receives the Supported Groups extension 293 with FFDHE codepoints in it, and which selects an FFDHE ciphersuite 294 MUST select one of the client's offered groups. The server indicates 295 the choice of group to the client by sending the group's parameters 296 as usual in the ServerKeyExchange as described in section 7.4.3 of 297 [RFC5246]. 299 A TLS server MUST NOT select a named FFDHE group that was not offered 300 by a compatible client. 302 A TLS server MUST NOT select an FFDHE ciphersuite if the client did 303 not offer one, even if the client offered an FFDHE group in the 304 Supported Groups extension. 306 If a non-anonymous FFDHE ciphersuite is selected, and the TLS client 307 has used this extension to offer an FFDHE group of comparable or 308 greater strength than the server's public key, the server SHOULD 309 select an FFDHE group at least as strong as the server's public key. 310 For example, if the server has a 3072-bit RSA key, and the client 311 offers only ffdhe2048 and ffdhe4096, the server SHOULD select 312 ffdhe4096. 314 When an FFDHE ciphersuite is selected, and the client sends a 315 ClientKeyExchange, the server MUST verify that 1 < dh_Yc < dh_p - 1. 316 If dh_Yc is out of range, the server MUST terminate the connection 317 with fatal handshake_failure(40) alert. 319 5. Optimizations 321 In a key exchange with a successfully negotiated known FFDHE group, 322 both peers know that the group in question uses a safe prime as a 323 modulus, and that the group in use is of size p-1 or (p-1)/2. This 324 allows at least three optimizations that can be used to improve 325 performance. 327 5.1. Checking the Peer's Public Key 329 Peers MUST validate each other's public key Y (dh_Ys offered by the 330 server or dh_Yc offered by the client) by ensuring that 1 < Y < p-1. 331 This simple check ensures that the remote peer is properly behaved 332 and isn't forcing the local system into the 2-element subgroup. 334 To reach the same assurance with an unknown group, the client would 335 need to verify the primality of the modulus, learn the factors of 336 p-1, and test both the generator g and Y against each factor to avoid 337 small subgroup attacks. 339 5.2. Short Exponents 341 Traditional Finite Field Diffie-Hellman has each peer choose their 342 secret exponent from the range [2,p-2]. Using exponentiation by 343 squaring, this means each peer must do roughly 2*log_2(p) 344 multiplications, twice (once for the generator and once for the 345 peer's public key). 347 Peers concerned with performance may also prefer to choose their 348 secret exponent from a smaller range, doing fewer multiplications, 349 while retaining the same level of overall security. Each named group 350 indicates its approximate security level, and provides a lower-bound 351 on the range of secret exponents that should preserve it. For 352 example, rather than doing 2*2*3072 multiplications for a ffdhe3072 353 handshake, each peer can choose to do 2*2*275 multiplications by 354 choosing their secret exponent from the range [2^274,2^275] (that is, 355 a m-bit integer where m is at least 275) and still keep the same 356 approximate security level. 358 A similar short-exponent approach is suggested in SSH's Diffie- 359 Hellman key exchange (See section 6.2 of [RFC4419]). 361 5.3. Table Acceleration 363 Peers wishing to further accelerate FFDHE key exchange can also pre- 364 compute a table of powers of the generator of a known group. This is 365 a memory vs. time tradeoff, and it only accelerates the first 366 exponentiation of the ephemeral DH exchange (the fixed-base 367 exponentiation). The variable-base exponentiation (using the peer's 368 public exponent as a base) still needs to be calculated as normal. 370 6. Operational Considerations 372 6.1. Preference Ordering 374 The ordering of named groups in the Supported Groups extension may 375 contain some ECDHE groups and some FFDHE groups. These SHOULD be 376 ranked in the order preferred by the client. 378 However, the ClientHello also contains list of desired ciphersuites, 379 also ranked in preference order. This presents the possibility of 380 conflicted preferences. For example, if the ClientHello contains a 381 CipherSuite with two choices in order 382 and the Supported Groups 384 Extension contains two choices in order then 385 there is a clear contradiction. Clients SHOULD NOT present such a 386 contradiction since it does not represent a sensible ordering. A 387 server that encounters such an contradiction when selecting between 388 an ECDHE or FFDHE key exchange mechanism while trying to respect 389 client preferences SHOULD give priority to the Supported Groups 390 extension (in the example case, it should select 391 TLS_ECDHE_RSA_WITH_AES_128_CBC_SHA with secp256r1), but MAY resolve 392 the contradiction any way it sees fit. 394 More subtly, clients MAY interleave preferences between ECDHE and 395 FFDHE groups, for example if stronger groups are preferred regardless 396 of cost, but weaker groups are acceptable, the Supported Groups 397 extension could consist of: 398 . In this example, with the 399 same CipherSuite offered as the previous example, a server configured 400 to respect client preferences and with support for all listed groups 401 SHOULD select TLS_DHE_RSA_WITH_AES_128_CBC_SHA with ffdhe8192. A 402 server configured to respect client preferences and with support for 403 only secp384p1 and ffdhe3072 SHOULD select 404 TLS_ECDHE_RSA_WITH_AES_128_CBC_SHA with secp384p1. 406 7. Acknowledgements 408 Thanks to Fedor Brunner, Dave Fergemann, Niels Ferguson, Sandy 409 Harris, Tero Kivinen, Watson Ladd, Nikos Mavrogiannopolous, Niels 410 Moeller, Bodo Moeller, Kenny Paterson, Eric Rescorla, Tom Ritter, 411 Rene Struik, Martin Thomson, Sean Turner, and other members of the 412 TLS Working Group for their comments and suggestions on this draft. 413 Any mistakes here are not theirs. 415 8. IANA Considerations 417 IANA maintains the registry currently known as EC Named Curves 418 (originally defined in [RFC4492] and updated by [RFC7027]) at [1]. 420 This document expands the semantics of this registry slightly, to 421 include groups based on finite fields in addition to groups based on 422 elliptic curves. IANA should add a range designation to that 423 registry, indicating that values from 256-511 (inclusive) are set 424 aside for "Finite Field Diffie-Hellman groups", and that all other 425 entries in the registry are "Elliptic curve groups". 427 This document allocates five well-defined codepoints in the registry 428 for specific Finite Field Diffie-Hellman groups defined in 429 Appendix A. 431 In addition, the four highest codepoints in this range (508-511, 432 inclusive) are designated for PRIVATE USE by peers who have 433 privately-developed Finite Field Diffie-Hellman groups that they wish 434 to signal internally. 436 The updated registry section should be as follows: 438 +---------------------+-------------+---------+-----------------+ 439 | Value | Description | DTLS-OK | Reference | 440 +---------------------+-------------+---------+-----------------+ 441 | 256 | ffdhe2048 | Y | [this document] | 442 | 257 | ffdhe3072 | Y | [this document] | 443 | 258 | ffdhe4096 | Y | [this document] | 444 | 259 | ffdhe6144 | Y | [this document] | 445 | 260 | ffdhe8192 | Y | [this document] | 446 | 508-511 (inclusive) | PRIVATE USE | - | - | 447 +---------------------+-------------+---------+-----------------+ 449 9. Security Considerations 451 9.1. Negotiation resistance to active attacks 453 Because the contents of the Supported Groups extension are hashed in 454 the finished message, an active MITM that tries to filter or omit 455 groups will cause the handshake to fail, but possibly not before 456 getting the peer to do something they would not otherwise have done. 458 An attacker who impersonates the server can try to do any of the 459 following: 461 Pretend that a non-compatible server is actually capable of this 462 extension, and select a group from the client's list, causing the 463 client to select a group it is willing to negotiate. It is 464 unclear how this would be an effective attack. 466 Pretend that a compatible server is actually non-compatible by 467 negotiating a non-FFDHE ciphersuite. This is no different than 468 MITM ciphersuite filtering. 470 Pretend that a compatible server is actually non-compatible by 471 negotiating a DHE ciphersuite, with a custom (perhaps weak) group 472 selected by the attacker. This is no worse than the current 473 scenario, and would require the attacker to be able to sign the 474 ServerDHParams, which should not be possible without access to the 475 server's secret key. 477 An attacker who impersonates the client can try to do the following: 479 Pretend that a compatible client is not compatible (e.g., by not 480 offering the Supported Groups extension, or by replacing the 481 Supported Groups extension with one that includes no FFDHE 482 groups). This could cause the server to negotiate a weaker DHE 483 group during the handshake, or to select a non-FFDHE ciphersuite, 484 but it would fail to complete during the final check of the 485 Finished message. 487 Pretend that a non-compatible client is compatible (e.g., by 488 adding the Supported Groups extension, or by adding FFDHE groups 489 to the extension). This could cause the server to select a 490 particular named group in the ServerKeyExchange, or to avoid 491 selecting an FFDHE ciphersuite. The peers would fail to compute 492 the final check of the Finished message. 494 Change the list of groups offered by the client (e.g., by removing 495 the stronger of the set of groups offered). This could cause the 496 server to negotiate a weaker group than desired, but again should 497 be caught by the check in the Finished message. 499 9.2. Group strength considerations 501 TLS implementations using FFDHE key exchange should consider the 502 strength of the group they negotiate. The strength of the selected 503 group is one of the factors that define the connection's resiliance 504 against attacks on the session's confidentiality and integrity, since 505 the session keys are derived from the DHE handshake. 507 While attacks on integrity must generally happen while the session is 508 in progress, attacks against session confidentiality can happen 509 significantly later, if the entire TLS session is stored for offline 510 analysis. Therefore, FFDHE groups should be selected by clients and 511 servers based on confidentiality guarantees they need. Sessions 512 which need extremely long-term confidentiality should prefer stronger 513 groups. 515 [ENISA] provides rough estimates of group resistance to attack, and 516 recommends that forward-looking implementations ("future systems") 517 should use FFDHE group sizes of at least 3072 bits. ffdhe3072 is 518 intended for use in these implementations. 520 Other sources (e.g., [NIST]) estimate the security levels of the DLOG 521 problem to be slightly more difficult than [ENISA]. This document's 522 suggested minimum exponent sizes in Appendix A for implementations 523 that use the short exponents optimization (Section 5.2) are 524 deliberately conservative to account for the range of these 525 estimates. 527 9.3. Finite-Field DHE only 529 Note that this document specifically targets only finite field-based 530 Diffie-Hellman ephemeral key exchange mechanisms. It does not cover 531 the non-ephemeral DH key exchange mechanisms, nor does it address 532 elliptic curve DHE (ECDHE) key exchange, which is defined in 533 [RFC4492]. 535 Measured by computational cost to the TLS peers, ECDHE appears today 536 to offer much a stronger key exchange mechanism than FFDHE. 538 9.4. Deprecating weak groups 540 Advances in hardware or in finite field cryptanalysis may cause some 541 of the negotiated groups to not provide the desired security margins, 542 as indicated by the estimated work factor of an adversary to discover 543 the premaster secret (and may therefore compromise the 544 confidentiality and integrity of the TLS session). 546 Revisions of this document should mark known-weak groups as 547 explicitly deprecated for use in TLS, and should update the estimated 548 work factor needed to break the group, if the cryptanalysis has 549 changed. Implementations that require strong confidentiality and 550 integrity guarantees should avoid using deprecated groups and should 551 be updated when the estimated security margins are updated. 553 9.5. Choice of groups 555 Other lists of named finite field Diffie-Hellman groups 556 [STRONGSWAN-IKE] exist. This draft chooses to not reuse them for 557 several reasons: 559 Using the same groups in multiple protocols increases the value 560 for an attacker with the resources to crack any single group. 562 The IKE groups include weak groups like MODP768 which are 563 unacceptable for secure TLS traffic. 565 Mixing group parameters across multiple implementations leaves 566 open the possibility of some sort of cross-protocol attack. This 567 shouldn't be relevant for ephemeral scenarios, and even with non- 568 ephemeral keying, services shouldn't share keys; however, using 569 different groups avoids these failure modes entirely. 571 9.6. Timing attacks 573 Any implementation of finite field Diffie-Hellman key exchange should 574 use constant-time modular-exponentiation implementations. This is 575 particularly true for those implementations that ever re-use DHE 576 secret keys (so-called "semi-static" ephemeral keying) or share DHE 577 secret keys across a multiple machines (e.g., in a load-balancer 578 situation). 580 9.7. Replay attacks from non-negotiated FFDHE 582 [SECURE-RESUMPTION], [CROSS-PROTOCOL], and [SSL3-ANALYSIS] all show a 583 malicious peer using a bad FFDHE group to maneuver a client into 584 selecting a pre-master secret of the peer's choice, which can be 585 replayed to another server using a non-FFDHE key exchange, and can 586 then be bootstrapped to replay client authentication. 588 To prevent this attack (barring the fixes proposed in 589 [SESSION-HASH]), a client would need not only to implement this 590 draft, but also to reject non-negotiated FFDHE ciphersuites whose 591 group structure it cannot afford to verify. Such a client would need 592 to abort the initial handshake and reconnect to the server in 593 question without listing any FFDHE ciphersuites on the subsequent 594 connection. 596 This tradeoff may be too costly for most TLS clients today, but may 597 be a reasonable choice for clients performing client certificate 598 authentication, or who have other reason to be concerned about 599 server-controlled pre-master secrets. 601 9.8. Forward Secrecy 603 One of the main reasons to prefer FFDHE ciphersuites is Forward 604 Secrecy, the ability to resist decryption even if when the endpoint's 605 long-term secret key (usually RSA) is revealed in the future. 607 This property depends on both sides of the connection discarding 608 their ephemeral keys promptly. Implementations should wipe their 609 FFDHE secret key material from memory as soon as it is no longer 610 needed, and should never store it in persistent storage. 612 Forward secrecy also depends on the strength of the Diffie-Hellman 613 group; using a very strong symmetric cipher like AES256 with a 614 forward-secret ciphersuite, but generating the keys with a much 615 weaker group like dhe2048 simply moves the adversary's cost from 616 attacking the symmetric cipher to attacking the dh_Ys or dh_Yc 617 ephemeral keyshares. 619 If the goal is to provide forward secrecy, attention should be paid 620 to all parts of the ciphersuite selection process, both key exchange 621 and symmetric cipher choice. 623 9.9. False Start 625 Clients capable of TLS False Start [FALSE-START] may receive a 626 proposed FFDHE group from a server that is attacker-controlled. In 627 particular, the attacker can modify the ClientHello to strip the 628 proposed FFDHE groups, which may cause the server to offer a weaker 629 FFDHE group than it should, and this will not be detected until 630 receipt of the server's Finished message. This could cause a client 631 using the False Start protocol modification to send data encrypted 632 under a weak key agreement. 634 Clients should have their own classification of FFDHE groups that are 635 "cryptographically strong" in the same sense described in the 636 description of symmetric ciphers in [FALSE-START], and SHOULD offer 637 at least one of these in the initial handshake if they contemplate 638 using the False Start protocol modification with an FFDHE 639 ciphersuite. 641 Compatible clients performing a full handshake MUST NOT use the False 642 Start protocol modification if the server selects an FFDHE 643 ciphersuite but sends a group that is not cryptographically strong 644 from the client's perspective. 646 10. Privacy Considerations 648 10.1. Client fingerprinting 650 This extension provides a few additional bits of information to 651 distinguish between classes of TLS clients (see e.g. 652 [PANOPTICLICK]). To minimize this sort of fingerprinting, clients 653 SHOULD support all named groups at or above their minimum security 654 threshhold. New named groups SHOULD NOT be added to the registry 655 without consideration of the cost of browser fingerprinting. 657 11. References 659 11.1. Normative References 661 [FALSE-START] 662 Langley, A., Modadugu, N., and B. Moeller, "Transport 663 Layer Security (TLS) False Start", Work in Progress, 664 draft-bmoeller-tls-falsestart-01, November 2014. 666 [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate 667 Requirement Levels", BCP 14, RFC 2119, March 1997. 669 [RFC4492] Blake-Wilson, S., Bolyard, N., Gupta, V., Hawk, C., and B. 670 Moeller, "Elliptic Curve Cryptography (ECC) Cipher Suites 671 for Transport Layer Security (TLS)", RFC 4492, May 2006. 673 [RFC5226] Narten, T. and H. Alvestrand, "Guidelines for Writing an 674 IANA Considerations Section in RFCs", BCP 26, RFC 5226, 675 May 2008. 677 [RFC5246] Dierks, T. and E. Rescorla, "The Transport Layer Security 678 (TLS) Protocol Version 1.2", RFC 5246, August 2008. 680 11.2. Informative References 682 [CROSS-PROTOCOL] 683 Mavrogiannopolous, N., Vercauteren, F., Velichkov, V., and 684 B. Preneel, "A Cross-Protocol Attack on the TLS Protocol", 685 October 2012, 686 . 689 [ECRYPTII] 690 European Network of Excellence in Cryptology II, "ECRYPT 691 II Yearly Report on Algorithms and Keysizes (2011-2012)", 692 September 2012, 693 . 695 [ENISA] European Union Agency for Network and Information Security 696 Agency, "Algorithms, Key Sizes and Parameters Report, 697 version 1.0", October 2013, 698 . 702 [NIST] National Institute of Standards and Technology, "NIST 703 Special Publication 800-57. Recommendation for key 704 management - Part 1: General (Revision 3)", 2012, 705 . 708 [PANOPTICLICK] 709 Electronic Frontier Foundation, "Panopticlick: How Unique 710 - and Trackable - Is Your Browser?", 2010, 711 . 713 [RFC3526] Kivinen, T. and M. Kojo, "More Modular Exponential (MODP) 714 Diffie-Hellman groups for Internet Key Exchange (IKE)", 715 RFC 3526, May 2003. 717 [RFC4419] Friedl, M., Provos, N., and W. Simpson, "Diffie-Hellman 718 Group Exchange for the Secure Shell (SSH) Transport Layer 719 Protocol", RFC 4419, March 2006. 721 [RFC7027] Merkle, J. and M. Lochter, "Elliptic Curve Cryptography 722 (ECC) Brainpool Curves for Transport Layer Security 723 (TLS)", RFC 7027, October 2013. 725 [SECURE-RESUMPTION] 726 Delignat-Lavaud, A., Bhargavan, K., and A. Pironti, 727 "Triple Handshakes Considered Harmful: Breaking and Fixing 728 Authentication over TLS", March 2014, . 731 [SESSION-HASH] 732 Bhargavan, K., Delignat-Lavaud, A., Pironti, A., Langley, 733 A., and M. Ray, "Triple Handshakes Considered Harmful: 734 Breaking and Fixing Authentication over TLS", March 2014, 735 . 738 [SSL3-ANALYSIS] 739 Schneier, B. and D. Wagner, "Analysis of the SSL 3.0 740 protocol", 1996, . 742 [STRONGSWAN-IKE] 743 Brunner, T. and A. Steffen, "Diffie Hellman Groups in 744 IKEv2 Cipher Suites", October 2013, 745 . 748 11.3. URIs 750 [1] https://www.iana.org/assignments/tls-parameters/tls- 751 parameters.xhtml#tls-parameters-8 753 Appendix A. Named Group Registry 755 Each description below indicates the group itself, its derivation, 756 its expected strength (estimated roughly from guidelines in 757 [ECRYPTII]), and whether it is recommended for use in TLS key 758 exchange at the given security level. It is not recommended to add 759 further finite field groups to the NamedCurves registry; any attempt 760 to do so should consider Section 10.1. 762 The primes in these finite field groups are all safe primes, that is, 763 a prime p is a safe prime when q = (p-1)/2 is also prime. Where e is 764 the base of the natural logarithm, and square brackets denote the 765 floor operation, the groups which initially populate this registry 766 are derived for a given bitlength b by finding the lowest positive 767 integer X that creates a safe prime p where: 769 p = 2^b - 2^{b-64} + {[2^{b-130} e] + X } * 2^64 - 1 771 New additions of FFDHE groups to this registry may use this same 772 derivation (e.g., with different bitlengths) or may choose their 773 parameters in a different way, but must be clear about how the 774 parameters were derived. 776 New additions of FFDHE groups MUST use a safe prime as the modulus to 777 enable the inexpensive peer verification described in Section 5.1. 779 A.1. ffdhe2048 781 The 2048-bit group has registry value 256, and is calculated from the 782 following formula: 784 The modulus is: p = 2^2048 - 2^1984 + {[2^1918 * e] + 560316 } * 2^64 785 - 1 787 The hexadecimal representation of p is: 789 FFFFFFFF FFFFFFFF ADF85458 A2BB4A9A AFDC5620 273D3CF1 790 D8B9C583 CE2D3695 A9E13641 146433FB CC939DCE 249B3EF9 791 7D2FE363 630C75D8 F681B202 AEC4617A D3DF1ED5 D5FD6561 792 2433F51F 5F066ED0 85636555 3DED1AF3 B557135E 7F57C935 793 984F0C70 E0E68B77 E2A689DA F3EFE872 1DF158A1 36ADE735 794 30ACCA4F 483A797A BC0AB182 B324FB61 D108A94B B2C8E3FB 795 B96ADAB7 60D7F468 1D4F42A3 DE394DF4 AE56EDE7 6372BB19 796 0B07A7C8 EE0A6D70 9E02FCE1 CDF7E2EC C03404CD 28342F61 797 9172FE9C E98583FF 8E4F1232 EEF28183 C3FE3B1B 4C6FAD73 798 3BB5FCBC 2EC22005 C58EF183 7D1683B2 C6F34A26 C1B2EFFA 799 886B4238 61285C97 FFFFFFFF FFFFFFFF 801 The generator is: g = 2 803 The group size is: q = (p-1)/2 805 The hexadecimal representation of q is: 807 7FFFFFFF FFFFFFFF D6FC2A2C 515DA54D 57EE2B10 139E9E78 808 EC5CE2C1 E7169B4A D4F09B20 8A3219FD E649CEE7 124D9F7C 809 BE97F1B1 B1863AEC 7B40D901 576230BD 69EF8F6A EAFEB2B0 810 9219FA8F AF833768 42B1B2AA 9EF68D79 DAAB89AF 3FABE49A 811 CC278638 707345BB F15344ED 79F7F439 0EF8AC50 9B56F39A 812 98566527 A41D3CBD 5E0558C1 59927DB0 E88454A5 D96471FD 813 DCB56D5B B06BFA34 0EA7A151 EF1CA6FA 572B76F3 B1B95D8C 814 8583D3E4 770536B8 4F017E70 E6FBF176 601A0266 941A17B0 815 C8B97F4E 74C2C1FF C7278919 777940C1 E1FF1D8D A637D6B9 816 9DDAFE5E 17611002 E2C778C1 BE8B41D9 6379A513 60D977FD 817 4435A11C 30942E4B FFFFFFFF FFFFFFFF 819 The estimated symmetric-equivalent strength of this group is 103 820 bits. 822 Peers using ffdhe2048 that want to optimize their key exchange with a 823 short exponent (Section 5.2) should choose a secret key of at least 824 225 bits. 826 A.2. ffdhe3072 828 The 3072-bit prime has registry value 257, and is calculated from the 829 following formula: 831 The modulus is: p = 2^3072 - 2^3008 + {[2^2942 * e] + 2625351} * 2^64 832 -1 834 The hexadecimal representation of p is: 836 FFFFFFFF FFFFFFFF ADF85458 A2BB4A9A AFDC5620 273D3CF1 837 D8B9C583 CE2D3695 A9E13641 146433FB CC939DCE 249B3EF9 838 7D2FE363 630C75D8 F681B202 AEC4617A D3DF1ED5 D5FD6561 839 2433F51F 5F066ED0 85636555 3DED1AF3 B557135E 7F57C935 840 984F0C70 E0E68B77 E2A689DA F3EFE872 1DF158A1 36ADE735 841 30ACCA4F 483A797A BC0AB182 B324FB61 D108A94B B2C8E3FB 842 B96ADAB7 60D7F468 1D4F42A3 DE394DF4 AE56EDE7 6372BB19 843 0B07A7C8 EE0A6D70 9E02FCE1 CDF7E2EC C03404CD 28342F61 844 9172FE9C E98583FF 8E4F1232 EEF28183 C3FE3B1B 4C6FAD73 845 3BB5FCBC 2EC22005 C58EF183 7D1683B2 C6F34A26 C1B2EFFA 846 886B4238 611FCFDC DE355B3B 6519035B BC34F4DE F99C0238 847 61B46FC9 D6E6C907 7AD91D26 91F7F7EE 598CB0FA C186D91C 848 AEFE1309 85139270 B4130C93 BC437944 F4FD4452 E2D74DD3 849 64F2E21E 71F54BFF 5CAE82AB 9C9DF69E E86D2BC5 22363A0D 850 ABC52197 9B0DEADA 1DBF9A42 D5C4484E 0ABCD06B FA53DDEF 851 3C1B20EE 3FD59D7C 25E41D2B 66C62E37 FFFFFFFF FFFFFFFF 853 The generator is: g = 2 854 The group size is: q = (p-1)/2 856 The hexadecimal representation of q is: 858 7FFFFFFF FFFFFFFF D6FC2A2C 515DA54D 57EE2B10 139E9E78 859 EC5CE2C1 E7169B4A D4F09B20 8A3219FD E649CEE7 124D9F7C 860 BE97F1B1 B1863AEC 7B40D901 576230BD 69EF8F6A EAFEB2B0 861 9219FA8F AF833768 42B1B2AA 9EF68D79 DAAB89AF 3FABE49A 862 CC278638 707345BB F15344ED 79F7F439 0EF8AC50 9B56F39A 863 98566527 A41D3CBD 5E0558C1 59927DB0 E88454A5 D96471FD 864 DCB56D5B B06BFA34 0EA7A151 EF1CA6FA 572B76F3 B1B95D8C 865 8583D3E4 770536B8 4F017E70 E6FBF176 601A0266 941A17B0 866 C8B97F4E 74C2C1FF C7278919 777940C1 E1FF1D8D A637D6B9 867 9DDAFE5E 17611002 E2C778C1 BE8B41D9 6379A513 60D977FD 868 4435A11C 308FE7EE 6F1AAD9D B28C81AD DE1A7A6F 7CCE011C 869 30DA37E4 EB736483 BD6C8E93 48FBFBF7 2CC6587D 60C36C8E 870 577F0984 C289C938 5A098649 DE21BCA2 7A7EA229 716BA6E9 871 B279710F 38FAA5FF AE574155 CE4EFB4F 743695E2 911B1D06 872 D5E290CB CD86F56D 0EDFCD21 6AE22427 055E6835 FD29EEF7 873 9E0D9077 1FEACEBE 12F20E95 B363171B FFFFFFFF FFFFFFFF 875 The estimated symmetric-equivalent strength of this group is 125 876 bits. 878 Peers using ffdhe3072 that want to optimize their key exchange with a 879 short exponent (Section 5.2) should choose a secret key of at least 880 275 bits. 882 A.3. ffdhe4096 884 The 4096-bit group has registry value 258, and is calculated from the 885 following formula: 887 The modulus is: p = 2^4096 - 2^4032 + {[2^3966 * e] + 5736041} * 2^64 888 - 1 890 The hexadecimal representation of p is: 892 FFFFFFFF FFFFFFFF ADF85458 A2BB4A9A AFDC5620 273D3CF1 893 D8B9C583 CE2D3695 A9E13641 146433FB CC939DCE 249B3EF9 894 7D2FE363 630C75D8 F681B202 AEC4617A D3DF1ED5 D5FD6561 895 2433F51F 5F066ED0 85636555 3DED1AF3 B557135E 7F57C935 896 984F0C70 E0E68B77 E2A689DA F3EFE872 1DF158A1 36ADE735 897 30ACCA4F 483A797A BC0AB182 B324FB61 D108A94B B2C8E3FB 898 B96ADAB7 60D7F468 1D4F42A3 DE394DF4 AE56EDE7 6372BB19 899 0B07A7C8 EE0A6D70 9E02FCE1 CDF7E2EC C03404CD 28342F61 900 9172FE9C E98583FF 8E4F1232 EEF28183 C3FE3B1B 4C6FAD73 901 3BB5FCBC 2EC22005 C58EF183 7D1683B2 C6F34A26 C1B2EFFA 902 886B4238 611FCFDC DE355B3B 6519035B BC34F4DE F99C0238 903 61B46FC9 D6E6C907 7AD91D26 91F7F7EE 598CB0FA C186D91C 904 AEFE1309 85139270 B4130C93 BC437944 F4FD4452 E2D74DD3 905 64F2E21E 71F54BFF 5CAE82AB 9C9DF69E E86D2BC5 22363A0D 906 ABC52197 9B0DEADA 1DBF9A42 D5C4484E 0ABCD06B FA53DDEF 907 3C1B20EE 3FD59D7C 25E41D2B 669E1EF1 6E6F52C3 164DF4FB 908 7930E9E4 E58857B6 AC7D5F42 D69F6D18 7763CF1D 55034004 909 87F55BA5 7E31CC7A 7135C886 EFB4318A ED6A1E01 2D9E6832 910 A907600A 918130C4 6DC778F9 71AD0038 092999A3 33CB8B7A 911 1A1DB93D 7140003C 2A4ECEA9 F98D0ACC 0A8291CD CEC97DCF 912 8EC9B55A 7F88A46B 4DB5A851 F44182E1 C68A007E 5E655F6A 913 FFFFFFFF FFFFFFFF 915 The generator is: g = 2 917 The group size is: q = (p-1)/2 919 The hexadecimal representation of q is: 921 7FFFFFFF FFFFFFFF D6FC2A2C 515DA54D 57EE2B10 139E9E78 922 EC5CE2C1 E7169B4A D4F09B20 8A3219FD E649CEE7 124D9F7C 923 BE97F1B1 B1863AEC 7B40D901 576230BD 69EF8F6A EAFEB2B0 924 9219FA8F AF833768 42B1B2AA 9EF68D79 DAAB89AF 3FABE49A 925 CC278638 707345BB F15344ED 79F7F439 0EF8AC50 9B56F39A 926 98566527 A41D3CBD 5E0558C1 59927DB0 E88454A5 D96471FD 927 DCB56D5B B06BFA34 0EA7A151 EF1CA6FA 572B76F3 B1B95D8C 928 8583D3E4 770536B8 4F017E70 E6FBF176 601A0266 941A17B0 929 C8B97F4E 74C2C1FF C7278919 777940C1 E1FF1D8D A637D6B9 930 9DDAFE5E 17611002 E2C778C1 BE8B41D9 6379A513 60D977FD 931 4435A11C 308FE7EE 6F1AAD9D B28C81AD DE1A7A6F 7CCE011C 932 30DA37E4 EB736483 BD6C8E93 48FBFBF7 2CC6587D 60C36C8E 933 577F0984 C289C938 5A098649 DE21BCA2 7A7EA229 716BA6E9 934 B279710F 38FAA5FF AE574155 CE4EFB4F 743695E2 911B1D06 935 D5E290CB CD86F56D 0EDFCD21 6AE22427 055E6835 FD29EEF7 936 9E0D9077 1FEACEBE 12F20E95 B34F0F78 B737A961 8B26FA7D 937 BC9874F2 72C42BDB 563EAFA1 6B4FB68C 3BB1E78E AA81A002 938 43FAADD2 BF18E63D 389AE443 77DA18C5 76B50F00 96CF3419 939 5483B005 48C09862 36E3BC7C B8D6801C 0494CCD1 99E5C5BD 940 0D0EDC9E B8A0001E 15276754 FCC68566 054148E6 E764BEE7 941 C764DAAD 3FC45235 A6DAD428 FA20C170 E345003F 2F32AFB5 942 7FFFFFFF FFFFFFFF 944 The estimated symmetric-equivalent strength of this group is 150 945 bits. 947 Peers using ffdhe4096 that want to optimize their key exchange with a 948 short exponent (Section 5.2) should choose a secret key of at least 949 325 bits. 951 A.4. ffdhe6144 953 The 6144-bit group has registry value 259, and is calculated from the 954 following formula: 956 The modulus is: p = 2^6144 - 2^6080 + {[2^6014 * e] + 15705020} * 957 2^64 - 1 959 The hexadecimal representation of p is: 961 FFFFFFFF FFFFFFFF ADF85458 A2BB4A9A AFDC5620 273D3CF1 962 D8B9C583 CE2D3695 A9E13641 146433FB CC939DCE 249B3EF9 963 7D2FE363 630C75D8 F681B202 AEC4617A D3DF1ED5 D5FD6561 964 2433F51F 5F066ED0 85636555 3DED1AF3 B557135E 7F57C935 965 984F0C70 E0E68B77 E2A689DA F3EFE872 1DF158A1 36ADE735 966 30ACCA4F 483A797A BC0AB182 B324FB61 D108A94B B2C8E3FB 967 B96ADAB7 60D7F468 1D4F42A3 DE394DF4 AE56EDE7 6372BB19 968 0B07A7C8 EE0A6D70 9E02FCE1 CDF7E2EC C03404CD 28342F61 969 9172FE9C E98583FF 8E4F1232 EEF28183 C3FE3B1B 4C6FAD73 970 3BB5FCBC 2EC22005 C58EF183 7D1683B2 C6F34A26 C1B2EFFA 971 886B4238 611FCFDC DE355B3B 6519035B BC34F4DE F99C0238 972 61B46FC9 D6E6C907 7AD91D26 91F7F7EE 598CB0FA C186D91C 973 AEFE1309 85139270 B4130C93 BC437944 F4FD4452 E2D74DD3 974 64F2E21E 71F54BFF 5CAE82AB 9C9DF69E E86D2BC5 22363A0D 975 ABC52197 9B0DEADA 1DBF9A42 D5C4484E 0ABCD06B FA53DDEF 976 3C1B20EE 3FD59D7C 25E41D2B 669E1EF1 6E6F52C3 164DF4FB 977 7930E9E4 E58857B6 AC7D5F42 D69F6D18 7763CF1D 55034004 978 87F55BA5 7E31CC7A 7135C886 EFB4318A ED6A1E01 2D9E6832 979 A907600A 918130C4 6DC778F9 71AD0038 092999A3 33CB8B7A 980 1A1DB93D 7140003C 2A4ECEA9 F98D0ACC 0A8291CD CEC97DCF 981 8EC9B55A 7F88A46B 4DB5A851 F44182E1 C68A007E 5E0DD902 982 0BFD64B6 45036C7A 4E677D2C 38532A3A 23BA4442 CAF53EA6 983 3BB45432 9B7624C8 917BDD64 B1C0FD4C B38E8C33 4C701C3A 984 CDAD0657 FCCFEC71 9B1F5C3E 4E46041F 388147FB 4CFDB477 985 A52471F7 A9A96910 B855322E DB6340D8 A00EF092 350511E3 986 0ABEC1FF F9E3A26E 7FB29F8C 183023C3 587E38DA 0077D9B4 987 763E4E4B 94B2BBC1 94C6651E 77CAF992 EEAAC023 2A281BF6 988 B3A739C1 22611682 0AE8DB58 47A67CBE F9C9091B 462D538C 989 D72B0374 6AE77F5E 62292C31 1562A846 505DC82D B854338A 990 E49F5235 C95B9117 8CCF2DD5 CACEF403 EC9D1810 C6272B04 991 5B3B71F9 DC6B80D6 3FDD4A8E 9ADB1E69 62A69526 D43161C1 992 A41D570D 7938DAD4 A40E329C D0E40E65 FFFFFFFF FFFFFFFF 994 The generator is: g = 2 996 The group size is: q = (p-1)/2 998 The hexadecimal representation of q is: 1000 7FFFFFFF FFFFFFFF D6FC2A2C 515DA54D 57EE2B10 139E9E78 1001 EC5CE2C1 E7169B4A D4F09B20 8A3219FD E649CEE7 124D9F7C 1002 BE97F1B1 B1863AEC 7B40D901 576230BD 69EF8F6A EAFEB2B0 1003 9219FA8F AF833768 42B1B2AA 9EF68D79 DAAB89AF 3FABE49A 1004 CC278638 707345BB F15344ED 79F7F439 0EF8AC50 9B56F39A 1005 98566527 A41D3CBD 5E0558C1 59927DB0 E88454A5 D96471FD 1006 DCB56D5B B06BFA34 0EA7A151 EF1CA6FA 572B76F3 B1B95D8C 1007 8583D3E4 770536B8 4F017E70 E6FBF176 601A0266 941A17B0 1008 C8B97F4E 74C2C1FF C7278919 777940C1 E1FF1D8D A637D6B9 1009 9DDAFE5E 17611002 E2C778C1 BE8B41D9 6379A513 60D977FD 1010 4435A11C 308FE7EE 6F1AAD9D B28C81AD DE1A7A6F 7CCE011C 1011 30DA37E4 EB736483 BD6C8E93 48FBFBF7 2CC6587D 60C36C8E 1012 577F0984 C289C938 5A098649 DE21BCA2 7A7EA229 716BA6E9 1013 B279710F 38FAA5FF AE574155 CE4EFB4F 743695E2 911B1D06 1014 D5E290CB CD86F56D 0EDFCD21 6AE22427 055E6835 FD29EEF7 1015 9E0D9077 1FEACEBE 12F20E95 B34F0F78 B737A961 8B26FA7D 1016 BC9874F2 72C42BDB 563EAFA1 6B4FB68C 3BB1E78E AA81A002 1017 43FAADD2 BF18E63D 389AE443 77DA18C5 76B50F00 96CF3419 1018 5483B005 48C09862 36E3BC7C B8D6801C 0494CCD1 99E5C5BD 1019 0D0EDC9E B8A0001E 15276754 FCC68566 054148E6 E764BEE7 1020 C764DAAD 3FC45235 A6DAD428 FA20C170 E345003F 2F06EC81 1021 05FEB25B 2281B63D 2733BE96 1C29951D 11DD2221 657A9F53 1022 1DDA2A19 4DBB1264 48BDEEB2 58E07EA6 59C74619 A6380E1D 1023 66D6832B FE67F638 CD8FAE1F 2723020F 9C40A3FD A67EDA3B 1024 D29238FB D4D4B488 5C2A9917 6DB1A06C 50077849 1A8288F1 1025 855F60FF FCF1D137 3FD94FC6 0C1811E1 AC3F1C6D 003BECDA 1026 3B1F2725 CA595DE0 CA63328F 3BE57CC9 77556011 95140DFB 1027 59D39CE0 91308B41 05746DAC 23D33E5F 7CE4848D A316A9C6 1028 6B9581BA 3573BFAF 31149618 8AB15423 282EE416 DC2A19C5 1029 724FA91A E4ADC88B C66796EA E5677A01 F64E8C08 63139582 1030 2D9DB8FC EE35C06B 1FEEA547 4D6D8F34 B1534A93 6A18B0E0 1031 D20EAB86 BC9C6D6A 5207194E 68720732 FFFFFFFF FFFFFFFF 1033 The estimated symmetric-equivalent strength of this group is 175 1034 bits. 1036 Peers using ffdhe6144 that want to optimize their key exchange with a 1037 short exponent (Section 5.2) should choose a secret key of at least 1038 375 bits. 1040 A.5. ffdhe8192 1042 The 8192-bit group has registry value 260, and is calculated from the 1043 following formula: 1045 The modulus is: p = 2^8192 - 2^8128 + {[2^8062 * e] + 10965728} * 1046 2^64 - 1 1047 The hexadecimal representation of p is: 1049 FFFFFFFF FFFFFFFF ADF85458 A2BB4A9A AFDC5620 273D3CF1 1050 D8B9C583 CE2D3695 A9E13641 146433FB CC939DCE 249B3EF9 1051 7D2FE363 630C75D8 F681B202 AEC4617A D3DF1ED5 D5FD6561 1052 2433F51F 5F066ED0 85636555 3DED1AF3 B557135E 7F57C935 1053 984F0C70 E0E68B77 E2A689DA F3EFE872 1DF158A1 36ADE735 1054 30ACCA4F 483A797A BC0AB182 B324FB61 D108A94B B2C8E3FB 1055 B96ADAB7 60D7F468 1D4F42A3 DE394DF4 AE56EDE7 6372BB19 1056 0B07A7C8 EE0A6D70 9E02FCE1 CDF7E2EC C03404CD 28342F61 1057 9172FE9C E98583FF 8E4F1232 EEF28183 C3FE3B1B 4C6FAD73 1058 3BB5FCBC 2EC22005 C58EF183 7D1683B2 C6F34A26 C1B2EFFA 1059 886B4238 611FCFDC DE355B3B 6519035B BC34F4DE F99C0238 1060 61B46FC9 D6E6C907 7AD91D26 91F7F7EE 598CB0FA C186D91C 1061 AEFE1309 85139270 B4130C93 BC437944 F4FD4452 E2D74DD3 1062 64F2E21E 71F54BFF 5CAE82AB 9C9DF69E E86D2BC5 22363A0D 1063 ABC52197 9B0DEADA 1DBF9A42 D5C4484E 0ABCD06B FA53DDEF 1064 3C1B20EE 3FD59D7C 25E41D2B 669E1EF1 6E6F52C3 164DF4FB 1065 7930E9E4 E58857B6 AC7D5F42 D69F6D18 7763CF1D 55034004 1066 87F55BA5 7E31CC7A 7135C886 EFB4318A ED6A1E01 2D9E6832 1067 A907600A 918130C4 6DC778F9 71AD0038 092999A3 33CB8B7A 1068 1A1DB93D 7140003C 2A4ECEA9 F98D0ACC 0A8291CD CEC97DCF 1069 8EC9B55A 7F88A46B 4DB5A851 F44182E1 C68A007E 5E0DD902 1070 0BFD64B6 45036C7A 4E677D2C 38532A3A 23BA4442 CAF53EA6 1071 3BB45432 9B7624C8 917BDD64 B1C0FD4C B38E8C33 4C701C3A 1072 CDAD0657 FCCFEC71 9B1F5C3E 4E46041F 388147FB 4CFDB477 1073 A52471F7 A9A96910 B855322E DB6340D8 A00EF092 350511E3 1074 0ABEC1FF F9E3A26E 7FB29F8C 183023C3 587E38DA 0077D9B4 1075 763E4E4B 94B2BBC1 94C6651E 77CAF992 EEAAC023 2A281BF6 1076 B3A739C1 22611682 0AE8DB58 47A67CBE F9C9091B 462D538C 1077 D72B0374 6AE77F5E 62292C31 1562A846 505DC82D B854338A 1078 E49F5235 C95B9117 8CCF2DD5 CACEF403 EC9D1810 C6272B04 1079 5B3B71F9 DC6B80D6 3FDD4A8E 9ADB1E69 62A69526 D43161C1 1080 A41D570D 7938DAD4 A40E329C CFF46AAA 36AD004C F600C838 1081 1E425A31 D951AE64 FDB23FCE C9509D43 687FEB69 EDD1CC5E 1082 0B8CC3BD F64B10EF 86B63142 A3AB8829 555B2F74 7C932665 1083 CB2C0F1C C01BD702 29388839 D2AF05E4 54504AC7 8B758282 1084 2846C0BA 35C35F5C 59160CC0 46FD8251 541FC68C 9C86B022 1085 BB709987 6A460E74 51A8A931 09703FEE 1C217E6C 3826E52C 1086 51AA691E 0E423CFC 99E9E316 50C1217B 624816CD AD9A95F9 1087 D5B80194 88D9C0A0 A1FE3075 A577E231 83F81D4A 3F2FA457 1088 1EFC8CE0 BA8A4FE8 B6855DFE 72B0A66E DED2FBAB FBE58A30 1089 FAFABE1C 5D71A87E 2F741EF8 C1FE86FE A6BBFDE5 30677F0D 1090 97D11D49 F7A8443D 0822E506 A9F4614E 011E2A94 838FF88C 1091 D68C8BB7 C5C6424C FFFFFFFF FFFFFFFF 1093 The generator is: g = 2 1094 The group size is: q = (p-1)/2 1096 The hexadecimal representation of q is: 1098 7FFFFFFF FFFFFFFF D6FC2A2C 515DA54D 57EE2B10 139E9E78 1099 EC5CE2C1 E7169B4A D4F09B20 8A3219FD E649CEE7 124D9F7C 1100 BE97F1B1 B1863AEC 7B40D901 576230BD 69EF8F6A EAFEB2B0 1101 9219FA8F AF833768 42B1B2AA 9EF68D79 DAAB89AF 3FABE49A 1102 CC278638 707345BB F15344ED 79F7F439 0EF8AC50 9B56F39A 1103 98566527 A41D3CBD 5E0558C1 59927DB0 E88454A5 D96471FD 1104 DCB56D5B B06BFA34 0EA7A151 EF1CA6FA 572B76F3 B1B95D8C 1105 8583D3E4 770536B8 4F017E70 E6FBF176 601A0266 941A17B0 1106 C8B97F4E 74C2C1FF C7278919 777940C1 E1FF1D8D A637D6B9 1107 9DDAFE5E 17611002 E2C778C1 BE8B41D9 6379A513 60D977FD 1108 4435A11C 308FE7EE 6F1AAD9D B28C81AD DE1A7A6F 7CCE011C 1109 30DA37E4 EB736483 BD6C8E93 48FBFBF7 2CC6587D 60C36C8E 1110 577F0984 C289C938 5A098649 DE21BCA2 7A7EA229 716BA6E9 1111 B279710F 38FAA5FF AE574155 CE4EFB4F 743695E2 911B1D06 1112 D5E290CB CD86F56D 0EDFCD21 6AE22427 055E6835 FD29EEF7 1113 9E0D9077 1FEACEBE 12F20E95 B34F0F78 B737A961 8B26FA7D 1114 BC9874F2 72C42BDB 563EAFA1 6B4FB68C 3BB1E78E AA81A002 1115 43FAADD2 BF18E63D 389AE443 77DA18C5 76B50F00 96CF3419 1116 5483B005 48C09862 36E3BC7C B8D6801C 0494CCD1 99E5C5BD 1117 0D0EDC9E B8A0001E 15276754 FCC68566 054148E6 E764BEE7 1118 C764DAAD 3FC45235 A6DAD428 FA20C170 E345003F 2F06EC81 1119 05FEB25B 2281B63D 2733BE96 1C29951D 11DD2221 657A9F53 1120 1DDA2A19 4DBB1264 48BDEEB2 58E07EA6 59C74619 A6380E1D 1121 66D6832B FE67F638 CD8FAE1F 2723020F 9C40A3FD A67EDA3B 1122 D29238FB D4D4B488 5C2A9917 6DB1A06C 50077849 1A8288F1 1123 855F60FF FCF1D137 3FD94FC6 0C1811E1 AC3F1C6D 003BECDA 1124 3B1F2725 CA595DE0 CA63328F 3BE57CC9 77556011 95140DFB 1125 59D39CE0 91308B41 05746DAC 23D33E5F 7CE4848D A316A9C6 1126 6B9581BA 3573BFAF 31149618 8AB15423 282EE416 DC2A19C5 1127 724FA91A E4ADC88B C66796EA E5677A01 F64E8C08 63139582 1128 2D9DB8FC EE35C06B 1FEEA547 4D6D8F34 B1534A93 6A18B0E0 1129 D20EAB86 BC9C6D6A 5207194E 67FA3555 1B568026 7B00641C 1130 0F212D18 ECA8D732 7ED91FE7 64A84EA1 B43FF5B4 F6E8E62F 1131 05C661DE FB258877 C35B18A1 51D5C414 AAAD97BA 3E499332 1132 E596078E 600DEB81 149C441C E95782F2 2A282563 C5BAC141 1133 1423605D 1AE1AFAE 2C8B0660 237EC128 AA0FE346 4E435811 1134 5DB84CC3 B523073A 28D45498 84B81FF7 0E10BF36 1C137296 1135 28D5348F 07211E7E 4CF4F18B 286090BD B1240B66 D6CD4AFC 1136 EADC00CA 446CE050 50FF183A D2BBF118 C1FC0EA5 1F97D22B 1137 8F7E4670 5D4527F4 5B42AEFF 39585337 6F697DD5 FDF2C518 1138 7D7D5F0E 2EB8D43F 17BA0F7C 60FF437F 535DFEF2 9833BF86 1139 CBE88EA4 FBD4221E 84117283 54FA30A7 008F154A 41C7FC46 1140 6B4645DB E2E32126 7FFFFFFF FFFFFFFF 1142 The estimated symmetric-equivalent strength of this group is 192 1143 bits. 1145 Peers using ffdhe8192 that want to optimize their key exchange with a 1146 short exponent (Section 5.2) should choose a secret key of at least 1147 400 bits. 1149 Author's Address 1151 Daniel Kahn Gillmor 1152 ACLU 1153 125 Broad Street, 18th Floor 1154 New York, NY 10004 1155 USA 1157 Email: dkg@fifthhorseman.net