idnits 2.17.00 (12 Aug 2021) /tmp/idnits61181/draft-ietf-tls-negotiated-ff-dhe-04.txt: Checking boilerplate required by RFC 5378 and the IETF Trust (see https://trustee.ietf.org/license-info): ---------------------------------------------------------------------------- No issues found here. Checking nits according to https://www.ietf.org/id-info/1id-guidelines.txt: ---------------------------------------------------------------------------- No issues found here. Checking nits according to https://www.ietf.org/id-info/checklist : ---------------------------------------------------------------------------- -- The draft header indicates that this document updates RFC4346, but the abstract doesn't seem to mention this, which it should. -- The draft header indicates that this document updates RFC5246, but the abstract doesn't seem to mention this, which it should. -- The draft header indicates that this document updates RFC2246, but the abstract doesn't seem to mention this, which it should. -- The draft header indicates that this document updates RFC4492, but the abstract doesn't seem to mention this, which it should. Miscellaneous warnings: ---------------------------------------------------------------------------- == The copyright year in the IETF Trust and authors Copyright Line does not match the current year (Using the creation date from RFC2246, updated by this document, for RFC5378 checks: 1996-12-03) -- The document seems to lack a disclaimer for pre-RFC5378 work, but may have content which was first submitted before 10 November 2008. If you have contacted all the original authors and they are all willing to grant the BCP78 rights to the IETF Trust, then this is fine, and you can ignore this comment. If not, you may need to add the pre-RFC5378 disclaimer. (See the Legal Provisions document at https://trustee.ietf.org/license-info for more information.) -- The document date (December 5, 2014) is 2723 days in the past. Is this intentional? Checking references for intended status: Informational ---------------------------------------------------------------------------- -- Looks like a reference, but probably isn't: '1' on line 658 ** Obsolete normative reference: RFC 4492 (Obsoleted by RFC 8422) ** Obsolete normative reference: RFC 5246 (Obsoleted by RFC 8446) Summary: 2 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 December 5, 2014 5 approved) 6 Intended status: Informational 7 Expires: June 8, 2015 9 Negotiated Finite Field Diffie-Hellman Ephemeral Parameters for TLS 10 draft-ietf-tls-negotiated-ff-dhe-04 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 June 8, 2015. 41 Copyright Notice 43 Copyright (c) 2014 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 4. Server Behavior . . . . . . . . . . . . . . . . . . . . . . . 6 64 5. Optimizations . . . . . . . . . . . . . . . . . . . . . . . . 7 65 5.1. Checking the Peer's Public Key . . . . . . . . . . . . . 7 66 5.2. Short Exponents . . . . . . . . . . . . . . . . . . . . . 7 67 5.3. Table Acceleration . . . . . . . . . . . . . . . . . . . 8 68 6. Operational Considerations . . . . . . . . . . . . . . . . . 8 69 6.1. Preference Ordering . . . . . . . . . . . . . . . . . . . 8 70 7. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 9 71 8. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 9 72 9. Security Considerations . . . . . . . . . . . . . . . . . . . 9 73 9.1. Negotiation resistance to active attacks . . . . . . . . 9 74 9.2. Group strength considerations . . . . . . . . . . . . . . 10 75 9.3. Finite-Field DHE only . . . . . . . . . . . . . . . . . . 11 76 9.4. Deprecating weak groups . . . . . . . . . . . . . . . . . 11 77 9.5. Choice of groups . . . . . . . . . . . . . . . . . . . . 11 78 9.6. Timing attacks . . . . . . . . . . . . . . . . . . . . . 12 79 9.7. Replay attacks from non-negotiated FFDHE . . . . . . . . 12 80 9.8. Forward Secrecy . . . . . . . . . . . . . . . . . . . . . 12 81 10. Privacy Considerations . . . . . . . . . . . . . . . . . . . 13 82 10.1. Client fingerprinting . . . . . . . . . . . . . . . . . 13 83 11. References . . . . . . . . . . . . . . . . . . . . . . . . . 13 84 11.1. Normative References . . . . . . . . . . . . . . . . . . 13 85 11.2. Informative References . . . . . . . . . . . . . . . . . 13 86 11.3. URIs . . . . . . . . . . . . . . . . . . . . . . . . . . 15 87 Appendix A. Named Group Registry . . . . . . . . . . . . . . . . 15 88 A.1. ffdhe2048 . . . . . . . . . . . . . . . . . . . . . . . . 15 89 A.2. ffdhe3072 . . . . . . . . . . . . . . . . . . . . . . . . 16 90 A.3. ffdhe4096 . . . . . . . . . . . . . . . . . . . . . . . . 18 91 A.4. ffdhe8192 . . . . . . . . . . . . . . . . . . . . . . . . 19 92 Author's Address . . . . . . . . . . . . . . . . . . . . . . . . 22 94 1. Introduction 96 Traditional TLS [RFC5246] offers a Diffie-Hellman ephemeral (DHE) key 97 exchange mode which provides Forward Secrecy for the connection. The 98 client offers a ciphersuite in the ClientHello that includes DHE, and 99 the server offers the client group parameters generator g and modulus 100 p. If the client does not consider the group strong enough (e.g. if 101 p is too small, or if p is not prime, or there are small subgroups), 102 or if it is unable to process the group for other reasons, the client 103 has no recourse but to terminate the connection. 105 Conversely, when a TLS server receives a suggestion for a DHE 106 ciphersuite from a client, it has no way of knowing what kinds of DH 107 groups the client is capable of handling, or what the client's 108 security requirements are for this key exchange session. For 109 example, some widely-distributed TLS clients are not capable of DH 110 groups where p > 1024 bits. Other TLS clients may by policy wish to 111 use DHE only if the server can offer a stronger group (and are 112 willing to use a non-PFS key-exchange mechanism otherwise). The 113 server has no way of knowing which type of client is connecting, but 114 must select DH parameters with insufficient knowledge. 116 Additionally, the DH parameters chosen by the server may have a known 117 structure which renders them secure against a small subgroup attack, 118 but a client receiving an arbitrary p and g has no efficient way to 119 verify that the structure of a new group is reasonable for use. 121 This modification to TLS solves these problems by using a section of 122 the "EC Named Curves" registry to select common DH groups with known 123 structure; defining the use of the "elliptic_curves(10)" extension 124 (described here as "Supported Groups" extension) for clients 125 advertising support for DHE with these groups. This document also 126 provides guidance for compliant peers to take advantage of the 127 additional security, availability, and efficiency offered. 129 The use of this mechanism by one compliant peer when interacting with 130 a non-compliant peer should have no detrimental effects. 132 1.1. Requirements Language 134 The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", 135 "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this 136 document are to be interpreted as described in [RFC2119]. 138 1.2. Vocabulary 140 The terms "DHE" or "FFDHE" are used in this document to refer to the 141 finite-field-based Diffie-Hellman ephemeral key exchange mechanism in 142 TLS. TLS also supports elliptic-curve-based Diffie-Hellman (ECDHE) 143 ephemeral key exchanges [RFC4492], but this document does not 144 document their use. A registry previously used only by ECHDE-capable 145 implementations is expanded in this document to cover FFDHE groups as 146 well. "FFDHE ciphersuites" is used in this document to refer 147 exclusively to ciphersuites with FFDHE key exchange mechanisms, but 148 note that these suites are typically labeled with a TLS_DHE_ prefix. 150 2. Named Group Overview 152 We use previously-unallocated codepoints within the extension 153 currently known as "elliptic_curves" (section 5.1.1. of [RFC4492]) to 154 indicate known finite field groups. The extension's semantics are 155 expanded from "Supported Elliptic Curves" to "Supported Groups". The 156 semantics of the extension's data type (enum NamedCurve) is also 157 expanded from "named curve" to "named group". 159 Codepoints in the NamedCurve registry with a high byte of 0x01 (that 160 is, between 256 and 511 inclusive) are set aside for FFDHE groups, 161 though only a small number of them are initially defined and we do 162 not expect many other FFDHE groups to be added to this range. No 163 codepoints outside of this range will be allocated to FFDHE groups. 164 The new code points for the NamedCurve registry are: 166 enum { 167 // other already defined elliptic curves (see RFC 4492) 168 ffdhe2048(256), ffdhe3072(257), ffdhe4096(258), 169 ffdhe8192(259), 170 // 171 } NamedCurve; 173 These additions to the Named Curve registry are described in detail 174 in Appendix A. They are all safe primes derived from the base of the 175 natural logarithm ("e"), with the high and low 64 bits set to 1 for 176 efficient Montgomery or Barrett reduction. 178 The use of the base of the natural logarithm here is as a "nothing- 179 up-my-sleeve" number. The goal is to guarantee that the bits in the 180 middle of the modulus are effectively random, while avoiding any 181 suspicion that the primes have secretly been selected to be weak 182 according to some secret criteria. [RFC3526] used pi for this value. 183 See Section 9.5 for reasons that this draft does not reuse pi. 185 3. Client Behavior 187 A TLS client that is capable of using strong finite field Diffie- 188 Hellman groups can advertise its capabilities and its preferences for 189 stronger key exchange by using this mechanism. 191 The compatible client that wants to be able to negotiate strong FFDHE 192 SHOULD send a "Supported Groups" extension (identified by type 193 elliptic_curves(10) in [RFC4492]) in the ClientHello, and include a 194 list of known FFDHE groups in the extension data, ordered from most 195 preferred to least preferred. If the client also supports and wants 196 to offer ECDHE key exchange, it MUST use a single "Supported Groups" 197 extension to include all supported groups (both ECDHE and FFDHE 198 groups). The ordering SHOULD be based on client preference, but see 199 Section 6.1 for more nuance. 201 A client that offers any of these values in the elliptic_curves 202 extension SHOULD ALSO include at least one FFDHE ciphersuite in the 203 Client Hello. 205 A client who offers a group MUST be able and willing to perform a DH 206 key exchange using that group. 208 A client that offers one or more FFDHE groups in the "Supported 209 Groups" extension and an FFDHE ciphersuite, and receives an FFDHE 210 ciphersuite from the server SHOULD take the following steps upon 211 receiving the ServerKeyExchange: 213 For non-anonymous ciphersuites where the offered Certificate is 214 valid and appropriate for the peer, validate the signature over 215 the ServerDHParams. If not valid, terminate the connection. 217 If the signature over ServerDHParams is valid, compare the 218 selected dh_p and dh_g with the FFDHE groups offered by the 219 client. If none of the offered groups match, the server is not 220 compatible with this draft. The client MAY decide to continue the 221 connection if the selected group is acceptable under local policy, 222 or it MAY decide to terminate the connection with a fatal 223 insufficient_security(71) alert. 225 If the selected group matches an offered FFDHE group exactly, the 226 the client MUST verify that dh_Ys is in the range 1 < dh_Ys < dh_p 227 - 1. If dh_Ys is not in this range, the client MUST terminate the 228 connection with a fatal handshake_failure(40) alert. 230 If the selected group matches an offered FFDHE group exactly, and 231 dh_Ys is in range, then the client SHOULD continue with the 232 connection as usual. 234 4. Server Behavior 236 If a compatible TLS server receives a Supported Groups extension from 237 a client that includes any FFDHE group (i.e. any codepoint between 238 256 and 511 inclusive, even if unknown to the server), and if none of 239 the client-proposed FFDHE groups are known and acceptable to the 240 server, then the server SHOULD NOT select an FFDHE ciphersuite. In 241 this case, the server SHOULD select an acceptable non-FFDHE 242 ciphersuite from the client's offered list. If the extension is 243 present with FFDHE groups, none of the client's offered groups are 244 acceptable by the server, and none of the client's proposed non-FFDHE 245 ciphersuites are acceptable to the server, the server SHOULD end the 246 connection with a fatal TLS alert of type insufficient_security(71). 248 If at least one FFDHE ciphersuite is present in the client 249 ciphersuite list, and the Supported Groups extension is present in 250 the ClientHello, but the extension does not include any FFDHE groups 251 (i.e. no codepoints between 256 and 511 inclusive), then the server 252 knows that the client is not compatible with this document. In this 253 scenario, a server MAY choose to select a non-FFDHE ciphersuite, or 254 MAY choose an FFDHE ciphersuite and offer an FFDHE group of its 255 choice to the client as part of a traditional ServerKeyExchange. 257 A compatible TLS server that receives the Supported Groups extension 258 with FFDHE codepoints in it, and which selects an FFDHE ciphersuite 259 MUST select one of the client's offered groups. The server indicates 260 the choice of group to the client by sending the group's parameters 261 as usual in the ServerKeyExchange as described in section 7.4.3 of 262 [RFC5246]. 264 A TLS server MUST NOT select a named FFDHE group that was not offered 265 by a compatible client. 267 A TLS server MUST NOT select an FFDHE ciphersuite if the client did 268 not offer one, even if the client offered an FFDHE group in the 269 Supported Groups extension. 271 If a non-anonymous FFDHE ciphersuite is chosen, and the TLS client 272 has used this extension to offer an FFDHE group of comparable or 273 greater strength than the server's public key, the server SHOULD 274 select an FFDHE group at least as strong as the server's public key. 275 For example, if the server has a 3072-bit RSA key, and the client 276 offers only ffdhe2048 and ffdhe4096, the server SHOULD select 277 ffdhe4096. 279 When a compatible server selects an FFDHE group from among a client's 280 Supported Groups, and the client sends a ClientKeyExchange, the 281 server MUST verify that 1 < dh_Yc < dh_p - 1. If it is out of range, 282 the server MUST terminate the connection with fatal 283 handshake_failure(40) alert. 285 5. Optimizations 287 In a key exchange with a successfully negotiated known FFDHE group, 288 both peers know that the group in question uses a safe prime as a 289 modulus, and that the group in use is of size p-1 or (p-1)/2. This 290 allows at least three optimizations that can be used to improve 291 performance. 293 5.1. Checking the Peer's Public Key 295 Peers MUST validate each other's public key Y (dh_Ys offered by the 296 server or dh_Yc offered by the client) by ensuring that 1 < Y < p-1. 297 This simple check ensures that the remote peer is properly behaved 298 and isn't forcing the local system into a small subgroup. 300 To reach the same assurance with an unknown group, the client would 301 need to verify the primality of the modulus, learn the factors of 302 p-1, and test both the generator g and Y against each factor to avoid 303 small subgroup attacks. 305 5.2. Short Exponents 307 Traditional Finite Field Diffie-Hellman has each peer choose their 308 secret exponent from the range [2,p-2]. Using exponentiation by 309 squaring, this means each peer must do roughly 2*log_2(p) 310 multiplications, twice (once for the generator and once for the 311 peer's public key). 313 Peers concerned with performance may also prefer to choose their 314 secret exponent from a smaller range, doing fewer multiplications, 315 while retaining the same level of overall security. Each named group 316 indicates its approximate security level, and provides a lower-bound 317 on the range of secret exponents that should preserve it. For 318 example, rather than doing 2*2*3072 multiplications for a ffdhe3072 319 handshake, each peer can choose to do 2*2*250 multiplications by 320 choosing their secret exponent from the range [2^249,2^250] (that is, 321 a m-bit integer where m is at least 224) and still keep the 322 approximate 125-bit security level. 324 A similar short-exponent approach is suggested in SSH's Diffie- 325 Hellman key exchange (See section 6.2 of [RFC4419]). 327 5.3. Table Acceleration 329 Peers wishing to further accelerate FFDHE key exchange can also pre- 330 compute a table of powers of the generator of a known group. This is 331 a memory vs. time tradeoff, and it only accelerates the first 332 exponentiation of the ephemeral DH exchange (the fixed-base 333 exponentiation). The variable-base exponentiation (using the peer's 334 public exponent as a base) still needs to be calculated as normal. 336 6. Operational Considerations 338 6.1. Preference Ordering 340 The ordering of named groups in the Supported Groups extension may 341 contain some ECDHE groups and some FFDHE groups. These SHOULD be 342 ranked in the order preferred by the client. 344 However, the ClientHello also contains list of desired ciphersuites, 345 also ranked in preference order. This presents the possibility of 346 conflicted preferences. For example, if the ClientHello contains a 347 CipherSuite with two choices in order 348 and the Supported Groups 350 Extension contains two choices in order then 351 there is a clear contradiction. Clients SHOULD NOT present such a 352 contradiction since it does not represent a sensible ordering. A 353 server that encounters such an contradiction when selecting between 354 an ECDHE or FFDHE key exchange mechanism while trying to respect 355 client preferences SHOULD give priority to the Supported Groups 356 extension (in the example case, it should select 357 TLS_ECDHE_RSA_WITH_AES_128_CBC_SHA with secp256r1), but MAY resolve 358 the contradiction any way it sees fit. 360 More subtly, clients MAY interleave preferences between ECDHE and 361 FFDHE groups, for example if stronger groups are preferred regardless 362 of cost, but weaker groups are acceptable, the Supported Groups 363 extension could consist of: 364 . In this example, with the 365 same CipherSuite offered as the previous example, a server configured 366 to respect client preferences and with support for all listed groups 367 SHOULD select TLS_DHE_RSA_WITH_AES_128_CBC_SHA with ffdhe8192. A 368 server configured to respect client preferences and with support for 369 only secp384p1 and ffdhe3072 SHOULD select 370 TLS_ECDHE_RSA_WITH_AES_128_CBC_SHA with secp384p1. 372 7. Acknowledgements 374 Thanks to Fedor Brunner, Dave Fergemann, Sandy Harris, Watson Ladd, 375 Nikos Mavrogiannopolous, Niels Moeller, Bodo Moeller, Kenny Paterson, 376 Eric Rescorla, Tom Ritter, Rene Struik, Martin Thomson, Sean Turner, 377 and other members of the TLS Working Group for their comments and 378 suggestions on this draft. Any mistakes here are not theirs. 380 8. IANA Considerations 382 IANA maintains the registry currently known as EC Named Curves 383 (originally defined in [RFC4492] and updated by [RFC7027]) at [1]. 385 This document expands the semantics of this registry slightly, to 386 include groups based on finite fields in addition to groups based on 387 elliptic curves. It should add a range designation to that registry, 388 indicating that values from 256-511 (inclusive) are set aside for 389 "Finite Field Diffie-Hellman groups", and that all other entries in 390 the registry are "Elliptic curve groups". 392 This document allocates five codepoints in the registry, as follows: 394 +-------+-------------+---------+-----------------+ 395 | Value | Description | DTLS-OK | Reference | 396 +-------+-------------+---------+-----------------+ 397 | 256 | ffdhe2048 | Y | [this document] | 398 | 257 | ffdhe3072 | Y | [this document] | 399 | 258 | ffdhe4096 | Y | [this document] | 400 | 259 | ffdhe8192 | Y | [this document] | 401 +-------+-------------+---------+-----------------+ 403 9. Security Considerations 405 9.1. Negotiation resistance to active attacks 407 Because the contents of the Supported Groups extension is hashed in 408 the finished message, an active MITM that tries to filter or omit 409 groups will cause the handshake to fail, but possibly not before 410 getting the peer to do something they would not otherwise have done. 412 An attacker who impersonates the server can try to do any of the 413 following: 415 Pretend that a non-compatible server is actually capable of this 416 extension, and select a group from the client's list, causing the 417 client to select a group it is willing to negotiate. It is 418 unclear how this would be an effective attack. 420 Pretend that a compatible server is actually non-compatible by 421 negotiating a non-FFDHE ciphersuite. This is no different than 422 MITM ciphersuite filtering. 424 Pretend that a compatible server is actually non-compatible by 425 negotiating a DHE ciphersuite, with a custom (perhaps weak) group 426 chosen by the attacker. This is no worse than the current 427 scenario, and would require the attacker to be able to sign the 428 ServerDHParams, which should not be possible without access to the 429 server's secret key. 431 An attacker who impersonates the client can try to do the following: 433 Pretend that a compatible client is not compatible (e.g. by not 434 offering the Supported Groups extension, or by replacing the 435 Supported Groups extension with one that includes no FFDHE 436 groups). This could cause the server to negotiate a weaker DHE 437 group during the handshake, or to select a non-FFDHE ciphersuite, 438 but it would fail to complete during the final check of the 439 Finished message. 441 Pretend that a non-compatible client is compatible (e.g. by . 442 This could cause the server to select a particular named group in 443 the ServerKeyExchange, or to avoid selecting an FFDHE ciphersuite. 444 The peers would fail to compute the final check of the Finished 445 message. 447 Change the list of groups offered by the client (e.g. by removing 448 the stronger of the set of groups offered). This could cause the 449 server to negotiate a weaker group than desired, but again should 450 be caught by the check in the Finished message. 452 9.2. Group strength considerations 454 TLS implementations using FFDHE key exchange should consider the 455 strength of the group they negotiate. The strength of the selected 456 group is one of the factors which defines the connection's resiliance 457 against attacks on the session's confidentiality and integrity, since 458 the session keys are derived from the DHE handshake. 460 While attacks on integrity must generally happen while the session is 461 in progress, attacks against session confidentiality can happen 462 significantly later, if the entire TLS session is stored for offline 463 analysis. Therefore, FFDHE groups should be selected by clients and 464 servers based on confidentiality guarantees they need. Sessions 465 which need extremely long-term confidentiality should prefer stronger 466 groups. 468 [ENISA] provides rough estimates of group resistance to attack, and 469 recommends that forward-looking implementations ("future systems") 470 should use FFDHE group sizes of at least 3072 bits. ffdhe3072 is 471 intended for use in these implementations. 473 9.3. Finite-Field DHE only 475 Note that this document specifically targets only finite field-based 476 Diffie-Hellman ephemeral key exchange mechanisms. It does not cover 477 the non-ephemeral DH key exchange mechanisms, nor does it address 478 elliptic curve DHE (ECDHE) key exchange, which is defined in 479 [RFC4492]. 481 Measured by computational cost to the TLS peers, ECDHE appears today 482 to offer much a stronger key exchange than FFDHE. 484 9.4. Deprecating weak groups 486 Advances in hardware or in finite field cryptanalysis may cause some 487 of the negotiated groups to not provide the desired security margins, 488 as indicated by the estimated work factor of an adversary to discover 489 the premaster secret (and may therefore compromise the 490 confidentiality and integrity of the TLS session). 492 Revisions of this document should mark known-weak groups as 493 explicitly deprecated for use in TLS, and should update the estimated 494 work factor needed to break the group, if the cryptanalysis has 495 changed. Implementations that require strong confidentiality and 496 integrity guarantees should avoid using deprecated groups and should 497 be updated when the estimated security margins are updated. 499 9.5. Choice of groups 501 Other lists of named finite field Diffie-Hellman groups 502 [STRONGSWAN-IKE] exist. This draft chooses to not reuse them for 503 several reasons: 505 Using the same groups in multiple protocols increases the value 506 for an attacker with the resources to crack any single group. 508 The IKE groups include weak groups like MODP768 which are 509 unacceptable for secure TLS traffic. 511 Mixing group parameters across multiple implementations leaves 512 open the possibility of some sort of cross-protocol attack. This 513 shouldn't be relevant for ephemeral scenarios, and even with non- 514 ephemeral keying, services shouldn't share keys; however, using 515 different groups avoids these failure modes entirely. 517 9.6. Timing attacks 519 Any implementation of finite field Diffie-Hellman key exchange should 520 use constant-time modular-exponentiation implementations. This is 521 particularly true for those implementations that ever re-use DHE 522 secret keys (so-called "semi-static" ephemeral keying) or share DHE 523 secret keys across a multiple machines (e.g. in a load-balancer 524 situation). 526 9.7. Replay attacks from non-negotiated FFDHE 528 [SECURE-RESUMPTION], [CROSS-PROTOCOL], and [SSL3-ANALYSIS] all show a 529 malicious peer using a bad FFDHE group to maneuver a client into 530 selecting a pre-master secret of the peer's choice, which can be 531 replayed to another server using a non-FFDHE key exchange, and can 532 then be bootstrapped to replay client authentication. 534 To prevent this attack (barring the fixes proposed in 535 [SESSION-HASH]), a client would need not only to implement this 536 draft, but also to reject non-negotiated FFDHE ciphersuites whose 537 group structure it cannot afford to verify. Such a client would need 538 to abort the initial handshake and reconnect to the server in 539 question without listing any FFDHE ciphersuites on the subsequent 540 connection. 542 This tradeoff may be too costly for most TLS clients today, but may 543 be a reasonable choice for clients performing client certificate 544 authentication, or who have other reason to be concerned about 545 server-controlled pre-master secrets. 547 9.8. Forward Secrecy 549 One of the main reasons to prefer FFDHE ciphersuites is Forward 550 Secrecy, the ability to resist decryption even if when the endpoint's 551 long-term secret key (usually RSA) is revealed in the future. 553 This property depends on both sides of the connection discarding 554 their ephemeral keys promptly. Implementations should wipe their 555 FFDHE secret key material from memory as soon as it is no longer 556 needed, and should never store it in persistent storage. 558 Forward secrecy also depends on the strength of the Diffie-Hellman 559 group; using a very strong symmetric cipher like AES256 with a 560 forward-secret ciphersuite, but generating the keys with a much 561 weaker group like dhe2048 simply moves the adversary's cost from 562 attacking the symmetric cipher to attacking the dh_Ys or dh_Yc 563 ephemeral keyshares. 565 If the goal is to provide forward secrecy, attention should be paid 566 to all parts of the ciphersuite selection process, both key exchange 567 and symmetric cipher choice. 569 10. Privacy Considerations 571 10.1. Client fingerprinting 573 This extension provides a few additional bits of information to 574 distinguish between classes of TLS clients (see e.g. 575 [PANOPTICLICK]). To minimize this sort of fingerprinting, clients 576 SHOULD support all named groups at or above their minimum security 577 threshhold. New named groups SHOULD NOT be added to the registry 578 without consideration of the cost of browser fingerprinting. 580 11. References 582 11.1. Normative References 584 [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate 585 Requirement Levels", BCP 14, RFC 2119, March 1997. 587 [RFC4492] Blake-Wilson, S., Bolyard, N., Gupta, V., Hawk, C., and B. 588 Moeller, "Elliptic Curve Cryptography (ECC) Cipher Suites 589 for Transport Layer Security (TLS)", RFC 4492, May 2006. 591 [RFC5246] Dierks, T. and E. Rescorla, "The Transport Layer Security 592 (TLS) Protocol Version 1.2", RFC 5246, August 2008. 594 11.2. Informative References 596 [CROSS-PROTOCOL] 597 Mavrogiannopolous, N., Vercauteren, F., Velichkov, V., and 598 B. Preneel, "A Cross-Protocol Attack on the TLS Protocol", 599 October 2012, 600 . 603 [ECRYPTII] 604 European Network of Excellence in Cryptology II, "ECRYPT 605 II Yearly Report on Algorithms and Keysizes (2011-2012)", 606 September 2012, 607 . 609 [ENISA] European Union Agency for Network and Information Security 610 Agency, "Algorithms, Key Sizes and Parameters Report, 611 version 1.0", October 2013, 612 . 616 [PANOPTICLICK] 617 Electronic Frontier Foundation, "Panopticlick: How Unique 618 - and Trackable - Is Your Browser?", 2010, 619 . 621 [RFC3526] Kivinen, T. and M. Kojo, "More Modular Exponential (MODP) 622 Diffie-Hellman groups for Internet Key Exchange (IKE)", 623 RFC 3526, May 2003. 625 [RFC4419] Friedl, M., Provos, N., and W. Simpson, "Diffie-Hellman 626 Group Exchange for the Secure Shell (SSH) Transport Layer 627 Protocol", RFC 4419, March 2006. 629 [RFC7027] Merkle, J. and M. Lochter, "Elliptic Curve Cryptography 630 (ECC) Brainpool Curves for Transport Layer Security 631 (TLS)", RFC 7027, October 2013. 633 [SECURE-RESUMPTION] 634 Delignat-Lavaud, A., Bhargavan, K., and A. Pironti, 635 "Triple Handshakes Considered Harmful: Breaking and Fixing 636 Authentication over TLS", March 2014, . 639 [SESSION-HASH] 640 Bhargavan, K., Delignat-Lavaud, A., Pironti, A., Langley, 641 A., and M. Ray, "Triple Handshakes Considered Harmful: 642 Breaking and Fixing Authentication over TLS", March 2014, 643 . 646 [SSL3-ANALYSIS] 647 Schneier, B. and D. Wagner, "Analysis of the SSL 3.0 648 protocol", 1996, . 650 [STRONGSWAN-IKE] 651 Brunner, T. and A. Steffen, "Diffie Hellman Groups in 652 IKEv2 Cipher Suites", October 2013, 653 . 656 11.3. URIs 658 [1] https://www.iana.org/assignments/tls-parameters/tls- 659 parameters.xhtml#tls-parameters-8 661 Appendix A. Named Group Registry 663 Each description below indicates the group itself, its derivation, 664 its expected strength (estimated roughly from guidelines in 665 [ECRYPTII]), and whether it is recommended for use in TLS key 666 exchange at the given security level. It is not recommended to add 667 further finite field groups to the NamedCurves registry; any attempt 668 to do so should consider Section 10.1. 670 The primes in these finite field groups are all safe primes, that is, 671 a prime p is a safe prime when q = (p-1)/2 is also prime. Where e is 672 the base of the natural logarithm, and square brackets denote the 673 floor operation, the groups which initially populate this registry 674 are derived for a given bitlength b by finding the lowest positive 675 integer X that creates a safe prime p where: 677 p = 2^b - 2^{b-64} + {[2^{b-130} e] + X } * 2^64 - 1 679 New additions of FFDHE groups to this registry may use this same 680 derivation (e.g. with different bitlengths) or may choose their 681 parameters in a different way, but must be clear about how the 682 parameters were derived. 684 New additions of FFDHE groups MUST use a safe prime as the modulus to 685 enable the inexpensive peer verification described in Section 5.1. 687 A.1. ffdhe2048 689 The 2048-bit group has registry value 256, and is calcluated from the 690 following formula: 692 The modulus is: p = 2^2048 - 2^1984 + {[2^1918 * e] + 560315 } * 2^64 693 - 1 695 The hexadecimal representation of p is: 697 FFFFFFFF FFFFFFFF ADF85458 A2BB4A9A AFDC5620 273D3CF1 698 D8B9C583 CE2D3695 A9E13641 146433FB CC939DCE 249B3EF9 699 7D2FE363 630C75D8 F681B202 AEC4617A D3DF1ED5 D5FD6561 700 2433F51F 5F066ED0 85636555 3DED1AF3 B557135E 7F57C935 701 984F0C70 E0E68B77 E2A689DA F3EFE872 1DF158A1 36ADE735 702 30ACCA4F 483A797A BC0AB182 B324FB61 D108A94B B2C8E3FB 703 B96ADAB7 60D7F468 1D4F42A3 DE394DF4 AE56EDE7 6372BB19 704 0B07A7C8 EE0A6D70 9E02FCE1 CDF7E2EC C03404CD 28342F61 705 9172FE9C E98583FF 8E4F1232 EEF28183 C3FE3B1B 4C6FAD73 706 3BB5FCBC 2EC22005 C58EF183 7D1683B2 C6F34A26 C1B2EFFA 707 886B4238 61285C97 FFFFFFFF FFFFFFFF 709 The generator is: g = 2 711 The group size is: q = (p-1)/2 713 The hexadecimal representation of q is: 715 7FFFFFFF FFFFFFFF D6FC2A2C 515DA54D 57EE2B10 139E9E78 716 EC5CE2C1 E7169B4A D4F09B20 8A3219FD E649CEE7 124D9F7C 717 BE97F1B1 B1863AEC 7B40D901 576230BD 69EF8F6A EAFEB2B0 718 9219FA8F AF833768 42B1B2AA 9EF68D79 DAAB89AF 3FABE49A 719 CC278638 707345BB F15344ED 79F7F439 0EF8AC50 9B56F39A 720 98566527 A41D3CBD 5E0558C1 59927DB0 E88454A5 D96471FD 721 DCB56D5B B06BFA34 0EA7A151 EF1CA6FA 572B76F3 B1B95D8C 722 8583D3E4 770536B8 4F017E70 E6FBF176 601A0266 941A17B0 723 C8B97F4E 74C2C1FF C7278919 777940C1 E1FF1D8D A637D6B9 724 9DDAFE5E 17611002 E2C778C1 BE8B41D9 6379A513 60D977FD 725 4435A11C 30942E4B FFFFFFFF FFFFFFFF 727 The estimated symmetric-equivalent strength of this group is 103 728 bits. 730 Peers using ffdhe2048 that want to optimize their key exchange with a 731 short exponent (Section 5.2) should choose a secret key of at least 732 206 bits. 734 A.2. ffdhe3072 736 The 3072-bit prime has registry value 257, and is calcluated from the 737 following formula: 739 The modulus is: p = 2^3072 - 2^3008 + {[2^2942 * e] + 2625351} * 2^64 740 -1 742 The hexadecimal representation of p is: 744 FFFFFFFF FFFFFFFF ADF85458 A2BB4A9A AFDC5620 273D3CF1 745 D8B9C583 CE2D3695 A9E13641 146433FB CC939DCE 249B3EF9 746 7D2FE363 630C75D8 F681B202 AEC4617A D3DF1ED5 D5FD6561 747 2433F51F 5F066ED0 85636555 3DED1AF3 B557135E 7F57C935 748 984F0C70 E0E68B77 E2A689DA F3EFE872 1DF158A1 36ADE735 749 30ACCA4F 483A797A BC0AB182 B324FB61 D108A94B B2C8E3FB 750 B96ADAB7 60D7F468 1D4F42A3 DE394DF4 AE56EDE7 6372BB19 751 0B07A7C8 EE0A6D70 9E02FCE1 CDF7E2EC C03404CD 28342F61 752 9172FE9C E98583FF 8E4F1232 EEF28183 C3FE3B1B 4C6FAD73 753 3BB5FCBC 2EC22005 C58EF183 7D1683B2 C6F34A26 C1B2EFFA 754 886B4238 611FCFDC DE355B3B 6519035B BC34F4DE F99C0238 755 61B46FC9 D6E6C907 7AD91D26 91F7F7EE 598CB0FA C186D91C 756 AEFE1309 85139270 B4130C93 BC437944 F4FD4452 E2D74DD3 757 64F2E21E 71F54BFF 5CAE82AB 9C9DF69E E86D2BC5 22363A0D 758 ABC52197 9B0DEADA 1DBF9A42 D5C4484E 0ABCD06B FA53DDEF 759 3C1B20EE 3FD59D7C 25E41D2B 66C62E37 FFFFFFFF FFFFFFFF 761 The generator is: g = 2 763 The group size is: q = (p-1)/2 765 The hexadecimal representation of q is: 767 7FFFFFFF FFFFFFFF D6FC2A2C 515DA54D 57EE2B10 139E9E78 768 EC5CE2C1 E7169B4A D4F09B20 8A3219FD E649CEE7 124D9F7C 769 BE97F1B1 B1863AEC 7B40D901 576230BD 69EF8F6A EAFEB2B0 770 9219FA8F AF833768 42B1B2AA 9EF68D79 DAAB89AF 3FABE49A 771 CC278638 707345BB F15344ED 79F7F439 0EF8AC50 9B56F39A 772 98566527 A41D3CBD 5E0558C1 59927DB0 E88454A5 D96471FD 773 DCB56D5B B06BFA34 0EA7A151 EF1CA6FA 572B76F3 B1B95D8C 774 8583D3E4 770536B8 4F017E70 E6FBF176 601A0266 941A17B0 775 C8B97F4E 74C2C1FF C7278919 777940C1 E1FF1D8D A637D6B9 776 9DDAFE5E 17611002 E2C778C1 BE8B41D9 6379A513 60D977FD 777 4435A11C 308FE7EE 6F1AAD9D B28C81AD DE1A7A6F 7CCE011C 778 30DA37E4 EB736483 BD6C8E93 48FBFBF7 2CC6587D 60C36C8E 779 577F0984 C289C938 5A098649 DE21BCA2 7A7EA229 716BA6E9 780 B279710F 38FAA5FF AE574155 CE4EFB4F 743695E2 911B1D06 781 D5E290CB CD86F56D 0EDFCD21 6AE22427 055E6835 FD29EEF7 782 9E0D9077 1FEACEBE 12F20E95 B363171B FFFFFFFF FFFFFFFF 784 The estimated symmetric-equivalent strength of this group is 125 785 bits. 787 Peers using ffdhe3072 that want to optimize their key exchange with a 788 short exponent (Section 5.2) should choose a secret key of at least 789 250 bits. 791 A.3. ffdhe4096 793 The 4096-bit group has registry value 258, and is calcluated from the 794 following formula: 796 The modulus is: p = 2^4096 - 2^4032 + {[2^3966 * e] + 5736041} * 2^64 797 - 1 799 The hexadecimal representation of p is: 801 FFFFFFFF FFFFFFFF ADF85458 A2BB4A9A AFDC5620 273D3CF1 802 D8B9C583 CE2D3695 A9E13641 146433FB CC939DCE 249B3EF9 803 7D2FE363 630C75D8 F681B202 AEC4617A D3DF1ED5 D5FD6561 804 2433F51F 5F066ED0 85636555 3DED1AF3 B557135E 7F57C935 805 984F0C70 E0E68B77 E2A689DA F3EFE872 1DF158A1 36ADE735 806 30ACCA4F 483A797A BC0AB182 B324FB61 D108A94B B2C8E3FB 807 B96ADAB7 60D7F468 1D4F42A3 DE394DF4 AE56EDE7 6372BB19 808 0B07A7C8 EE0A6D70 9E02FCE1 CDF7E2EC C03404CD 28342F61 809 9172FE9C E98583FF 8E4F1232 EEF28183 C3FE3B1B 4C6FAD73 810 3BB5FCBC 2EC22005 C58EF183 7D1683B2 C6F34A26 C1B2EFFA 811 886B4238 611FCFDC DE355B3B 6519035B BC34F4DE F99C0238 812 61B46FC9 D6E6C907 7AD91D26 91F7F7EE 598CB0FA C186D91C 813 AEFE1309 85139270 B4130C93 BC437944 F4FD4452 E2D74DD3 814 64F2E21E 71F54BFF 5CAE82AB 9C9DF69E E86D2BC5 22363A0D 815 ABC52197 9B0DEADA 1DBF9A42 D5C4484E 0ABCD06B FA53DDEF 816 3C1B20EE 3FD59D7C 25E41D2B 669E1EF1 6E6F52C3 164DF4FB 817 7930E9E4 E58857B6 AC7D5F42 D69F6D18 7763CF1D 55034004 818 87F55BA5 7E31CC7A 7135C886 EFB4318A ED6A1E01 2D9E6832 819 A907600A 918130C4 6DC778F9 71AD0038 092999A3 33CB8B7A 820 1A1DB93D 7140003C 2A4ECEA9 F98D0ACC 0A8291CD CEC97DCF 821 8EC9B55A 7F88A46B 4DB5A851 F44182E1 C68A007E 5E655F6A 822 FFFFFFFF FFFFFFFF 824 The generator is: g = 2 826 The group size is: q = (p-1)/2 828 The hexadecimal representation of q is: 830 7FFFFFFF FFFFFFFF D6FC2A2C 515DA54D 57EE2B10 139E9E78 831 EC5CE2C1 E7169B4A D4F09B20 8A3219FD E649CEE7 124D9F7C 832 BE97F1B1 B1863AEC 7B40D901 576230BD 69EF8F6A EAFEB2B0 833 9219FA8F AF833768 42B1B2AA 9EF68D79 DAAB89AF 3FABE49A 834 CC278638 707345BB F15344ED 79F7F439 0EF8AC50 9B56F39A 835 98566527 A41D3CBD 5E0558C1 59927DB0 E88454A5 D96471FD 836 DCB56D5B B06BFA34 0EA7A151 EF1CA6FA 572B76F3 B1B95D8C 837 8583D3E4 770536B8 4F017E70 E6FBF176 601A0266 941A17B0 838 C8B97F4E 74C2C1FF C7278919 777940C1 E1FF1D8D A637D6B9 839 9DDAFE5E 17611002 E2C778C1 BE8B41D9 6379A513 60D977FD 840 4435A11C 308FE7EE 6F1AAD9D B28C81AD DE1A7A6F 7CCE011C 841 30DA37E4 EB736483 BD6C8E93 48FBFBF7 2CC6587D 60C36C8E 842 577F0984 C289C938 5A098649 DE21BCA2 7A7EA229 716BA6E9 843 B279710F 38FAA5FF AE574155 CE4EFB4F 743695E2 911B1D06 844 D5E290CB CD86F56D 0EDFCD21 6AE22427 055E6835 FD29EEF7 845 9E0D9077 1FEACEBE 12F20E95 B34F0F78 B737A961 8B26FA7D 846 BC9874F2 72C42BDB 563EAFA1 6B4FB68C 3BB1E78E AA81A002 847 43FAADD2 BF18E63D 389AE443 77DA18C5 76B50F00 96CF3419 848 5483B005 48C09862 36E3BC7C B8D6801C 0494CCD1 99E5C5BD 849 0D0EDC9E B8A0001E 15276754 FCC68566 054148E6 E764BEE7 850 C764DAAD 3FC45235 A6DAD428 FA20C170 E345003F 2F32AFB5 851 7FFFFFFF FFFFFFFF 853 The estimated symmetric-equivalent strength of this group is 150 854 bits. 856 Peers using ffdhe4096 that want to optimize their key exchange with a 857 short exponent (Section 5.2) should choose a secret key of at least 858 300 bits. 860 A.4. ffdhe8192 862 The 8192-bit group has registry value 259, and is calcluated from the 863 following formula: 865 The modulus is: p = 2^8192 - 2^8128 + {[2^8062 * e] + 10965728} * 866 2^64 - 1 868 The hexadecimal representation of p is: 870 FFFFFFFF FFFFFFFF ADF85458 A2BB4A9A AFDC5620 273D3CF1 871 D8B9C583 CE2D3695 A9E13641 146433FB CC939DCE 249B3EF9 872 7D2FE363 630C75D8 F681B202 AEC4617A D3DF1ED5 D5FD6561 873 2433F51F 5F066ED0 85636555 3DED1AF3 B557135E 7F57C935 874 984F0C70 E0E68B77 E2A689DA F3EFE872 1DF158A1 36ADE735 875 30ACCA4F 483A797A BC0AB182 B324FB61 D108A94B B2C8E3FB 876 B96ADAB7 60D7F468 1D4F42A3 DE394DF4 AE56EDE7 6372BB19 877 0B07A7C8 EE0A6D70 9E02FCE1 CDF7E2EC C03404CD 28342F61 878 9172FE9C E98583FF 8E4F1232 EEF28183 C3FE3B1B 4C6FAD73 879 3BB5FCBC 2EC22005 C58EF183 7D1683B2 C6F34A26 C1B2EFFA 880 886B4238 611FCFDC DE355B3B 6519035B BC34F4DE F99C0238 881 61B46FC9 D6E6C907 7AD91D26 91F7F7EE 598CB0FA C186D91C 882 AEFE1309 85139270 B4130C93 BC437944 F4FD4452 E2D74DD3 883 64F2E21E 71F54BFF 5CAE82AB 9C9DF69E E86D2BC5 22363A0D 884 ABC52197 9B0DEADA 1DBF9A42 D5C4484E 0ABCD06B FA53DDEF 885 3C1B20EE 3FD59D7C 25E41D2B 669E1EF1 6E6F52C3 164DF4FB 886 7930E9E4 E58857B6 AC7D5F42 D69F6D18 7763CF1D 55034004 887 87F55BA5 7E31CC7A 7135C886 EFB4318A ED6A1E01 2D9E6832 888 A907600A 918130C4 6DC778F9 71AD0038 092999A3 33CB8B7A 889 1A1DB93D 7140003C 2A4ECEA9 F98D0ACC 0A8291CD CEC97DCF 890 8EC9B55A 7F88A46B 4DB5A851 F44182E1 C68A007E 5E0DD902 891 0BFD64B6 45036C7A 4E677D2C 38532A3A 23BA4442 CAF53EA6 892 3BB45432 9B7624C8 917BDD64 B1C0FD4C B38E8C33 4C701C3A 893 CDAD0657 FCCFEC71 9B1F5C3E 4E46041F 388147FB 4CFDB477 894 A52471F7 A9A96910 B855322E DB6340D8 A00EF092 350511E3 895 0ABEC1FF F9E3A26E 7FB29F8C 183023C3 587E38DA 0077D9B4 896 763E4E4B 94B2BBC1 94C6651E 77CAF992 EEAAC023 2A281BF6 897 B3A739C1 22611682 0AE8DB58 47A67CBE F9C9091B 462D538C 898 D72B0374 6AE77F5E 62292C31 1562A846 505DC82D B854338A 899 E49F5235 C95B9117 8CCF2DD5 CACEF403 EC9D1810 C6272B04 900 5B3B71F9 DC6B80D6 3FDD4A8E 9ADB1E69 62A69526 D43161C1 901 A41D570D 7938DAD4 A40E329C CFF46AAA 36AD004C F600C838 902 1E425A31 D951AE64 FDB23FCE C9509D43 687FEB69 EDD1CC5E 903 0B8CC3BD F64B10EF 86B63142 A3AB8829 555B2F74 7C932665 904 CB2C0F1C C01BD702 29388839 D2AF05E4 54504AC7 8B758282 905 2846C0BA 35C35F5C 59160CC0 46FD8251 541FC68C 9C86B022 906 BB709987 6A460E74 51A8A931 09703FEE 1C217E6C 3826E52C 907 51AA691E 0E423CFC 99E9E316 50C1217B 624816CD AD9A95F9 908 D5B80194 88D9C0A0 A1FE3075 A577E231 83F81D4A 3F2FA457 909 1EFC8CE0 BA8A4FE8 B6855DFE 72B0A66E DED2FBAB FBE58A30 910 FAFABE1C 5D71A87E 2F741EF8 C1FE86FE A6BBFDE5 30677F0D 911 97D11D49 F7A8443D 0822E506 A9F4614E 011E2A94 838FF88C 912 D68C8BB7 C5C6424C FFFFFFFF FFFFFFFF 914 The generator is: g = 2 916 The group size is: q = (p-1)/2 917 The hexadecimal representation of q is: 919 7FFFFFFF FFFFFFFF D6FC2A2C 515DA54D 57EE2B10 139E9E78 920 EC5CE2C1 E7169B4A D4F09B20 8A3219FD E649CEE7 124D9F7C 921 BE97F1B1 B1863AEC 7B40D901 576230BD 69EF8F6A EAFEB2B0 922 9219FA8F AF833768 42B1B2AA 9EF68D79 DAAB89AF 3FABE49A 923 CC278638 707345BB F15344ED 79F7F439 0EF8AC50 9B56F39A 924 98566527 A41D3CBD 5E0558C1 59927DB0 E88454A5 D96471FD 925 DCB56D5B B06BFA34 0EA7A151 EF1CA6FA 572B76F3 B1B95D8C 926 8583D3E4 770536B8 4F017E70 E6FBF176 601A0266 941A17B0 927 C8B97F4E 74C2C1FF C7278919 777940C1 E1FF1D8D A637D6B9 928 9DDAFE5E 17611002 E2C778C1 BE8B41D9 6379A513 60D977FD 929 4435A11C 308FE7EE 6F1AAD9D B28C81AD DE1A7A6F 7CCE011C 930 30DA37E4 EB736483 BD6C8E93 48FBFBF7 2CC6587D 60C36C8E 931 577F0984 C289C938 5A098649 DE21BCA2 7A7EA229 716BA6E9 932 B279710F 38FAA5FF AE574155 CE4EFB4F 743695E2 911B1D06 933 D5E290CB CD86F56D 0EDFCD21 6AE22427 055E6835 FD29EEF7 934 9E0D9077 1FEACEBE 12F20E95 B34F0F78 B737A961 8B26FA7D 935 BC9874F2 72C42BDB 563EAFA1 6B4FB68C 3BB1E78E AA81A002 936 43FAADD2 BF18E63D 389AE443 77DA18C5 76B50F00 96CF3419 937 5483B005 48C09862 36E3BC7C B8D6801C 0494CCD1 99E5C5BD 938 0D0EDC9E B8A0001E 15276754 FCC68566 054148E6 E764BEE7 939 C764DAAD 3FC45235 A6DAD428 FA20C170 E345003F 2F06EC81 940 05FEB25B 2281B63D 2733BE96 1C29951D 11DD2221 657A9F53 941 1DDA2A19 4DBB1264 48BDEEB2 58E07EA6 59C74619 A6380E1D 942 66D6832B FE67F638 CD8FAE1F 2723020F 9C40A3FD A67EDA3B 943 D29238FB D4D4B488 5C2A9917 6DB1A06C 50077849 1A8288F1 944 855F60FF FCF1D137 3FD94FC6 0C1811E1 AC3F1C6D 003BECDA 945 3B1F2725 CA595DE0 CA63328F 3BE57CC9 77556011 95140DFB 946 59D39CE0 91308B41 05746DAC 23D33E5F 7CE4848D A316A9C6 947 6B9581BA 3573BFAF 31149618 8AB15423 282EE416 DC2A19C5 948 724FA91A E4ADC88B C66796EA E5677A01 F64E8C08 63139582 949 2D9DB8FC EE35C06B 1FEEA547 4D6D8F34 B1534A93 6A18B0E0 950 D20EAB86 BC9C6D6A 5207194E 67FA3555 1B568026 7B00641C 951 0F212D18 ECA8D732 7ED91FE7 64A84EA1 B43FF5B4 F6E8E62F 952 05C661DE FB258877 C35B18A1 51D5C414 AAAD97BA 3E499332 953 E596078E 600DEB81 149C441C E95782F2 2A282563 C5BAC141 954 1423605D 1AE1AFAE 2C8B0660 237EC128 AA0FE346 4E435811 955 5DB84CC3 B523073A 28D45498 84B81FF7 0E10BF36 1C137296 956 28D5348F 07211E7E 4CF4F18B 286090BD B1240B66 D6CD4AFC 957 EADC00CA 446CE050 50FF183A D2BBF118 C1FC0EA5 1F97D22B 958 8F7E4670 5D4527F4 5B42AEFF 39585337 6F697DD5 FDF2C518 959 7D7D5F0E 2EB8D43F 17BA0F7C 60FF437F 535DFEF2 9833BF86 960 CBE88EA4 FBD4221E 84117283 54FA30A7 008F154A 41C7FC46 961 6B4645DB E2E32126 7FFFFFFF FFFFFFFF 963 The estimated symmetric-equivalent strength of this group is 192 964 bits. 966 Peers using ffdhe8192 that want to optimize their key exchange with a 967 short exponent (Section 5.2) should choose a secret key of at least 968 384 bits. 970 Author's Address 972 Daniel Kahn Gillmor 973 ACLU 974 125 Broad Street, 18th Floor 975 New York, NY 10004 976 USA 978 Email: dkg@fifthhorseman.net