Negotiated Finite Field Diffie-Hellman Ephemeral Parameters for TLS
draft-ietf-tls-negotiated-ff-dhe-01

Internet Engineering Task Force                               D. Gillmor
Internet-Draft                                                      ACLU
Intended status: Informational                           August 27, 2014
Expires: February 28, 2015

  Negotiated Finite Field Diffie-Hellman Ephemeral Parameters for TLS
                  draft-ietf-tls-negotiated-ff-dhe-01

Abstract

   Traditional finite-field-based Diffie-Hellman (DH) key exchange
   during the 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 establishing
   a registry of DH parameters with known structure and a mechanism for
   peers to indicate support for these groups.

Status of This Memo

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   Copyright (c) 2014 IETF Trust and the persons identified as the
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   include Simplified BSD License text as described in Section 4.e of
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Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
     1.1.  Requirements Language . . . . . . . . . . . . . . . . . .   3
     1.2.  Vocabulary  . . . . . . . . . . . . . . . . . . . . . . .   3
   2.  Client Behavior . . . . . . . . . . . . . . . . . . . . . . .   4
   3.  Server Behavior . . . . . . . . . . . . . . . . . . . . . . .   4
     3.1.  ServerDHParams changes  . . . . . . . . . . . . . . . . .   5
   4.  Optimizations . . . . . . . . . . . . . . . . . . . . . . . .   5
     4.1.  Checking the Peer's Public Key  . . . . . . . . . . . . .   6
     4.2.  Short Exponents . . . . . . . . . . . . . . . . . . . . .   6
     4.3.  Table Acceleration  . . . . . . . . . . . . . . . . . . .   6
   5.  Open Questions  . . . . . . . . . . . . . . . . . . . . . . .   7
     5.1.  Server Indication of support  . . . . . . . . . . . . . .   7
     5.2.  Normalizing Weak Groups . . . . . . . . . . . . . . . . .   7
     5.3.  Arbitrary Groups  . . . . . . . . . . . . . . . . . . . .   7
   6.  Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .   8
   7.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .   8
   8.  Security Considerations . . . . . . . . . . . . . . . . . . .   8
     8.1.  Negotiation resistance to active attacks  . . . . . . . .   8
     8.2.  DHE only  . . . . . . . . . . . . . . . . . . . . . . . .   9
     8.3.  Deprecating weak groups . . . . . . . . . . . . . . . . .   9
     8.4.  Choice of groups  . . . . . . . . . . . . . . . . . . . .  10
     8.5.  Timing attacks  . . . . . . . . . . . . . . . . . . . . .  10
     8.6.  Replay attacks from non-negotiated FF DHE . . . . . . . .  10
   9.  Privacy Considerations  . . . . . . . . . . . . . . . . . . .  11
     9.1.  Client fingerprinting . . . . . . . . . . . . . . . . . .  11
   10. References  . . . . . . . . . . . . . . . . . . . . . . . . .  11
     10.1.  Normative References . . . . . . . . . . . . . . . . . .  11
     10.2.  Informative References . . . . . . . . . . . . . . . . .  11
   Appendix A.  Named Group Registry . . . . . . . . . . . . . . . .  13
     A.1.  ffdhe2432 . . . . . . . . . . . . . . . . . . . . . . . .  13
     A.2.  ffdhe3072 . . . . . . . . . . . . . . . . . . . . . . . .  14
     A.3.  ffdhe4096 . . . . . . . . . . . . . . . . . . . . . . . .  16
     A.4.  ffdhe6144 . . . . . . . . . . . . . . . . . . . . . . . .  17
     A.5.  ffdhe8192 . . . . . . . . . . . . . . . . . . . . . . . .  19
   Author's Address  . . . . . . . . . . . . . . . . . . . . . . . .  22

1.  Introduction

   Traditional TLS [RFC5246] offers a Diffie-Hellman ephemeral (DHE) key
   exchange mode which provides Perfect Forward Secrecy for the
   connection.  The client offers a ciphersuite in the ClientHello that
   includes DHE, and the server offers the client group parameters g and

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   p.  If the client does not consider the group strong enough (e.g. if
   p is too small, or if p is not prime, or there are small subgroups),
   or if it is unable to process it for other reasons, it has no
   recourse but to terminate the connection.

   Conversely, when a TLS server receives a suggestion for a DHE
   ciphersuite 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.  Some
   widely-distributed TLS clients are not capable of DH groups where p >
   1024.  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
   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 chosen by the server may have a known
   structure which renders them secure against a small subgroup attack,
   but a client receiving an arbitrary p has no efficient way to verify
   that the structure of a new group is reasonable for use.

   This extension solves these problems with a registry of groups of
   known reasonable structure, an extension for clients to advertise
   support for them and servers to select them, and guidance for
   compliant peers to take advantage of the additional security,
   availability, and efficiency offered.

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

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 term "DHE" is used in this document to refer to the finite-field-
   based Diffie-Hellman ephemeral key exchange mechanism in TLS.  TLS
   also supports elliptic-curve-based Diffie-Hellman (ECDHE) ephemeral
   key exchanges, but this document does not discuss their use.
   Mentions of DHE here refer strictly to finite-field-based DHE, and
   not to ECDHE.

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2.  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 client SHOULD send an extension of type
   "negotiated_ff_dhe_groups" in the ClientHello, indicating a list of
   known finite field Diffie-Hellman groups, ordered from most preferred
   to least preferred.

   The "extension_data" field of this extension SHALL contain
   "FiniteFieldDHEGroups" where:

         enum {
             ffdhe2432(0), ffdhe3072(1), ffdhe4096(2),
             ffdhe6144(3), ffdhe8192(4), (255)
         } FiniteFieldDHEGroup;

         struct {
             FiniteFieldDHEGroup finite_field_dhe_group_list<1..2^8-1>;
         } FiniteFieldDHEGroups;

   A client that offers this extension SHOULD include at least one DHE-
   key-exchange ciphersuite in the Client Hello.

   The known groups defined by the FiniteFieldDHEGroup registry are
   listed in Appendix A.  These 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 8.4 for reasons that this draft does not reuse pi.

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

3.  Server Behavior

   A TLS server MUST NOT send the NegotiatedDHParams extension to a
   client that does not offer it first.

   A compatible TLS server that receives this extension from a client
   SHOULD NOT select a DHE ciphersuite if it is unwilling to use one of

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   the DH groups named by the client.  In this case, it SHOULD select an
   acceptable non-DHE ciphersuite from the client's offered list.  If
   the extension is present, none of the client's offered groups are
   acceptable by the server, and none of the client's proposed non-DHE
   ciphersuites are acceptable to the server, the server SHOULD end the
   connection with a fatal TLS alert of type insufficient_security.

   A compatible TLS server that receives this extension from a client
   and selects a DHE-key-exchange ciphersuite selects one of the offered
   groups and indicates it to the client in the ServerHello by sending a
   "negotiated_ff_dhe_groups" extension.  The "extension_data" field of
   this extension on the server side should be a single one-byte value
   FiniteFieldDHEGroup.

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

   If a non-anonymous DHE ciphersuite is chosen, and the TLS client has
   used this extension to offer a DHE group of comparable or greater
   strength than the server's public key, the server SHOULD select a DHE
   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
   ffdhe2432 and ffdhe4096, the server SHOULD select ffdhe4096.

3.1.  ServerDHParams changes

   When the server sends the "negotiated_ff_dhe_groups" extension in the
   ServerHello, the ServerDHParams member of the subsequent
   ServerKeyExchange message should indicate a one-byte zero value (0)
   in place of dh_g and the identifier of the named group in place of
   dh_p, represented as a one-byte value.  dh_Ys must be transmitted as
   normal.

   This re-purposing of dh_p and dh_g is unambiguous: there are no
   groups with a generator of 0, and no implementation should accept a
   modulus of size < 9 bits.  This change serves two purposes:

      The size of the handshake is reduced (significantly, in the case
      of a large prime modulus).

      The signed struct should not be re-playable in a subsequent key
      exchange that does not indicate named DH groups.

4.  Optimizations

   In a successfully negotiated finite field DH group key exchange, both
   peers know that the group in question uses a safe prime as a modulus,

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   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.

4.1.  Checking the Peer's Public Key

   Peers should validate the 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 a small 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.

4.2.  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*2432 multiplications for a ffdhe2432
   handshake, each peer can choose to do 2*2*224 multiplications by
   choosing their secret exponent in the range [2,2^224] and still keep
   the approximate 112-bit security level.

   A similar short-exponent approach is suggested in SSH's Diffie-
   Hellman key exchange (See section 6.2 of [RFC4419]).

4.3.  Table Acceleration

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

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5.  Open Questions

   [This section should be removed, and questions resolved, before any
   formalization of this draft]

5.1.  Server Indication of support

   Some servers will support this extension, but for whatever reason
   decide to not negotiate a ciphersuite with DHE key exchange at all.
   Some possible reasons include:

      The client indicated that a server-supported non-DHE ciphersuite
      was preferred over all DHE ciphersuites, and the server honors
      that preference.

      The server prefers a client-supported non-DHE ciphersuite over all
      DHE ciphersuites, and selects it unilaterally.

      The server would have chosen a DHE ciphersuite, but none of the
      client's offered groups are acceptable to the server,

   Clients will not know that such a server supports the extension.

   Should we offer a way for a server to indicate its support for this
   extension to a compatible client in this case?

   Should the server have a way to advertise that it supports this
   extension even if the client does not offer it?

5.2.  Normalizing Weak Groups

   Is there any reason to include a weak group in the list of groups?
   Most DHE-capable peers can already handle 1024-bit DHE, and therefore
   1024-bit DHE does not need to be negotiated.  Properly-chosen
   2432-bit DH groups should be roughly equivalent to 112-bit security.
   And future implementations should use sizes of at least 3072 bits
   according to [ENISA].

5.3.  Arbitrary Groups

   This spec currently doesn't indicate any support for groups other
   than the named groups.  Other DHE specifications have moved away from
   staticly-named groups with the explicitly-stated rationale of
   reducing the incentive for precomputation-driven attacks on any
   specific group (e.g. section 1 of [RFC4419]).  However, arbitrary
   large groups are expensive to transmit over the network and it is
   computationally infeasible for the client to verify their structure
   during a key exchange.  If we instead allow the server to propose

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   arbitrary groups, we could make it a MUST that the generated groups
   use safe prime moduli, while still allowing clients to signal support
   (and desire) for large groups.  This leaves the client in the
   position of relying on the server to choose a strong modulus, though.

   Note that in several known attacks against TLS and SSL
   [SECURE-RESUMPTION] [CROSS-PROTOCOL] [SSL3-ANALYSIS], a malicious
   server uses a deliberately broken finite field DHE group to
   impersonate the client to a different server.

6.  Acknowledgements

   Thanks to Fedor Brunner, Dave Fergemann, Sandy Harris, Watson Ladd,
   Nikos Mavrogiannopolous, Niels Moeller, Kenny Paterson, and Tom
   Ritter for their comments and suggestions on this draft.  Any
   mistakes here are not theirs.

7.  IANA Considerations

   This document defines a new TLS extension, "negotiated_dh_group",
   assigned a value of XXX from the TLS ExtensionType registry defined
   in section 12 of [RFC5246].  This value is used as the extension
   number for the extensions in both the client hello message and the
   server hello message.

   Appendix A defines a TLS Finite Field DHE Named Group Registry.  Each
   entry in this registry 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.  This registry may be updated
   by the addition of new finite field groups, and by reassessments of
   the security level or utility to TLS of any already present group.
   Updates are made by IETF Review [RFC5226], and should consider
   Section 9.1.

8.  Security Considerations

8.1.  Negotiation resistance to active attacks

   Because the contents of this extension is hashed in the finished
   message, an active MITM that tries to filter or omit groups will
   cause the handshake to fail, but possibly not before getting the peer
   to do something they would not otherwise have done.

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

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      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.

      Pretend that a compatible server is actually non-compatible by
      negotiating a non-DHE ciphersuite.  This is no different than MITM
      ciphersuite filtering.

      Pretend that a compatible server is actually non-compatible by
      negotiating a DHE ciphersuite and no extension, with an explicit
      (perhaps weak) group chosen by the server.  [XXX what are the
      worst consequences in this case?  What might the client leak
      before it notices that the handshake fails?  XXX]

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

      Pretend that a compatible client is not compliant (e.g. by not
      offering this extension).  This could cause the server to
      negotiate a weaker DHE group during the handshake, but it would
      fail to complete during the final check of the Finished message.

      Pretend that a non-compatible client is compatible.  This could
      cause the server to send what appears to be an extremely odd
      ServerDHParams (see Section 3.1), and the check in the Finished
      message would fail.  It is not clear how this could be an attack.

      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.  DHE only

   Note that this extension specifically targets only finite field-based
   Diffie-Hellman ephemeral key exchange mechanisms.  It does not cover
   the non-ephemeral DH key exchange mechanisms, nor does it cover
   elliptic curve-based DHE key exchange, which has its own list of
   named groups.

8.3.  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 therefore compromise the confidentiality
   and integrity of the TLS session).

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   Revisions of this extension or updates 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.

8.4.  Choice of groups

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

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

      The IKE groups include weak groups like MODP768 which are
      unacceptable for secure TLS traffic.

      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.

      Other lists of named FF DHE groups are not collected in a single
      IANA registry, or are mixed with non-FF DHE groups, which makes
      them inconvenient for re-use in a TLS DHE key exchange context.

8.5.  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 re-use 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).

8.6.  Replay attacks from non-negotiated FF DHE

   [SECURE-RESUMPTION] shows a malicious peer using a bad FF DHE group
   to maneuver a client into selecting a pre-master secret of the peer's
   choice, which can be replayed to another server using a non-DHE key
   exchange, and can then be bootstrapped to replay client
   authentication.

   To prevent this attack (barring the fixes proposed in
   [SESSION-HASH]), a client would need not only to implement this

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   draft, but also to reject non-negotiated FF DHE ciphersuites 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 FF DHE ciphersuites on the subsequent
   connection.

   This tradeoff may be too costly for most TLS clients today, but may
   be a reasonable choice for clients performing client certificate
   authentication, or who have other reason to be concerned about
   server-controlled pre-master secrets.

9.  Privacy Considerations

9.1.  Client fingerprinting

   This extension provides a few additional bits of information to
   distinguish between classes of TLS clients (see e.g.
   [PANOPTICLICK]).  To minimize this sort of fingerprinting, clients
   SHOULD support all named groups at or above their minimum security
   threshhold.  New named groups SHOULD NOT be added to the 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, March 1997.

10.2.  Informative References

   [CROSS-PROTOCOL]
              Mavrogiannopolous, N., Vercauteren, F., Velichkov, V., and
              B. Preneel, "A Cross-Protocol Attack on the TLS Protocol",
              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)",
              September 2012,
              <http://www.ecrypt.eu.org/documents/D.SPA.20.pdf>.

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   [ENISA]    European Union Agency for Network and Information Security
              Agency, "Algorithms, Key Sizes and Parameters Report,
              version 1.0", October 2013,
              <http://www.enisa.europa.eu/activities/identity-and-
              trust/library/deliverables/
              algorithms-key-sizes-and-parameters-report>.

   [PANOPTICLICK]
              Electronic Frontier Foundation, "Panopticlick: How Unique
              - and Trackable - Is Your Browser?", 2010,
              <https://panopticlick.eff.org/>.

   [RFC3526]  Kivinen, T. and M. Kojo, "More Modular Exponential (MODP)
              Diffie-Hellman groups for Internet Key Exchange (IKE)",
              RFC 3526, May 2003.

   [RFC4419]  Friedl, M., Provos, N., and W. Simpson, "Diffie-Hellman
              Group Exchange for the Secure Shell (SSH) Transport Layer
              Protocol", RFC 4419, March 2006.

   [RFC5226]  Narten, T. and H. Alvestrand, "Guidelines for Writing an
              IANA Considerations Section in RFCs", BCP 26, RFC 5226,
              May 2008.

   [RFC5246]  Dierks, T. and E. Rescorla, "The Transport Layer Security
              (TLS) Protocol Version 1.2", RFC 5246, August 2008.

   [SECURE-RESUMPTION]
              Delignat-Lavaud, A., Bhargavan, K., and A. Pironti,
              "Triple Handshakes Considered Harmful: Breaking and Fixing
              Authentication over TLS", March 2014, <https://secure-
              resumption.com/>.

   [SESSION-HASH]
              Bhargavan, K., Delignat-Lavaud, A., Pironti, A., Langley,
              A., and M. Ray, "Triple Handshakes Considered Harmful:
              Breaking and Fixing Authentication over TLS", March 2014,
              <https://secure-resumption.com/draft-bhargavan-tls-
              session-hash-00.txt>.

   [SSL3-ANALYSIS]
              Schneier, B. and D. Wagner, "Analysis of the SSL 3.0
              protocol", 1996, <https://www.schneier.com/paper-ssl.pdf>.

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

Appendix A.  Named Group Registry

   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 which initially populate this registry
   are derived for a given bitlength 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 to this registry may use this same derivation (e.g.
   with different bitlengths) or may choose their parameters in a
   different way, but must be clear about how the parameters were
   derived.

A.1.  ffdhe2432

   The 2432-bit group has registry value 0, and is calcluated from the
   following formula:

   The modulus is: p = 2^2432 - 2^2368 + {[2^2302 * e] + 2111044} * 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 8533C8B3 FFFFFFFF FFFFFFFF

   The generator is: g = 2

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   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 C299E459 FFFFFFFF FFFFFFFF

   The estimated symmetric-equivalent strength of this group is 112
   bits.

   Peers using ffdhe2432 that want to optimize their key exchange with a
   short exponent (Section 4.2) should choose a secret key of at least
   224 bits.

A.2.  ffdhe3072

   The 3072-bit prime has registry value 1, and is calcluated from the
   following formula:

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

   The hexadecimal representation of p is:

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    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 4.2) should choose a secret key of at least
   250 bits.

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A.3.  ffdhe4096

   The 4096-bit group has registry value 2, and is calcluated 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

   The hexadecimal representation of q is:

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    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 4.2) should choose a secret key of at least
   300 bits.

A.4.  ffdhe6144

   The 6144-bit group has registry value 3, and is calcluated from the
   following formula:

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

   The hexadecimal representation of p is:

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    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

   The hexadecimal representation of q is:

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    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 4.2) should choose a secret key of at least
   350 bits.

A.5.  ffdhe8192

   The 8192-bit group has registry value 4, and is calcluated from the
   following formula:

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

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   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

Gillmor                 Expires February 28, 2015              [Page 20]
Internet-Draft          Negotiated-FF-DHE-for-TLS            August 2014

   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

Gillmor                 Expires February 28, 2015              [Page 21]
Internet-Draft          Negotiated-FF-DHE-for-TLS            August 2014

   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 4.2) should choose a secret key of at least
   384 bits.

Author's Address

   Daniel Kahn Gillmor
   ACLU
   125 Broad Street, 18th Floor
   New York, NY  10004
   USA

   Email: dkg@fifthhorseman.net

Gillmor                 Expires February 28, 2015              [Page 22]