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RFC 7919

Proposed STD
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Negotiated Finite Field Diffie-Hellman Ephemeral Parameters for Transport Layer Security (TLS)

Updates:    2246    4346    4492    5246


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Internet Engineering Task Force (IETF)                        D. Gillmor
Request for Comments: 7919                                          ACLU
Updates: 2246, 4346, 4492, 5246                              August 2016
Category: Standards Track
ISSN: 2070-1721


      Negotiated Finite Field Diffie-Hellman Ephemeral Parameters
                   for Transport Layer Security (TLS)

Abstract

   Traditional finite-field-based Diffie-Hellman (DH) key exchange
   during the Transport Layer Security (TLS) handshake suffers from a
   number of security, interoperability, and efficiency shortcomings.
   These shortcomings arise from lack of clarity about which DH group
   parameters TLS servers should offer and clients should accept.  This
   document offers a solution to these shortcomings for compatible peers
   by using a section of the TLS "Supported Groups Registry" (renamed
   from "EC Named Curve Registry" by this document) to establish common
   finite field DH parameters with known structure and a mechanism for
   peers to negotiate support for these groups.

   This document updates TLS versions 1.0 (RFC 2246), 1.1 (RFC 4346),
   and 1.2 (RFC 5246), as well as the TLS Elliptic Curve Cryptography
   (ECC) extensions (RFC 4492).

Status of This Memo

   This is an Internet Standards Track document.

   This document is a product of the Internet Engineering Task Force
   (IETF).  It represents the consensus of the IETF community.  It has
   received public review and has been approved for publication by the
   Internet Engineering Steering Group (IESG).  Further information on
   Internet Standards is available in Section 2 of RFC 7841.

   Information about the current status of this document, any errata,
   and how to provide feedback on it may be obtained at
   http://www.rfc-editor.org/info/rfc7919.

[Page 2] 
Copyright Notice

   Copyright (c) 2016 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents
   (http://trustee.ietf.org/license-info) in effect on the date of
   publication of this document.  Please review these documents
   carefully, as they describe your rights and restrictions with respect
   to this document.  Code Components extracted from this document must
   include Simplified BSD License text as described in Section 4.e of
   the Trust Legal Provisions and are provided without warranty as
   described in the Simplified BSD License.

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Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   4
     1.1.  Requirements Language . . . . . . . . . . . . . . . . . .   5
     1.2.  Vocabulary  . . . . . . . . . . . . . . . . . . . . . . .   5
   2.  Named Group Overview  . . . . . . . . . . . . . . . . . . . .   5
   3.  Client Behavior . . . . . . . . . . . . . . . . . . . . . . .   6
     3.1.  Client Local Policy on Custom Groups  . . . . . . . . . .   7
   4.  Server Behavior . . . . . . . . . . . . . . . . . . . . . . .   8
   5.  Optimizations . . . . . . . . . . . . . . . . . . . . . . . .   9
     5.1.  Checking the Peer's Public Key  . . . . . . . . . . . . .   9
     5.2.  Short Exponents . . . . . . . . . . . . . . . . . . . . .   9
     5.3.  Table Acceleration  . . . . . . . . . . . . . . . . . . .  10
   6.  Operational Considerations  . . . . . . . . . . . . . . . . .  10
     6.1.  Preference Ordering . . . . . . . . . . . . . . . . . . .  10
   7.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .  11
   8.  Security Considerations . . . . . . . . . . . . . . . . . . .  12
     8.1.  Negotiation Resistance to Active Attacks  . . . . . . . .  12
     8.2.  Group Strength Considerations . . . . . . . . . . . . . .  13
     8.3.  Finite Field DHE Only . . . . . . . . . . . . . . . . . .  13
     8.4.  Deprecating Weak Groups . . . . . . . . . . . . . . . . .  14
     8.5.  Choice of Groups  . . . . . . . . . . . . . . . . . . . .  14
     8.6.  Timing Attacks  . . . . . . . . . . . . . . . . . . . . .  14
     8.7.  Replay Attacks from Non-negotiated FFDHE  . . . . . . . .  15
     8.8.  Forward Secrecy . . . . . . . . . . . . . . . . . . . . .  15
     8.9.  False Start . . . . . . . . . . . . . . . . . . . . . . .  15
   9.  Privacy Considerations  . . . . . . . . . . . . . . . . . . .  16
     9.1.  Client Fingerprinting . . . . . . . . . . . . . . . . . .  16
   10. References  . . . . . . . . . . . . . . . . . . . . . . . . .  16
     10.1.  Normative References . . . . . . . . . . . . . . . . . .  16
     10.2.  Informative References . . . . . . . . . . . . . . . . .  17
   Appendix A.  Supported Groups Registry (Formerly "EC Named Curve
                Registry") . . . . . . . . . . . . . . . . . . . . .  19
     A.1.  ffdhe2048 . . . . . . . . . . . . . . . . . . . . . . . .  19
     A.2.  ffdhe3072 . . . . . . . . . . . . . . . . . . . . . . . .  20
     A.3.  ffdhe4096 . . . . . . . . . . . . . . . . . . . . . . . .  22
     A.4.  ffdhe6144 . . . . . . . . . . . . . . . . . . . . . . . .  23
     A.5.  ffdhe8192 . . . . . . . . . . . . . . . . . . . . . . . .  26
   Acknowledgements  . . . . . . . . . . . . . . . . . . . . . . . .  29
   Author's Address  . . . . . . . . . . . . . . . . . . . . . . . .  29

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1.  Introduction

   Traditional TLS [RFC5246] offers a Diffie-Hellman Ephemeral (DHE) key
   exchange mode that provides forward secrecy for the connection.  The
   client offers a cipher suite in the ClientHello that includes DHE,
   and the server offers the client group parameters generator g and
   modulus p.  If the client does not consider the group strong enough
   (e.g., p is too small, p is not prime, or there are small subgroups
   that cannot be easily avoided) or if it is unable to process the
   group for other reasons, the client has no recourse but to terminate
   the connection.

   Conversely, when a TLS server receives a suggestion for a DHE cipher
   suite from a client, it has no way of knowing what kinds of DH groups
   the client is capable of handling or what the client's security
   requirements are for this key exchange session.  For example, some
   widely distributed TLS clients are not capable of DH groups where p >
   1024 bits.  Other TLS clients may, by policy, wish to use DHE only if
   the server can offer a stronger group (and are willing to use a non-
   PFS (Perfect Forward Secrecy) key exchange mechanism otherwise).  The
   server has no way of knowing which type of client is connecting but
   must select DH parameters with insufficient knowledge.

   Additionally, the DH parameters selected by the server may have a
   known structure that renders them secure against a small subgroup
   attack, but a client receiving an arbitrary p and g has no efficient
   way to verify that the structure of a new group is reasonable for
   use.

   This modification to TLS solves these problems by using a section of
   the "Supported Groups Registry" (renamed from "EC Named Curve
   Registry" by this document) to select common DH groups with known
   structure and defining the use of the "elliptic_curves(10)" extension
   (described here as the Supported Groups extension) for clients
   advertising support for DHE with these groups.  This document also
   provides guidance for compatible peers to take advantage of the
   additional security, availability, and efficiency offered.

   The use of this mechanism by one compatible peer when interacting
   with a non-compatible peer should have no detrimental effects.

   This document updates TLS versions 1.0 [RFC2246], 1.1 [RFC4346], and
   1.2 [RFC5246], as well as the TLS ECC extensions [RFC4492].

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1.1.  Requirements Language

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
   document are to be interpreted as described in [RFC2119].

1.2.  Vocabulary

   The terms "DHE" or "FFDHE" are used in this document to refer to the
   finite-field-based Diffie-Hellman ephemeral key exchange mechanism in
   TLS.  TLS also supports Elliptic Curve Diffie-Hellman Ephemeral
   (ECDHE) key exchanges [RFC4492], but this document does not document
   their use.  A registry previously used only by ECDHE-capable
   implementations is expanded in this document to cover FFDHE groups as
   well.  "FFDHE cipher suites" is used in this document to refer
   exclusively to cipher suites with FFDHE key exchange mechanisms, but
   note that these suites are typically labeled with a TLS_DHE_ prefix.

2.  Named Group Overview

   We use previously unallocated codepoints within the extension
   currently known as "elliptic_curves" (Section 5.1.1. of [RFC4492]) to
   indicate known finite field groups.  The extension's semantics are
   expanded from "Supported Elliptic Curves" to "Supported Groups".  The
   enum datatype used in the extension has been renamed from NamedCurve
   to NamedGroup.  Its semantics are likewise expanded from "named
   curve" to "named group".

   Additionally, we explicitly relax the requirement about when the
   Supported Groups extension can be sent.  This extension MAY be sent
   by the client when either FFDHE or ECDHE cipher suites are listed.

   Codepoints in the "Supported Groups Registry" with a high byte of
   0x01 (that is, between 256 and 511, inclusive) are set aside for
   FFDHE groups, though only a small number of them are initially
   defined and we do not expect many other FFDHE groups to be added to
   this range.  No codepoints outside of this range will be allocated to
   FFDHE groups.  The new codepoints for the "Supported Groups Registry"
   are:

           enum {
           // other already defined elliptic curves (see RFC 4492)
               ffdhe2048(256), ffdhe3072(257), ffdhe4096(258),
               ffdhe6144(259), ffdhe8192(260),
           //
           } NamedGroup;

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   These additions to the "Supported Groups Registry" are described in
   detail in Appendix A.  They are all safe primes derived from the base
   of the natural logarithm ("e"), with the high and low 64 bits set to
   1 for efficient Montgomery or Barrett reduction.

   The use of the base of the natural logarithm here is as a "nothing-
   up-my-sleeve" number.  The goal is to guarantee that the bits in the
   middle of the modulus are effectively random, while avoiding any
   suspicion that the primes have secretly been selected to be weak
   according to some secret criteria.  [RFC3526] used pi for this value.
   See Section 8.5 for reasons that this document does not reuse pi.

3.  Client Behavior

   A TLS client that is capable of using strong finite field Diffie-
   Hellman groups can advertise its capabilities and its preferences for
   stronger key exchange by using this mechanism.

   The compatible client that wants to be able to negotiate strong FFDHE
   sends a Supported Groups extension (identified by type
   elliptic_curves(10) in [RFC4492]) in the ClientHello and includes a
   list of known FFDHE groups in the extension data, ordered from most
   preferred to least preferred.  If the client also supports and wants
   to offer ECDHE key exchange, it MUST use a single Supported Groups
   extension to include all supported groups (both ECDHE and FFDHE
   groups).  The ordering SHOULD be based on client preference, but see
   Section 6.1 for more nuance.

   A client that offers a Supported Groups extension containing an FFDHE
   group SHOULD also include at least one FFDHE cipher suite in the
   ClientHello.

   A client that offers a group MUST be able and willing to perform a DH
   key exchange using that group.

   A client that offers one or more FFDHE groups in the Supported Groups
   extension and an FFDHE cipher suite and that receives an FFDHE cipher
   suite from the server SHOULD take the following steps upon receiving
   the ServerKeyExchange:

   o  For non-anonymous cipher suites where the offered certificate is
      valid and appropriate for the peer, validate the signature over
      the ServerDHParams.  If not valid, terminate the connection.

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   o  If the signature over ServerDHParams is valid, compare the
      selected dh_p and dh_g with the FFDHE groups offered by the
      client.  If none of the offered groups match, the server is not
      compatible with this document.  The client MAY decide to continue
      the connection if the selected group is acceptable under local
      policy, or it MAY decide to terminate the connection with a fatal
      insufficient_security(71) alert.

   o  If the client continues (either because the server offered a
      matching group or because local policy permits the offered custom
      group), the client MUST verify that dh_Ys is in the range 1 <
      dh_Ys < dh_p - 1.  If dh_Ys is not in this range, the client MUST
      terminate the connection with a fatal handshake_failure(40) alert.

   o  If dh_Ys is in range, then the client SHOULD continue with the
      connection as usual.

3.1.  Client Local Policy on Custom Groups

   Compatible clients that are willing to accept FFDHE cipher suites
   from non-compatible servers may have local policy about what custom
   FFDHE groups they are willing to accept.  This local policy presents
   a risk to clients, who may accept weakly protected communications
   from misconfigured servers.

   This document cannot enumerate all possible safe local policy (the
   safest may be to simply reject all custom groups), but compatible
   clients that accept some custom groups from the server MUST do at
   least cursory checks on group size and may take other properties into
   consideration as well.

   A compatible client that accepts FFDHE cipher suites using custom
   groups from non-compatible servers MUST reject any group with |dh_p|
   < 768 bits and SHOULD reject any group with |dh_p| < 1024 bits.

   A compatible client that rejects a non-compatible server's custom
   group may decide to retry the connection while omitting all FFDHE
   cipher suites from the ClientHello.  A client SHOULD only use this
   approach if it successfully verified the server's expected signature
   over the ServerDHParams, to avoid being forced by an active attacker
   into a non-preferred cipher suite.

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4.  Server Behavior

   If a compatible TLS server receives a Supported Groups extension from
   a client that includes any FFDHE group (i.e., any codepoint between
   256 and 511, inclusive, even if unknown to the server), and if none
   of the client-proposed FFDHE groups are known and acceptable to the
   server, then the server MUST NOT select an FFDHE cipher suite.  In
   this case, the server SHOULD select an acceptable non-FFDHE cipher
   suite from the client's offered list.  If the extension is present
   with FFDHE groups, none of the client's offered groups are acceptable
   by the server, and none of the client's proposed non-FFDHE cipher
   suites are acceptable to the server, the server MUST end the
   connection with a fatal TLS alert of type insufficient_security(71).

   If at least one FFDHE cipher suite is present in the client cipher
   suite list and the Supported Groups extension is either absent from
   the ClientHello entirely or contains no FFDHE groups (i.e., no
   codepoints between 256 and 511, inclusive), then the server knows
   that the client is not compatible with this document.  In this
   scenario, a server MAY select a non-FFDHE cipher suite, or it MAY
   select an FFDHE cipher suite and offer an FFDHE group of its choice
   to the client as part of a traditional ServerKeyExchange.

   A compatible TLS server that receives the Supported Groups extension
   with FFDHE codepoints in it and that selects an FFDHE cipher suite
   MUST select one of the client's offered groups.  The server indicates
   the choice of group to the client by sending the group's parameters
   as usual in the ServerKeyExchange as described in Section 7.4.3 of
   [RFC5246].

   A TLS server MUST NOT select a named FFDHE group that was not offered
   by a compatible client.

   A TLS server MUST NOT select an FFDHE cipher suite if the client did
   not offer one, even if the client offered an FFDHE group in the
   Supported Groups extension.

   If a non-anonymous FFDHE cipher suite is selected and the TLS client
   has used this extension to offer an FFDHE group of comparable or
   greater strength than the server's public key, the server SHOULD
   select an FFDHE group at least as strong as the server's public key.
   For example, if the server has a 3072-bit RSA key and the client
   offers only ffdhe2048 and ffdhe4096, the server SHOULD select
   ffdhe4096.

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   When an FFDHE cipher suite is selected and the client sends a
   ClientKeyExchange, the server MUST verify that 1 < dh_Yc < dh_p - 1.
   If dh_Yc is out of range, the server MUST terminate the connection
   with a fatal handshake_failure(40) alert.

5.  Optimizations

   In a key exchange with a successfully negotiated known FFDHE group,
   both peers know that the group in question uses a safe prime as a
   modulus and that the group in use is of size p-1 or (p-1)/2.  This
   allows at least three optimizations that can be used to improve
   performance.

5.1.  Checking the Peer's Public Key

   Peers MUST validate each other's public key Y (dh_Ys offered by the
   server or dh_Yc offered by the client) by ensuring that 1 < Y < p-1.
   This simple check ensures that the remote peer is properly behaved
   and isn't forcing the local system into the 2-element subgroup.

   To reach the same assurance with an unknown group, the client would
   need to verify the primality of the modulus, learn the factors of
   p-1, and test both the generator g and Y against each factor to avoid
   small subgroup attacks.

5.2.  Short Exponents

   Traditional finite field Diffie-Hellman has each peer choose their
   secret exponent from the range [2, p-2].  Using exponentiation by
   squaring, this means each peer must do roughly 2*log_2(p)
   multiplications, twice (once for the generator and once for the
   peer's public key).

   Peers concerned with performance may also prefer to choose their
   secret exponent from a smaller range, doing fewer multiplications,
   while retaining the same level of overall security.  Each named group
   indicates its approximate security level and provides a lower bound
   on the range of secret exponents that should preserve it.  For
   example, rather than doing 2*2*3072 multiplications for an ffdhe3072
   handshake, each peer can choose to do 2*2*275 multiplications by
   choosing their secret exponent from the range [2^274, 2^275] (that
   is, an m-bit integer where m is at least 275) and still keep the same
   approximate security level.

   A similar short-exponent approach is suggested in a Secure SHell
   (SSH) Diffie-Hellman key exchange (see Section 6.2 of [RFC4419]).

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5.3.  Table Acceleration

   Peers wishing to further accelerate FFDHE key exchange can also pre-
   compute a table of powers of the generator of a known group.  This is
   a memory vs. time trade-off, and it only accelerates the first
   exponentiation of the ephemeral DH exchange (the fixed-base
   exponentiation).  The variable-base exponentiation (using the peer's
   public exponent as a base) still needs to be calculated as normal.

6.  Operational Considerations

6.1.  Preference Ordering

   The ordering of named groups in the Supported Groups extension may
   contain some ECDHE groups and some FFDHE groups.  These SHOULD be
   ranked in the order preferred by the client.

   However, the ClientHello also contains a list of desired cipher
   suites, also ranked in preference order.  This presents the
   possibility of conflicted preferences.  For example, if the
   ClientHello contains a cipher_suite field with two choices in order
   <TLS_DHE_RSA_WITH_AES_128_CBC_SHA,
   TLS_ECDHE_RSA_WITH_AES_128_CBC_SHA> and the Supported Groups
   extension contains two choices in order <secp256r1,ffdhe3072>, then
   there is a clear contradiction.  Clients SHOULD NOT present such a
   contradiction since it does not represent a sensible ordering.  A
   server that encounters such a contradiction when selecting between an
   ECDHE or FFDHE key exchange mechanism while trying to respect client
   preferences SHOULD give priority to the Supported Groups extension
   (in the example case, it should select
   TLS_ECDHE_RSA_WITH_AES_128_CBC_SHA with secp256r1) but MAY resolve
   the contradiction any way it sees fit.

   More subtly, clients MAY interleave preferences between ECDHE and
   FFDHE groups; for example, if stronger groups are preferred
   regardless of cost, but weaker groups are acceptable, the Supported
   Groups extension could consist of
   <ffdhe8192,secp384p1,ffdhe3072,secp256r1>.  In this example, with the
   same cipher_suite field offered as the previous example, a server
   configured to respect client preferences and with support for all
   listed groups SHOULD select TLS_DHE_RSA_WITH_AES_128_CBC_SHA with
   ffdhe8192.  A server configured to respect client preferences and
   with support for only secp384p1 and ffdhe3072 SHOULD select
   TLS_ECDHE_RSA_WITH_AES_128_CBC_SHA with secp384p1.

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7.  IANA Considerations

   This document renames the "EC Named Curve Registry" (originally
   defined in [RFC4492] and updated by [RFC7027]) to the "Supported
   Groups Registry".  See
   <https://www.iana.org/assignments/tls-parameters>.

   This document expands the semantics of this registry to include
   groups based on finite fields in addition to groups based on elliptic
   curves.  IANA has added a range designation to the registry,
   indicating that values from 256-511 (inclusive) are set aside for
   "Finite Field Diffie-Hellman groups" and that all other entries in
   the registry are "Elliptic curve groups".

   This document allocates five well-defined codepoints in the registry
   for specific finite field Diffie-Hellman groups defined in
   Appendix A.

   In addition, the four highest codepoints in the range 508-511,
   inclusive, are designated for Private Use [RFC5226] by peers who have
   privately developed finite field Diffie-Hellman groups that they wish
   to signal internally.

   The updated registry section is as follows:

   +---------+-------------+---------+-----------+
   | Value   | Description | DTLS-OK | Reference |
   +---------+-------------+---------+-----------+
   | 256     | ffdhe2048   | Y       | RFC 7919  |
   | 257     | ffdhe3072   | Y       | RFC 7919  |
   | 258     | ffdhe4096   | Y       | RFC 7919  |
   | 259     | ffdhe6144   | Y       | RFC 7919  |
   | 260     | ffdhe8192   | Y       | RFC 7919  |
   | 508-511 | Private Use | -       | RFC 7919  |
   +---------+-------------+---------+-----------+

   IANA has also renamed the "elliptic_curves" extension as
   "supported_groups" and added a reference to this document in the
   "ExtensionType Values" registry.  See
   <http://www.iana.org/assignments/tls-extensiontype-values>.

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8.  Security Considerations

8.1.  Negotiation Resistance to Active Attacks

   Because the contents of the Supported Groups extension are hashed in
   the Finished message, an active Man in the Middle (MITM) that tries
   to filter or omit groups will cause the handshake to fail, but
   possibly not before getting the peer to do something it would not
   otherwise have done.

   An attacker who impersonates the server can try to do any of the
   following:

   o  Pretend that a non-compatible server is actually capable of this
      extension and select a group from the client's list, causing the
      client to select a group it is willing to negotiate.  It is
      unclear how this would be an effective attack.

   o  Pretend that a compatible server is actually non-compatible by
      negotiating a non-FFDHE cipher suite.  This is no different than
      MITM cipher suite filtering.

   o  Pretend that a compatible server is actually non-compatible by
      negotiating a DHE cipher suite with a custom (perhaps weak) group
      selected by the attacker.  This is no worse than the current
      scenario and would require the attacker to be able to sign the
      ServerDHParams, which should not be possible without access to the
      server's secret key.

   An attacker who impersonates the client can try to do the following:

   o  Pretend that a compatible client is not compatible (e.g., by not
      offering the Supported Groups extension or by replacing the
      Supported Groups extension with one that includes no FFDHE
      groups).  This could cause the server to negotiate a weaker DHE
      group during the handshake or to select a non-FFDHE cipher suite,
      but it would fail to complete during the final check of the
      Finished message.

   o  Pretend that a non-compatible client is compatible (e.g., by
      adding the Supported Groups extension or by adding FFDHE groups to
      the extension).  This could cause the server to select a
      particular named group in the ServerKeyExchange or to avoid
      selecting an FFDHE cipher suite.  The peers would fail to compute
      the final check of the Finished message.

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   o  Change the list of groups offered by the client (e.g., by removing
      the stronger of the set of groups offered).  This could cause the
      server to negotiate a weaker group than desired, but again, should
      be caught by the check in the Finished message.

8.2.  Group Strength Considerations

   TLS implementations using FFDHE key exchange should consider the
   strength of the group they negotiate.  The strength of the selected
   group is one of the factors that define the connection's resilience
   against attacks on the session's confidentiality and integrity, since
   the session keys are derived from the DHE handshake.

   While attacks on integrity must generally happen while the session is
   in progress, attacks against session confidentiality can happen
   significantly later if the entire TLS session is stored for offline
   analysis.  Therefore, FFDHE groups should be selected by clients and
   servers based on confidentiality guarantees they need.  Sessions that
   need extremely long-term confidentiality should prefer stronger
   groups.

   [ENISA] provides rough estimates of group resistance to attack and
   recommends that forward-looking implementations ("future systems")
   should use FFDHE group sizes of at least 3072 bits.  ffdhe3072 is
   intended for use in these implementations.

   Other sources (e.g., [NIST]) estimate the security levels of the
   Discrete Log (DLOG) problem to be slightly more difficult than
   [ENISA].  This document's suggested minimum exponent sizes in
   Appendix A for implementations that use the short-exponent
   optimization (Section 5.2) are deliberately conservative to account
   for the range of these estimates.

8.3.  Finite Field DHE Only

   Note that this document specifically targets only finite field
   Diffie-Hellman ephemeral key exchange mechanisms.  It does not cover
   the non-ephemeral DH key exchange mechanisms, nor does it address
   ECDHE key exchange, which is defined in [RFC4492].

   Measured by computational cost to the TLS peers, ECDHE appears today
   to offer a much stronger key exchange mechanism than FFDHE.

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8.4.  Deprecating Weak Groups

   Advances in hardware or in finite field cryptanalysis may cause some
   of the negotiated groups to not provide the desired security margins,
   as indicated by the estimated work factor of an adversary to discover
   the premaster secret (and may therefore compromise the
   confidentiality and integrity of the TLS session).

   Revisions of this document should mark known weak groups as
   explicitly deprecated for use in TLS and should update the estimated
   work factor needed to break the group if the cryptanalysis has
   changed.  Implementations that require strong confidentiality and
   integrity guarantees should avoid using deprecated groups and should
   be updated when the estimated security margins are updated.

8.5.  Choice of Groups

   Other lists of named finite field Diffie-Hellman groups
   [STRONGSWAN-IKE] exist.  This document chooses to not reuse them for
   several reasons:

   o  Using the same groups in multiple protocols increases the value
      for an attacker with the resources to crack any single group.

   o  The Internet Key Exchange Protocol (IKE) groups include weak
      groups like MODP768 that are unacceptable for secure TLS traffic.

   o  Mixing group parameters across multiple implementations leaves
      open the possibility of some sort of cross-protocol attack.  This
      shouldn't be relevant for ephemeral scenarios, and even with non-
      ephemeral keying, services shouldn't share keys; however, using
      different groups avoids these failure modes entirely.

8.6.  Timing Attacks

   Any implementation of finite field Diffie-Hellman key exchange should
   use constant-time modular-exponentiation implementations.  This is
   particularly true for those implementations that ever reuse DHE
   secret keys (so-called "semi-static" ephemeral keying) or share DHE
   secret keys across a multiple machines (e.g., in a load-balancer
   situation).

Top      ToC       Page 15 
8.7.  Replay Attacks from Non-negotiated FFDHE

   [SECURE-RESUMPTION], [CROSS-PROTOCOL], and [SSL3-ANALYSIS] all show a
   malicious peer using a bad FFDHE group to maneuver a client into
   selecting a premaster secret of the peer's choice, which can be
   replayed to another server using a non-FFDHE key exchange and can
   then be bootstrapped to replay client authentication.

   To prevent this attack (barring the session hash fix documented in
   [RFC7627]), a client would need not only to implement this document,
   but also to reject non-negotiated FFDHE cipher suites whose group
   structure it cannot afford to verify.  Such a client would need to
   abort the initial handshake and reconnect to the server in question
   without listing any FFDHE cipher suites on the subsequent connection.

   This trade-off may be too costly for most TLS clients today but may
   be a reasonable choice for clients performing client certificate
   authentication or for clients who have other reasons to be concerned
   about server-controlled premaster secrets.

8.8.  Forward Secrecy

   One of the main reasons to prefer FFDHE ciphersuites is forward
   secrecy, the ability to resist decryption even if the endpoint's
   long-term secret key (usually RSA) is revealed in the future.

   This property depends on both sides of the connection discarding
   their ephemeral keys promptly.  Implementations should wipe their
   FFDHE secret key material from memory as soon as it is no longer
   needed and should never store it in persistent storage.

   Forward secrecy also depends on the strength of the Diffie-Hellman
   group; using a very strong symmetric cipher like AES256 with a
   forward-secret cipher suite but generating the keys with a much
   weaker group like dhe2048 simply moves the adversary's cost from
   attacking the symmetric cipher to attacking the dh_Ys or dh_Yc
   ephemeral key shares.

   If the goal is to provide forward secrecy, attention should be paid
   to all parts of the cipher suite selection process, both key exchange
   and symmetric cipher choice.

8.9.  False Start

   Clients capable of TLS False Start [RFC7918] may receive a proposed
   FFDHE group from a server that is attacker controlled.  In
   particular, the attacker can modify the ClientHello to strip the
   proposed FFDHE groups, which may cause the server to offer a weaker

Top      ToC       Page 16 
   FFDHE group than it should, and this will not be detected until
   receipt of the server's Finished message.  This could cause a client
   using the False Start protocol modification to send data encrypted
   under a weak key agreement.

   Clients should have their own classification of FFDHE groups that are
   "cryptographically strong" in the same sense described in the
   description of symmetric ciphers in [RFC7918] and SHOULD offer at
   least one of these in the initial handshake if they contemplate using
   the False Start protocol modification with an FFDHE cipher suite.

   Compatible clients performing a full handshake MUST NOT use the False
   Start protocol modification if the server selects an FFDHE cipher
   suite but sends a group that is not cryptographically strong from the
   client's perspective.

9.  Privacy Considerations

9.1.  Client Fingerprinting

   This extension provides a few additional bits of information to
   distinguish between classes of TLS clients (e.g., see
   [PANOPTICLICK]).  To minimize this sort of fingerprinting, clients
   SHOULD support all named groups at or above their minimum security
   threshold.  New groups SHOULD NOT be added to the "Supported Groups
   Registry" without consideration of the cost of browser
   fingerprinting.

10.  References

10.1.  Normative References

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997,
              <http://www.rfc-editor.org/info/rfc2119>.

   [RFC4492]  Blake-Wilson, S., Bolyard, N., Gupta, V., Hawk, C., and B.
              Moeller, "Elliptic Curve Cryptography (ECC) Cipher Suites
              for Transport Layer Security (TLS)", RFC 4492,
              DOI 10.17487/RFC4492, May 2006,
              <http://www.rfc-editor.org/info/rfc4492>.

   [RFC5226]  Narten, T. and H. Alvestrand, "Guidelines for Writing an
              IANA Considerations Section in RFCs", BCP 26, RFC 5226,
              DOI 10.17487/RFC5226, May 2008,
              <http://www.rfc-editor.org/info/rfc5226>.

Top      ToC       Page 17 
   [RFC5246]  Dierks, T. and E. Rescorla, "The Transport Layer Security
              (TLS) Protocol Version 1.2", RFC 5246,
              DOI 10.17487/RFC5246, August 2008,
              <http://www.rfc-editor.org/info/rfc5246>.

   [RFC7918]  Langley, A., Modadugu, N., and B. Moeller, "Transport
              Layer Security (TLS) False Start", DOI 10.17487/RFC7918,
              June 2016, <http://www.rfc-editor.org/info/rfc7918>.

10.2.  Informative References

   [CROSS-PROTOCOL]
              Mavrogiannopolous, N., Vercauteren, F., Velichkov, V., and
              B. Preneel, "A Cross-Protocol Attack on the TLS Protocol",
              In Proceedings of the 2012 ACM Conference on Computer and
              Communications Security, DOI 10.1145/2382196.2382206,
              October 2012, <http://www.cosic.esat.kuleuven.be/
              publications/article-2216.pdf>.

   [ECRYPTII]
              European Network of Excellence in Cryptology II, "ECRYPT
              II Yearly Report on Algorithms and Keysizes (2011-2012)",
              Revision 1.0, September 2012,
              <http://www.ecrypt.eu.org/ecrypt2/documents/D.SPA.20.pdf>.

   [ENISA]    European Union Agency for Network and Information Security
              (ENISA), "Algorithms, Key Sizes and Parameters Report -
              2014", November 2014,
              <https://www.enisa.europa.eu/publications/
              algorithms-key-size-and-parameters-report-2014>.

   [NIST]     National Institute of Standards and Technology,
              "Recommendation for Key Management - Part 1: General",
              NIST Special Publication 800-57, Revision 4,
              DOI 10.6028/NIST.SP.800-57pt1r4, January 2016,
              <http://nvlpubs.nist.gov/nistpubs/SpecialPublications/
              NIST.SP.800-57pt1r4.pdf>.

   [PANOPTICLICK]
              Electronic Frontier Foundation, "Panopticlick: Is your
              browser safe against tracking?",
              <https://panopticlick.eff.org/>.

   [RFC2246]  Dierks, T. and C. Allen, "The TLS Protocol Version 1.0",
              RFC 2246, DOI 10.17487/RFC2246, January 1999,
              <http://www.rfc-editor.org/info/rfc2246>.

Top      ToC       Page 18 
   [RFC3526]  Kivinen, T. and M. Kojo, "More Modular Exponential (MODP)
              Diffie-Hellman groups for Internet Key Exchange (IKE)",
              RFC 3526, DOI 10.17487/RFC3526, May 2003,
              <http://www.rfc-editor.org/info/rfc3526>.

   [RFC4346]  Dierks, T. and E. Rescorla, "The Transport Layer Security
              (TLS) Protocol Version 1.1", RFC 4346,
              DOI 10.17487/RFC4346, April 2006,
              <http://www.rfc-editor.org/info/rfc4346>.

   [RFC4419]  Friedl, M., Provos, N., and W. Simpson, "Diffie-Hellman
              Group Exchange for the Secure Shell (SSH) Transport Layer
              Protocol", RFC 4419, DOI 10.17487/RFC4419, March 2006,
              <http://www.rfc-editor.org/info/rfc4419>.

   [RFC7027]  Merkle, J. and M. Lochter, "Elliptic Curve Cryptography
              (ECC) Brainpool Curves for Transport Layer Security
              (TLS)", RFC 7027, DOI 10.17487/RFC7027, October 2013,
              <http://www.rfc-editor.org/info/rfc7027>.

   [RFC7627]  Bhargavan, K., Ed., Delignat-Lavaud, A., Pironti, A.,
              Langley, A., and M. Ray, "Transport Layer Security (TLS)
              Session Hash and Extended Master Secret Extension",
              RFC 7627, DOI 10.17487/RFC7627, September 2015,
              <http://www.rfc-editor.org/info/rfc7627>.

   [SECURE-RESUMPTION]
              Delignat-Lavaud, A., Bhargavan, K., and A. Pironti,
              "Triple Handshakes Considered Harmful: Breaking and Fixing
              Authentication over TLS", 2014 IEEE Symposium on
              Security and Privacy, DOI 10.1109/SP.2014.14, March 2014,
              <https://secure-resumption.com/>.

   [SSL3-ANALYSIS]
              Schneier, B. and D. Wagner, "Analysis of the SSL 3.0
              protocol", In Proceedings of the Second UNIX Workshop
              on Electronic Commerce, 1996,
              <https://www.schneier.com/paper-ssl.pdf>.

   [STRONGSWAN-IKE]
              Brunner, T. and A. Steffen, "IKEv2 Cipher Suites: Diffie
              Hellman Groups", October 2013,
              <https://wiki.strongswan.org/projects/strongswan/wiki/
              IKEv2CipherSuites#Diffie-Hellman-Groups>.

Top      ToC       Page 19 
Appendix A.  Supported Groups Registry (Formerly "EC Named Curve
             Registry")

   Each description below indicates the group itself, its derivation,
   its expected strength (estimated roughly from guidelines in
   [ECRYPTII]), and whether it is recommended for use in TLS key
   exchange at the given security level.  It is not recommended to add
   further finite field groups to the "Supported Groups Registry"; any
   attempt to do so should consider Section 9.1.

   The primes in these finite field groups are all safe primes; that is,
   a prime p is a safe prime when q = (p-1)/2 is also prime.  Where e is
   the base of the natural logarithm and square brackets denote the
   floor operation, the groups that initially populate this registry are
   derived for a given bit length b by finding the lowest positive
   integer X that creates a safe prime p where:

    p = 2^b - 2^{b-64} + {[2^{b-130} e] + X } * 2^64 - 1

   New additions of FFDHE groups to this registry may use this same
   derivation (e.g., with different bit lengths) or may choose their
   parameters in a different way, but they must be clear about how the
   parameters were derived.

   New additions of FFDHE groups MUST use a safe prime as the modulus to
   enable the inexpensive peer verification described in Section 5.1.

A.1.  ffdhe2048

   The 2048-bit group has registry value 256 and is calculated from the
   following formula:

   The modulus is:

   p = 2^2048 - 2^1984 + {[2^1918 * e] + 560316 } * 2^64 - 1

Top      ToC       Page 20 
   The hexadecimal representation of p is:

    FFFFFFFF FFFFFFFF ADF85458 A2BB4A9A AFDC5620 273D3CF1
    D8B9C583 CE2D3695 A9E13641 146433FB CC939DCE 249B3EF9
    7D2FE363 630C75D8 F681B202 AEC4617A D3DF1ED5 D5FD6561
    2433F51F 5F066ED0 85636555 3DED1AF3 B557135E 7F57C935
    984F0C70 E0E68B77 E2A689DA F3EFE872 1DF158A1 36ADE735
    30ACCA4F 483A797A BC0AB182 B324FB61 D108A94B B2C8E3FB
    B96ADAB7 60D7F468 1D4F42A3 DE394DF4 AE56EDE7 6372BB19
    0B07A7C8 EE0A6D70 9E02FCE1 CDF7E2EC C03404CD 28342F61
    9172FE9C E98583FF 8E4F1232 EEF28183 C3FE3B1B 4C6FAD73
    3BB5FCBC 2EC22005 C58EF183 7D1683B2 C6F34A26 C1B2EFFA
    886B4238 61285C97 FFFFFFFF FFFFFFFF

   The generator is: g = 2

   The group size is: q = (p-1)/2

   The hexadecimal representation of q is:

    7FFFFFFF FFFFFFFF D6FC2A2C 515DA54D 57EE2B10 139E9E78
    EC5CE2C1 E7169B4A D4F09B20 8A3219FD E649CEE7 124D9F7C
    BE97F1B1 B1863AEC 7B40D901 576230BD 69EF8F6A EAFEB2B0
    9219FA8F AF833768 42B1B2AA 9EF68D79 DAAB89AF 3FABE49A
    CC278638 707345BB F15344ED 79F7F439 0EF8AC50 9B56F39A
    98566527 A41D3CBD 5E0558C1 59927DB0 E88454A5 D96471FD
    DCB56D5B B06BFA34 0EA7A151 EF1CA6FA 572B76F3 B1B95D8C
    8583D3E4 770536B8 4F017E70 E6FBF176 601A0266 941A17B0
    C8B97F4E 74C2C1FF C7278919 777940C1 E1FF1D8D A637D6B9
    9DDAFE5E 17611002 E2C778C1 BE8B41D9 6379A513 60D977FD
    4435A11C 30942E4B FFFFFFFF FFFFFFFF

   The estimated symmetric-equivalent strength of this group is 103
   bits.

   Peers using ffdhe2048 that want to optimize their key exchange with a
   short exponent (Section 5.2) should choose a secret key of at least
   225 bits.

A.2.  ffdhe3072

   The 3072-bit prime has registry value 257 and is calculated from the
   following formula:

   The modulus is:

   p = 2^3072 - 2^3008 + {[2^2942 * e] + 2625351} * 2^64 - 1

Top      ToC       Page 21 
   The hexadecimal representation of p is:

    FFFFFFFF FFFFFFFF ADF85458 A2BB4A9A AFDC5620 273D3CF1
    D8B9C583 CE2D3695 A9E13641 146433FB CC939DCE 249B3EF9
    7D2FE363 630C75D8 F681B202 AEC4617A D3DF1ED5 D5FD6561
    2433F51F 5F066ED0 85636555 3DED1AF3 B557135E 7F57C935
    984F0C70 E0E68B77 E2A689DA F3EFE872 1DF158A1 36ADE735
    30ACCA4F 483A797A BC0AB182 B324FB61 D108A94B B2C8E3FB
    B96ADAB7 60D7F468 1D4F42A3 DE394DF4 AE56EDE7 6372BB19
    0B07A7C8 EE0A6D70 9E02FCE1 CDF7E2EC C03404CD 28342F61
    9172FE9C E98583FF 8E4F1232 EEF28183 C3FE3B1B 4C6FAD73
    3BB5FCBC 2EC22005 C58EF183 7D1683B2 C6F34A26 C1B2EFFA
    886B4238 611FCFDC DE355B3B 6519035B BC34F4DE F99C0238
    61B46FC9 D6E6C907 7AD91D26 91F7F7EE 598CB0FA C186D91C
    AEFE1309 85139270 B4130C93 BC437944 F4FD4452 E2D74DD3
    64F2E21E 71F54BFF 5CAE82AB 9C9DF69E E86D2BC5 22363A0D
    ABC52197 9B0DEADA 1DBF9A42 D5C4484E 0ABCD06B FA53DDEF
    3C1B20EE 3FD59D7C 25E41D2B 66C62E37 FFFFFFFF FFFFFFFF

   The generator is: g = 2

   The group size is: q = (p-1)/2

   The hexadecimal representation of q is:

    7FFFFFFF FFFFFFFF D6FC2A2C 515DA54D 57EE2B10 139E9E78
    EC5CE2C1 E7169B4A D4F09B20 8A3219FD E649CEE7 124D9F7C
    BE97F1B1 B1863AEC 7B40D901 576230BD 69EF8F6A EAFEB2B0
    9219FA8F AF833768 42B1B2AA 9EF68D79 DAAB89AF 3FABE49A
    CC278638 707345BB F15344ED 79F7F439 0EF8AC50 9B56F39A
    98566527 A41D3CBD 5E0558C1 59927DB0 E88454A5 D96471FD
    DCB56D5B B06BFA34 0EA7A151 EF1CA6FA 572B76F3 B1B95D8C
    8583D3E4 770536B8 4F017E70 E6FBF176 601A0266 941A17B0
    C8B97F4E 74C2C1FF C7278919 777940C1 E1FF1D8D A637D6B9
    9DDAFE5E 17611002 E2C778C1 BE8B41D9 6379A513 60D977FD
    4435A11C 308FE7EE 6F1AAD9D B28C81AD DE1A7A6F 7CCE011C
    30DA37E4 EB736483 BD6C8E93 48FBFBF7 2CC6587D 60C36C8E
    577F0984 C289C938 5A098649 DE21BCA2 7A7EA229 716BA6E9
    B279710F 38FAA5FF AE574155 CE4EFB4F 743695E2 911B1D06
    D5E290CB CD86F56D 0EDFCD21 6AE22427 055E6835 FD29EEF7
    9E0D9077 1FEACEBE 12F20E95 B363171B FFFFFFFF FFFFFFFF

   The estimated symmetric-equivalent strength of this group is 125
   bits.

   Peers using ffdhe3072 that want to optimize their key exchange with a
   short exponent (Section 5.2) should choose a secret key of at least
   275 bits.

Top      ToC       Page 22 
A.3.  ffdhe4096

   The 4096-bit group has registry value 258 and is calculated from the
   following formula:

   The modulus is:

   p = 2^4096 - 2^4032 + {[2^3966 * e] + 5736041} * 2^64 - 1

   The hexadecimal representation of p is:

    FFFFFFFF FFFFFFFF ADF85458 A2BB4A9A AFDC5620 273D3CF1
    D8B9C583 CE2D3695 A9E13641 146433FB CC939DCE 249B3EF9
    7D2FE363 630C75D8 F681B202 AEC4617A D3DF1ED5 D5FD6561
    2433F51F 5F066ED0 85636555 3DED1AF3 B557135E 7F57C935
    984F0C70 E0E68B77 E2A689DA F3EFE872 1DF158A1 36ADE735
    30ACCA4F 483A797A BC0AB182 B324FB61 D108A94B B2C8E3FB
    B96ADAB7 60D7F468 1D4F42A3 DE394DF4 AE56EDE7 6372BB19
    0B07A7C8 EE0A6D70 9E02FCE1 CDF7E2EC C03404CD 28342F61
    9172FE9C E98583FF 8E4F1232 EEF28183 C3FE3B1B 4C6FAD73
    3BB5FCBC 2EC22005 C58EF183 7D1683B2 C6F34A26 C1B2EFFA
    886B4238 611FCFDC DE355B3B 6519035B BC34F4DE F99C0238
    61B46FC9 D6E6C907 7AD91D26 91F7F7EE 598CB0FA C186D91C
    AEFE1309 85139270 B4130C93 BC437944 F4FD4452 E2D74DD3
    64F2E21E 71F54BFF 5CAE82AB 9C9DF69E E86D2BC5 22363A0D
    ABC52197 9B0DEADA 1DBF9A42 D5C4484E 0ABCD06B FA53DDEF
    3C1B20EE 3FD59D7C 25E41D2B 669E1EF1 6E6F52C3 164DF4FB
    7930E9E4 E58857B6 AC7D5F42 D69F6D18 7763CF1D 55034004
    87F55BA5 7E31CC7A 7135C886 EFB4318A ED6A1E01 2D9E6832
    A907600A 918130C4 6DC778F9 71AD0038 092999A3 33CB8B7A
    1A1DB93D 7140003C 2A4ECEA9 F98D0ACC 0A8291CD CEC97DCF
    8EC9B55A 7F88A46B 4DB5A851 F44182E1 C68A007E 5E655F6A
    FFFFFFFF FFFFFFFF

   The generator is: g = 2

   The group size is: q = (p-1)/2

Top      ToC       Page 23 
   The hexadecimal representation of q is:

    7FFFFFFF FFFFFFFF D6FC2A2C 515DA54D 57EE2B10 139E9E78
    EC5CE2C1 E7169B4A D4F09B20 8A3219FD E649CEE7 124D9F7C
    BE97F1B1 B1863AEC 7B40D901 576230BD 69EF8F6A EAFEB2B0
    9219FA8F AF833768 42B1B2AA 9EF68D79 DAAB89AF 3FABE49A
    CC278638 707345BB F15344ED 79F7F439 0EF8AC50 9B56F39A
    98566527 A41D3CBD 5E0558C1 59927DB0 E88454A5 D96471FD
    DCB56D5B B06BFA34 0EA7A151 EF1CA6FA 572B76F3 B1B95D8C
    8583D3E4 770536B8 4F017E70 E6FBF176 601A0266 941A17B0
    C8B97F4E 74C2C1FF C7278919 777940C1 E1FF1D8D A637D6B9
    9DDAFE5E 17611002 E2C778C1 BE8B41D9 6379A513 60D977FD
    4435A11C 308FE7EE 6F1AAD9D B28C81AD DE1A7A6F 7CCE011C
    30DA37E4 EB736483 BD6C8E93 48FBFBF7 2CC6587D 60C36C8E
    577F0984 C289C938 5A098649 DE21BCA2 7A7EA229 716BA6E9
    B279710F 38FAA5FF AE574155 CE4EFB4F 743695E2 911B1D06
    D5E290CB CD86F56D 0EDFCD21 6AE22427 055E6835 FD29EEF7
    9E0D9077 1FEACEBE 12F20E95 B34F0F78 B737A961 8B26FA7D
    BC9874F2 72C42BDB 563EAFA1 6B4FB68C 3BB1E78E AA81A002
    43FAADD2 BF18E63D 389AE443 77DA18C5 76B50F00 96CF3419
    5483B005 48C09862 36E3BC7C B8D6801C 0494CCD1 99E5C5BD
    0D0EDC9E B8A0001E 15276754 FCC68566 054148E6 E764BEE7
    C764DAAD 3FC45235 A6DAD428 FA20C170 E345003F 2F32AFB5
    7FFFFFFF FFFFFFFF

   The estimated symmetric-equivalent strength of this group is 150
   bits.

   Peers using ffdhe4096 that want to optimize their key exchange with a
   short exponent (Section 5.2) should choose a secret key of at least
   325 bits.

A.4.  ffdhe6144

   The 6144-bit group has registry value 259 and is calculated from the
   following formula:

   The modulus is:

   p = 2^6144 - 2^6080 + {[2^6014 * e] + 15705020} * 2^64 - 1

Top      ToC       Page 24 
   The hexadecimal representation of p is:

    FFFFFFFF FFFFFFFF ADF85458 A2BB4A9A AFDC5620 273D3CF1
    D8B9C583 CE2D3695 A9E13641 146433FB CC939DCE 249B3EF9
    7D2FE363 630C75D8 F681B202 AEC4617A D3DF1ED5 D5FD6561
    2433F51F 5F066ED0 85636555 3DED1AF3 B557135E 7F57C935
    984F0C70 E0E68B77 E2A689DA F3EFE872 1DF158A1 36ADE735
    30ACCA4F 483A797A BC0AB182 B324FB61 D108A94B B2C8E3FB
    B96ADAB7 60D7F468 1D4F42A3 DE394DF4 AE56EDE7 6372BB19
    0B07A7C8 EE0A6D70 9E02FCE1 CDF7E2EC C03404CD 28342F61
    9172FE9C E98583FF 8E4F1232 EEF28183 C3FE3B1B 4C6FAD73
    3BB5FCBC 2EC22005 C58EF183 7D1683B2 C6F34A26 C1B2EFFA
    886B4238 611FCFDC DE355B3B 6519035B BC34F4DE F99C0238
    61B46FC9 D6E6C907 7AD91D26 91F7F7EE 598CB0FA C186D91C
    AEFE1309 85139270 B4130C93 BC437944 F4FD4452 E2D74DD3
    64F2E21E 71F54BFF 5CAE82AB 9C9DF69E E86D2BC5 22363A0D
    ABC52197 9B0DEADA 1DBF9A42 D5C4484E 0ABCD06B FA53DDEF
    3C1B20EE 3FD59D7C 25E41D2B 669E1EF1 6E6F52C3 164DF4FB
    7930E9E4 E58857B6 AC7D5F42 D69F6D18 7763CF1D 55034004
    87F55BA5 7E31CC7A 7135C886 EFB4318A ED6A1E01 2D9E6832
    A907600A 918130C4 6DC778F9 71AD0038 092999A3 33CB8B7A
    1A1DB93D 7140003C 2A4ECEA9 F98D0ACC 0A8291CD CEC97DCF
    8EC9B55A 7F88A46B 4DB5A851 F44182E1 C68A007E 5E0DD902
    0BFD64B6 45036C7A 4E677D2C 38532A3A 23BA4442 CAF53EA6
    3BB45432 9B7624C8 917BDD64 B1C0FD4C B38E8C33 4C701C3A
    CDAD0657 FCCFEC71 9B1F5C3E 4E46041F 388147FB 4CFDB477
    A52471F7 A9A96910 B855322E DB6340D8 A00EF092 350511E3
    0ABEC1FF F9E3A26E 7FB29F8C 183023C3 587E38DA 0077D9B4
    763E4E4B 94B2BBC1 94C6651E 77CAF992 EEAAC023 2A281BF6
    B3A739C1 22611682 0AE8DB58 47A67CBE F9C9091B 462D538C
    D72B0374 6AE77F5E 62292C31 1562A846 505DC82D B854338A
    E49F5235 C95B9117 8CCF2DD5 CACEF403 EC9D1810 C6272B04
    5B3B71F9 DC6B80D6 3FDD4A8E 9ADB1E69 62A69526 D43161C1
    A41D570D 7938DAD4 A40E329C D0E40E65 FFFFFFFF FFFFFFFF

   The generator is: g = 2

   The group size is: q = (p-1)/2

Top      ToC       Page 25 
   The hexadecimal representation of q is:

    7FFFFFFF FFFFFFFF D6FC2A2C 515DA54D 57EE2B10 139E9E78
    EC5CE2C1 E7169B4A D4F09B20 8A3219FD E649CEE7 124D9F7C
    BE97F1B1 B1863AEC 7B40D901 576230BD 69EF8F6A EAFEB2B0
    9219FA8F AF833768 42B1B2AA 9EF68D79 DAAB89AF 3FABE49A
    CC278638 707345BB F15344ED 79F7F439 0EF8AC50 9B56F39A
    98566527 A41D3CBD 5E0558C1 59927DB0 E88454A5 D96471FD
    DCB56D5B B06BFA34 0EA7A151 EF1CA6FA 572B76F3 B1B95D8C
    8583D3E4 770536B8 4F017E70 E6FBF176 601A0266 941A17B0
    C8B97F4E 74C2C1FF C7278919 777940C1 E1FF1D8D A637D6B9
    9DDAFE5E 17611002 E2C778C1 BE8B41D9 6379A513 60D977FD
    4435A11C 308FE7EE 6F1AAD9D B28C81AD DE1A7A6F 7CCE011C
    30DA37E4 EB736483 BD6C8E93 48FBFBF7 2CC6587D 60C36C8E
    577F0984 C289C938 5A098649 DE21BCA2 7A7EA229 716BA6E9
    B279710F 38FAA5FF AE574155 CE4EFB4F 743695E2 911B1D06
    D5E290CB CD86F56D 0EDFCD21 6AE22427 055E6835 FD29EEF7
    9E0D9077 1FEACEBE 12F20E95 B34F0F78 B737A961 8B26FA7D
    BC9874F2 72C42BDB 563EAFA1 6B4FB68C 3BB1E78E AA81A002
    43FAADD2 BF18E63D 389AE443 77DA18C5 76B50F00 96CF3419
    5483B005 48C09862 36E3BC7C B8D6801C 0494CCD1 99E5C5BD
    0D0EDC9E B8A0001E 15276754 FCC68566 054148E6 E764BEE7
    C764DAAD 3FC45235 A6DAD428 FA20C170 E345003F 2F06EC81
    05FEB25B 2281B63D 2733BE96 1C29951D 11DD2221 657A9F53
    1DDA2A19 4DBB1264 48BDEEB2 58E07EA6 59C74619 A6380E1D
    66D6832B FE67F638 CD8FAE1F 2723020F 9C40A3FD A67EDA3B
    D29238FB D4D4B488 5C2A9917 6DB1A06C 50077849 1A8288F1
    855F60FF FCF1D137 3FD94FC6 0C1811E1 AC3F1C6D 003BECDA
    3B1F2725 CA595DE0 CA63328F 3BE57CC9 77556011 95140DFB
    59D39CE0 91308B41 05746DAC 23D33E5F 7CE4848D A316A9C6
    6B9581BA 3573BFAF 31149618 8AB15423 282EE416 DC2A19C5
    724FA91A E4ADC88B C66796EA E5677A01 F64E8C08 63139582
    2D9DB8FC EE35C06B 1FEEA547 4D6D8F34 B1534A93 6A18B0E0
    D20EAB86 BC9C6D6A 5207194E 68720732 FFFFFFFF FFFFFFFF

   The estimated symmetric-equivalent strength of this group is 175
   bits.

   Peers using ffdhe6144 that want to optimize their key exchange with a
   short exponent (Section 5.2) should choose a secret key of at least
   375 bits.

Top      ToC       Page 26 
A.5.  ffdhe8192

   The 8192-bit group has registry value 260 and is calculated from the
   following formula:

   The modulus is:

   p = 2^8192 - 2^8128 + {[2^8062 * e] + 10965728} * 2^64 - 1

Top      ToC       Page 27 
   The hexadecimal representation of p is:

    FFFFFFFF FFFFFFFF ADF85458 A2BB4A9A AFDC5620 273D3CF1
    D8B9C583 CE2D3695 A9E13641 146433FB CC939DCE 249B3EF9
    7D2FE363 630C75D8 F681B202 AEC4617A D3DF1ED5 D5FD6561
    2433F51F 5F066ED0 85636555 3DED1AF3 B557135E 7F57C935
    984F0C70 E0E68B77 E2A689DA F3EFE872 1DF158A1 36ADE735
    30ACCA4F 483A797A BC0AB182 B324FB61 D108A94B B2C8E3FB
    B96ADAB7 60D7F468 1D4F42A3 DE394DF4 AE56EDE7 6372BB19
    0B07A7C8 EE0A6D70 9E02FCE1 CDF7E2EC C03404CD 28342F61
    9172FE9C E98583FF 8E4F1232 EEF28183 C3FE3B1B 4C6FAD73
    3BB5FCBC 2EC22005 C58EF183 7D1683B2 C6F34A26 C1B2EFFA
    886B4238 611FCFDC DE355B3B 6519035B BC34F4DE F99C0238
    61B46FC9 D6E6C907 7AD91D26 91F7F7EE 598CB0FA C186D91C
    AEFE1309 85139270 B4130C93 BC437944 F4FD4452 E2D74DD3
    64F2E21E 71F54BFF 5CAE82AB 9C9DF69E E86D2BC5 22363A0D
    ABC52197 9B0DEADA 1DBF9A42 D5C4484E 0ABCD06B FA53DDEF
    3C1B20EE 3FD59D7C 25E41D2B 669E1EF1 6E6F52C3 164DF4FB
    7930E9E4 E58857B6 AC7D5F42 D69F6D18 7763CF1D 55034004
    87F55BA5 7E31CC7A 7135C886 EFB4318A ED6A1E01 2D9E6832
    A907600A 918130C4 6DC778F9 71AD0038 092999A3 33CB8B7A
    1A1DB93D 7140003C 2A4ECEA9 F98D0ACC 0A8291CD CEC97DCF
    8EC9B55A 7F88A46B 4DB5A851 F44182E1 C68A007E 5E0DD902
    0BFD64B6 45036C7A 4E677D2C 38532A3A 23BA4442 CAF53EA6
    3BB45432 9B7624C8 917BDD64 B1C0FD4C B38E8C33 4C701C3A
    CDAD0657 FCCFEC71 9B1F5C3E 4E46041F 388147FB 4CFDB477
    A52471F7 A9A96910 B855322E DB6340D8 A00EF092 350511E3
    0ABEC1FF F9E3A26E 7FB29F8C 183023C3 587E38DA 0077D9B4
    763E4E4B 94B2BBC1 94C6651E 77CAF992 EEAAC023 2A281BF6
    B3A739C1 22611682 0AE8DB58 47A67CBE F9C9091B 462D538C
    D72B0374 6AE77F5E 62292C31 1562A846 505DC82D B854338A
    E49F5235 C95B9117 8CCF2DD5 CACEF403 EC9D1810 C6272B04
    5B3B71F9 DC6B80D6 3FDD4A8E 9ADB1E69 62A69526 D43161C1
    A41D570D 7938DAD4 A40E329C CFF46AAA 36AD004C F600C838
    1E425A31 D951AE64 FDB23FCE C9509D43 687FEB69 EDD1CC5E
    0B8CC3BD F64B10EF 86B63142 A3AB8829 555B2F74 7C932665
    CB2C0F1C C01BD702 29388839 D2AF05E4 54504AC7 8B758282
    2846C0BA 35C35F5C 59160CC0 46FD8251 541FC68C 9C86B022
    BB709987 6A460E74 51A8A931 09703FEE 1C217E6C 3826E52C
    51AA691E 0E423CFC 99E9E316 50C1217B 624816CD AD9A95F9
    D5B80194 88D9C0A0 A1FE3075 A577E231 83F81D4A 3F2FA457
    1EFC8CE0 BA8A4FE8 B6855DFE 72B0A66E DED2FBAB FBE58A30
    FAFABE1C 5D71A87E 2F741EF8 C1FE86FE A6BBFDE5 30677F0D
    97D11D49 F7A8443D 0822E506 A9F4614E 011E2A94 838FF88C
    D68C8BB7 C5C6424C FFFFFFFF FFFFFFFF

   The generator is: g = 2

Top      ToC       Page 28 
   The group size is: q = (p-1)/2

   The hexadecimal representation of q is:

    7FFFFFFF FFFFFFFF D6FC2A2C 515DA54D 57EE2B10 139E9E78
    EC5CE2C1 E7169B4A D4F09B20 8A3219FD E649CEE7 124D9F7C
    BE97F1B1 B1863AEC 7B40D901 576230BD 69EF8F6A EAFEB2B0
    9219FA8F AF833768 42B1B2AA 9EF68D79 DAAB89AF 3FABE49A
    CC278638 707345BB F15344ED 79F7F439 0EF8AC50 9B56F39A
    98566527 A41D3CBD 5E0558C1 59927DB0 E88454A5 D96471FD
    DCB56D5B B06BFA34 0EA7A151 EF1CA6FA 572B76F3 B1B95D8C
    8583D3E4 770536B8 4F017E70 E6FBF176 601A0266 941A17B0
    C8B97F4E 74C2C1FF C7278919 777940C1 E1FF1D8D A637D6B9
    9DDAFE5E 17611002 E2C778C1 BE8B41D9 6379A513 60D977FD
    4435A11C 308FE7EE 6F1AAD9D B28C81AD DE1A7A6F 7CCE011C
    30DA37E4 EB736483 BD6C8E93 48FBFBF7 2CC6587D 60C36C8E
    577F0984 C289C938 5A098649 DE21BCA2 7A7EA229 716BA6E9
    B279710F 38FAA5FF AE574155 CE4EFB4F 743695E2 911B1D06
    D5E290CB CD86F56D 0EDFCD21 6AE22427 055E6835 FD29EEF7
    9E0D9077 1FEACEBE 12F20E95 B34F0F78 B737A961 8B26FA7D
    BC9874F2 72C42BDB 563EAFA1 6B4FB68C 3BB1E78E AA81A002
    43FAADD2 BF18E63D 389AE443 77DA18C5 76B50F00 96CF3419
    5483B005 48C09862 36E3BC7C B8D6801C 0494CCD1 99E5C5BD
    0D0EDC9E B8A0001E 15276754 FCC68566 054148E6 E764BEE7
    C764DAAD 3FC45235 A6DAD428 FA20C170 E345003F 2F06EC81
    05FEB25B 2281B63D 2733BE96 1C29951D 11DD2221 657A9F53
    1DDA2A19 4DBB1264 48BDEEB2 58E07EA6 59C74619 A6380E1D
    66D6832B FE67F638 CD8FAE1F 2723020F 9C40A3FD A67EDA3B
    D29238FB D4D4B488 5C2A9917 6DB1A06C 50077849 1A8288F1
    855F60FF FCF1D137 3FD94FC6 0C1811E1 AC3F1C6D 003BECDA
    3B1F2725 CA595DE0 CA63328F 3BE57CC9 77556011 95140DFB
    59D39CE0 91308B41 05746DAC 23D33E5F 7CE4848D A316A9C6
    6B9581BA 3573BFAF 31149618 8AB15423 282EE416 DC2A19C5
    724FA91A E4ADC88B C66796EA E5677A01 F64E8C08 63139582
    2D9DB8FC EE35C06B 1FEEA547 4D6D8F34 B1534A93 6A18B0E0
    D20EAB86 BC9C6D6A 5207194E 67FA3555 1B568026 7B00641C
    0F212D18 ECA8D732 7ED91FE7 64A84EA1 B43FF5B4 F6E8E62F
    05C661DE FB258877 C35B18A1 51D5C414 AAAD97BA 3E499332
    E596078E 600DEB81 149C441C E95782F2 2A282563 C5BAC141
    1423605D 1AE1AFAE 2C8B0660 237EC128 AA0FE346 4E435811
    5DB84CC3 B523073A 28D45498 84B81FF7 0E10BF36 1C137296
    28D5348F 07211E7E 4CF4F18B 286090BD B1240B66 D6CD4AFC
    EADC00CA 446CE050 50FF183A D2BBF118 C1FC0EA5 1F97D22B
    8F7E4670 5D4527F4 5B42AEFF 39585337 6F697DD5 FDF2C518
    7D7D5F0E 2EB8D43F 17BA0F7C 60FF437F 535DFEF2 9833BF86
    CBE88EA4 FBD4221E 84117283 54FA30A7 008F154A 41C7FC46
    6B4645DB E2E32126 7FFFFFFF FFFFFFFF

Top      ToC       Page 29 
   The estimated symmetric-equivalent strength of this group is 192
   bits.

   Peers using ffdhe8192 that want to optimize their key exchange with a
   short exponent (Section 5.2) should choose a secret key of at least
   400 bits.

Acknowledgements

   Thanks to Fedor Brunner, Dave Fergemann, Niels Ferguson, Sandy
   Harris, Tero Kivinen, Watson Ladd, Nikos Mavrogiannopolous, Niels
   Moeller, Bodo Moeller, Kenny Paterson, Eric Rescorla, Tom Ritter,
   Rene Struik, Martin Thomson, Sean Turner, and other members of the
   TLS Working Group for their comments and suggestions on this
   document.  Any mistakes here are not theirs.

Author's Address

   Daniel Kahn Gillmor
   ACLU
   125 Broad Street, 18th Floor
   New York, NY  10004
   United States of America

   Email: dkg@fifthhorseman.net