Internet DRAFT - draft-ietf-tls-negotiated-dl-dhe
draft-ietf-tls-negotiated-dl-dhe
Internet Engineering Task Force D. Gillmor
Internet-Draft ACLU
Intended status: Informational July 21, 2014
Expires: January 22, 2015
Negotiated Discrete Log Diffie-Hellman Ephemeral Parameters for TLS
draft-ietf-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
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This Internet-Draft will expire on January 22, 2015.
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
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include Simplified BSD License text as described in Section 4.e of
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described in the Simplified BSD License.
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 . . . . . . . . . . . . . . . . . . . . . . . . 6
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 DL 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 . . . . . . . . . . . . . . . . 12
A.1. dldhe2432 . . . . . . . . . . . . . . . . . . . . . . . . 13
A.2. dldhe3072 . . . . . . . . . . . . . . . . . . . . . . . . 13
A.3. dldhe4096 . . . . . . . . . . . . . . . . . . . . . . . . 14
A.4. dldhe6144 . . . . . . . . . . . . . . . . . . . . . . . . 15
A.5. dldhe8192 . . . . . . . . . . . . . . . . . . . . . . . . 17
Author's Address . . . . . . . . . . . . . . . . . . . . . . . . 19
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 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].
1.2. Vocabulary
The term "DHE" is used in this document to refer to the discrete-
logarithm-based Diffie-Hellman ephemeral key exchange mechanism in
TLS. TLS also supports elliptic-curve-based Diffie Hellman ephemeral
key exchanges, but this document does not discuss their use.
Mentions of DHE here refer strictly to discrete-log-based DHE, and
not to ECDHE.
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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 a list of
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 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 that they 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.
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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.
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
dldhe2432 and dldhe4096, the server SHOULD select dldhe4096.
3.1. ServerDHParams changes
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 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.
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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 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 Y against each factor.
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,
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 dldhe2432
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 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 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 Discrete Log 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 discrete log 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 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.
8.3. 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 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 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.
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 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 a TLS DHE key exchange context.
8.5. 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
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 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
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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. 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
[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>.
[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.
[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>.
[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 discrete log 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
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different way, but must be clear about how the parameters were
derived.
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 + {[2^2302 * e] + 2111044} * 2^64
- 1
Its hexadecimal representation 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
The group size is (p-1)/2
The estimated symmetric-equivalent strength of this group is 112
bits.
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 + {[2^2942 * e] + 2625351} * 2^64 -1
Its hexadecimal representation 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: (p-1)/2
The estimated symmetric-equivalent strength of this group is 125
bits.
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
250 bits.
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 + {[2^3966 * e] + 5736041} * 2^64
- 1
Its hexadecimal representation 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 5E655F6A
FFFFFFFF FFFFFFFF
The base is: g = 2
The group size is: (p-1)/2
The estimated symmetric-equivalent strength of this group is 150
bits.
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
300 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 + {[2^6014 * e] + 15705020} *
2^64 - 1
Its hexadecimal representation 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: 2
The group size is: (p-1)/2
The estimated symmetric-equivalent strength of this group is 175
bits.
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
350 bits.
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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 + {[2^8062 * e] + 10965728} *
2^64 - 1
Its hexadecimal representation 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 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 base is: g = 2
The group size is: (p-1)/2
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The estimated symmetric-equivalent strength of this group is 192
bits.
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
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 January 22, 2015 [Page 19]
