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Versions: 00 01 02 draft-ietf-tls-negotiated-dl-dhe

Internet Engineering Task Force                               D. Gillmor
Internet-Draft                                                      ACLU
Intended status: Informational                            March 28, 2014
Expires: September 29, 2014


  Negotiated Discrete Log Diffie-Hellman Ephemeral Parameters for TLS
                 draft-gillmor-tls-negotiated-dl-dhe-00

Abstract

   Traditional discrete logarithm-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

   This Internet-Draft is submitted in full conformance with the
   provisions of BCP 78 and BCP 79.

   Internet-Drafts are working documents of the Internet Engineering
   Task Force (IETF).  Note that other groups may also distribute
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   Internet-Drafts are draft documents valid for a maximum of six months
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   This Internet-Draft will expire on September 29, 2014.

Copyright Notice

   Copyright (c) 2014 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



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   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.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
     1.1.  Requirements Language . . . . . . . . . . . . . . . . . .   3
   2.  Client Behavior . . . . . . . . . . . . . . . . . . . . . . .   3
   3.  Server Behavior . . . . . . . . . . . . . . . . . . . . . . .   4
   4.  Optimizations . . . . . . . . . . . . . . . . . . . . . . . .   5
     4.1.  Checking the Peer's Public Key  . . . . . . . . . . . . .   5
     4.2.  Short Exponents . . . . . . . . . . . . . . . . . . . . .   5
     4.3.  Table Acceleration  . . . . . . . . . . . . . . . . . . .   6
   5.  Open Questions  . . . . . . . . . . . . . . . . . . . . . . .   6
     5.1.  Server Indication of support  . . . . . . . . . . . . . .   6
     5.2.  Normalizing Weak Groups . . . . . . . . . . . . . . . . .   7
     5.3.  Arbitrary Groups  . . . . . . . . . . . . . . . . . . . .   7
   6.  Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .   7
   7.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .   7
   8.  Security Considerations . . . . . . . . . . . . . . . . . . .   8
     8.1.  Negotiation resistance to active attacks  . . . . . . . .   8
     8.2.  DHE only  . . . . . . . . . . . . . . . . . . . . . . . .   8
     8.3.  Client fingerprinting . . . . . . . . . . . . . . . . . .   9
     8.4.  Deprecating weak groups . . . . . . . . . . . . . . . . .   9
     8.5.  Choice of groups  . . . . . . . . . . . . . . . . . . . .   9
     8.6.  Timing attacks  . . . . . . . . . . . . . . . . . . . . .  10
     8.7.  Replay attacks from non-negotiated DL DHE . . . . . . . .  10
   9.  References  . . . . . . . . . . . . . . . . . . . . . . . . .  10
     9.1.  Normative References  . . . . . . . . . . . . . . . . . .  10
     9.2.  Informative References  . . . . . . . . . . . . . . . . .  10
   Appendix A.  Named Group Registry . . . . . . . . . . . . . . . .  11
     A.1.  dldhe2432 . . . . . . . . . . . . . . . . . . . . . . . .  11
     A.2.  dldhe3072 . . . . . . . . . . . . . . . . . . . . . . . .  12
     A.3.  dldhe4096 . . . . . . . . . . . . . . . . . . . . . . . .  13
     A.4.  dldhe6144 . . . . . . . . . . . . . . . . . . . . . . . .  13
     A.5.  dldhe8192 . . . . . . . . . . . . . . . . . . . . . . . .  14
   Author's Address  . . . . . . . . . . . . . . . . . . . . . . . .  14

1.  Introduction











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   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
   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 DHE parameters
   with insufficient knowledge.

   Additionally, the DH parameters chosen by the server may have a known
   structure which renders them secure against 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].

2.  Client Behavior

   A TLS client that is capable of using strong discrete log 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_dl_dhe_groups" in the ClientHello, indicating an list of



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   known discrete log Diffie-Hellman groups, ordered from most preferred
   to least preferred.

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

         enum {
             dldhe2432(0), dldhe3072(1), dldhe4096(2),
             dldhe6144(3), dldhe8192(4), (255)
         } DiscreteLogDHEGroup;

         struct {
             DiscreteLogDHEGroup discrete_log_dhe_group_list<1..2^8-1>;
         } DiscreteLogDHEGroups;

   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 DiscreteLogDHEGroup registry are
   listed in Appendix A.  These are all safe primes, designed to be
   sparse, and with the high and low 64 bits set to 1 for efficient
   Montgomery or Barrett reduction.

   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
   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_dl_dhe_groups" extension.  The "extension_data" field of
   this extension on the server side should be a single one-byte value
   DiscreteLogDHEGroup.





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

   When the server sends the "negotiated_dl_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 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 discrete log DH group key exchange, 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 two 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 its factors, and
   test Y against each of its factors.

4.2.  Short Exponents

   Traditional Discrete Log 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,



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   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*2048 multiplications for a dldhe2048
   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 used 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).

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?



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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
   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 at least one known attack against TLS
   [SECURE-RESUMPTION], a malicious server uses a deliberately broken
   discrete log DHE group to impersonate the client to a different
   server.

6.  Acknowledgements

   Thanks to Tom Ritter and Nikos Mavrogiannopolous and Niels Moeller
   and Kenny Paterson for their comments and suggestions on the idea for
   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.

   This extension also defines a registry of TLS named Discrete Log DH
   groups, derived initially from some of the IKE DH groups [RFC3526],
   indicating the advised strength of each group and whether it is
   recommended for use in TLS.  These recommendations may be updated by
   future revisions.



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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 the following:

      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.  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 discrete log-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.



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8.3.  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.

8.4.  Deprecating weak groups

   Advances in hardware or in discrete log cryptanalysis may cause some
   of the negotiated groups to not provide the desired security margins,
   as indicated by number of years to protect the premaster secret (and
   therefore the confidentiality and integrity of the TLS session)
   against a powerful adversary.

   Revisions of this extension or updates should mark known-weak groups
   as explicitly deprecated, and implementations that require strong
   confidentiality and integrity guarantees should avoid using any
   deprecated groups.

8.5.  Choice of groups

   Other lists of named discrete log 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.

      The IKE groups do not have sparse moduli, which makes modular
      exponentiation less efficient.

      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 reuse keys; however, using
      different groups avoids these failure modes entirely.

      The DL DHE groups are not collected in a single IANA registry, or
      are mixed with non-DL DHE groups, which makes them inconvenient
      for re-use in TLS.





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8.6.  Timing attacks

   Any implementation of discrete log Diffie-Hellman key exchange should
   use constant-time modular-exponentiation implementations.  This is
   particularly true for those implementations that ever re-use DHE
   parameters (so-called "semi-static" ephemeral keying).

8.7.  Replay attacks from non-negotiated DL DHE

   [SECURE-RESUMPTION] shows a malicious peer using a bad DL 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
   draft, but also to reject non-negotiated DL 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 DL 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.  References

9.1.  Normative References

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119, March 1997.

9.2.  Informative References

   [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/>.




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   [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.

   [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>.

   [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

A.1.  dldhe2432

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

   The modulus is: p = 2^2432 - 2^2368 + 2332920 * 2^64 - 1

   Its hexadecimal representation is:

   FFFFFFFF FFFFFFFF 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000



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   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 002398F7 FFFFFFFF FFFFFFFF

   The generator is: g = 2

   The group size is (p-1)/2

   Peers using dldhe2432 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.  dldhe3072

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

   p = 2^3072 - 2^3008 + 425754 * 2^64 -1

   Its hexadecimal representation is:

   FFFFFFFF FFFFFFFF 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00067F19 FFFFFFFF FFFFFFFF

   The generator is: g = 2

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

   Peers using dldhe3072 that want to optimize their key exchange with a
   short exponent (Section 4.2) should choose a secret key of at least
   256 bits.



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A.3.  dldhe4096

   The 4096-bit group has registry value 2, and is calcluated from the
   following formula:

   The modulus is: p = 2^4096 - 2^4032 + 341664 * 2^64 - 1

   Its hexadecimal representation is:

   FFFFFFFF FFFFFFFF 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 00000000
   00000000 00000000 00000000 00000000 00000000 0005369F
   FFFFFFFF FFFFFFFF

   The base is: g = 2

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

   Peers using dldhe4096 that want to optimize their key exchange with a
   short exponent (Section 4.2) should choose a secret key of at least
   XXX bits.

A.4.  dldhe6144

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

   The modulus is: p = 2^6144 - 2^6080 + XXX * 2^64 - 1

   Its hexadecimal representation is:



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   XXX ...still calculating... XXX

   The generator is: 2

   Peers using dldhe6144 that want to optimize their key exchange with a
   short exponent (Section 4.2) should choose a secret key of at least
   XXX bits.

A.5.  dldhe8192

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

   The modulus is: p = 2^8192 - 2^8128 + XXX * 2^64 - 1

   Its hexadecimal representation is:

   XXX ...still calculating... XXX

   The base is: g = 2

   Peers using dldhe8192 that want to optimize their key exchange with a
   short exponent (Section 4.2) should choose a secret key of at least
   XXX bits.

Author's Address

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

   Email: dkg@fifthhorseman.net

















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