| Internet Engineering Task Force | D. Gillmor |
| Internet-Draft | ACLU |
| Updates: 4492, 5246, 4346, 2246 (if | December 19, 2014 |
| approved) | |
| Intended status: Informational | |
| Expires: June 22, 2015 |
Negotiated Finite Field Diffie-Hellman Ephemeral Parameters for TLS
draft-ietf-tls-negotiated-ff-dhe-05
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 using a section of the TLS "EC Named Curve Registry" to establish common finite-field DH parameters with known structure and a mechanism for peers to negotiate support for these groups.
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 working documents as Internet-Drafts. The list of current Internet-Drafts is at http://datatracker.ietf.org/drafts/current/.
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This Internet-Draft will expire on June 22, 2015.
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Traditional TLS [RFC5246] offers a Diffie-Hellman ephemeral (DHE) key exchange mode which provides Forward Secrecy for the connection. The client offers a ciphersuite in the ClientHello that includes DHE, and the server offers the client group parameters generator g and modulus p. If the client does not consider the group strong enough (e.g. if p is too small, or if p is not prime, or there are small subgroups), or if it is unable to process the group for other reasons, the client has no recourse but to terminate the connection.
Conversely, when a TLS server receives a suggestion for a DHE 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. For example, some widely-distributed TLS clients are not capable of DH groups where p > 1024 bits. Other TLS clients may by policy wish to use DHE only if the server can offer a stronger group (and are willing to use a non-PFS 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 and g has no efficient way to verify that the structure of a new group is reasonable for use.
This modification to TLS solves these problems by using a section of the "EC Named Curves" registry to select common DH groups with known structure; defining the use of the "elliptic_curves(10)" extension (described here as "Supported Groups" extension) for clients advertising support for DHE with these groups. This document also provides guidance for compliant peers to take advantage of the additional security, availability, and efficiency offered.
The use of this mechanism by one compliant peer when interacting with a non-compliant peer should have no detrimental effects.
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].
The terms "DHE" or "FFDHE" are used in this document to refer to the finite-field-based Diffie-Hellman ephemeral key exchange mechanism in TLS. TLS also supports elliptic-curve-based Diffie-Hellman (ECDHE) ephemeral key exchanges [RFC4492], but this document does not document their use. A registry previously used only by ECHDE-capable implementations is expanded in this document to cover FFDHE groups as well. "FFDHE ciphersuites" is used in this document to refer exclusively to ciphersuites with FFDHE key exchange mechanisms, but note that these suites are typically labeled with a TLS_DHE_ prefix.
We use previously-unallocated codepoints within the extension currently known as "elliptic_curves" (section 5.1.1. of [RFC4492]) to indicate known finite field groups. The extension's semantics are expanded from "Supported Elliptic Curves" to "Supported Groups". The semantics of the extension's data type (enum NamedCurve) is also expanded from "named curve" to "named group".
Codepoints in the NamedCurve registry with a high byte of 0x01 (that is, between 256 and 511 inclusive) are set aside for FFDHE groups, though only a small number of them are initially defined and we do not expect many other FFDHE groups to be added to this range. No codepoints outside of this range will be allocated to FFDHE groups. The new code points for the NamedCurve registry are:
enum {
// other already defined elliptic curves (see RFC 4492)
ffdhe2048(256), ffdhe3072(257), ffdhe4096(258),
ffdhe8192(259),
//
} NamedCurve;
These additions to the Named Curve registry are described in detail in Appendix A. They are all safe primes derived from the base of the natural logarithm ("e"), with the high and low 64 bits set to 1 for efficient Montgomery or Barrett reduction.
The use of the base of the natural logarithm here is as a "nothing-up-my-sleeve" number. The goal is to guarantee that the bits in the middle of the modulus are effectively random, while avoiding any suspicion that the primes have secretly been selected to be weak according to some secret criteria. [RFC3526] used pi for this value. See Section 9.5 for reasons that this draft does not reuse pi.
A TLS client that is capable of using strong finite field Diffie-Hellman groups can advertise its capabilities and its preferences for stronger key exchange by using this mechanism.
The compatible client that wants to be able to negotiate strong FFDHE SHOULD send a "Supported Groups" extension (identified by type elliptic_curves(10) in [RFC4492]) in the ClientHello, and include a list of known FFDHE groups in the extension data, ordered from most preferred to least preferred. If the client also supports and wants to offer ECDHE key exchange, it MUST use a single "Supported Groups" extension to include all supported groups (both ECDHE and FFDHE groups). The ordering SHOULD be based on client preference, but see Section 6.1 for more nuance.
A client that offers any of these values in the elliptic_curves extension SHOULD ALSO include at least one FFDHE ciphersuite in the Client Hello.
A client who offers a group MUST be able and willing to perform a DH key exchange using that group.
A client that offers one or more FFDHE groups in the "Supported Groups" extension and an FFDHE ciphersuite, and receives an FFDHE ciphersuite from the server SHOULD take the following steps upon receiving the ServerKeyExchange:
If a compatible TLS server receives a Supported Groups extension from a client that includes any FFDHE group (i.e. any codepoint between 256 and 511 inclusive, even if unknown to the server), and if none of the client-proposed FFDHE groups are known and acceptable to the server, then the server SHOULD NOT select an FFDHE ciphersuite. In this case, the server SHOULD select an acceptable non-FFDHE ciphersuite from the client's offered list. If the extension is present with FFDHE groups, none of the client's offered groups are acceptable by the server, and none of the client's proposed non-FFDHE ciphersuites are acceptable to the server, the server SHOULD end the connection with a fatal TLS alert of type insufficient_security(71).
If at least one FFDHE ciphersuite is present in the client ciphersuite list, and the Supported Groups extension is present in the ClientHello, but the extension does not include any FFDHE groups (i.e. no codepoints between 256 and 511 inclusive), then the server knows that the client is not compatible with this document. In this scenario, a server MAY choose to select a non-FFDHE ciphersuite, or MAY choose an FFDHE ciphersuite and offer an FFDHE group of its choice to the client as part of a traditional ServerKeyExchange.
A compatible TLS server that receives the Supported Groups extension with FFDHE codepoints in it, and which selects an FFDHE ciphersuite MUST select one of the client's offered groups. The server indicates the choice of group to the client by sending the group's parameters as usual in the ServerKeyExchange as described in section 7.4.3 of [RFC5246].
A TLS server MUST NOT select a named FFDHE group that was not offered by a compatible client.
A TLS server MUST NOT select an FFDHE ciphersuite if the client did not offer one, even if the client offered an FFDHE group in the Supported Groups extension.
If a non-anonymous FFDHE ciphersuite is chosen, and the TLS client has used this extension to offer an FFDHE group of comparable or greater strength than the server's public key, the server SHOULD select an FFDHE group at least as strong as the server's public key. For example, if the server has a 3072-bit RSA key, and the client offers only ffdhe2048 and ffdhe4096, the server SHOULD select ffdhe4096.
When a compatible server selects an FFDHE group from among a client's Supported Groups, and the client sends a ClientKeyExchange, the server MUST verify that 1 < dh_Yc < dh_p - 1. If it is out of range, the server MUST terminate the connection with fatal handshake_failure(40) alert.
In a key exchange with a successfully negotiated known FFDHE group, both peers know that the group in question uses a safe prime as a modulus, and that the group in use is of size p-1 or (p-1)/2. This allows at least three optimizations that can be used to improve performance.
Peers MUST validate each other's public key Y (dh_Ys offered by the server or dh_Yc offered by the client) by ensuring that 1 < Y < p-1. This simple check ensures that the remote peer is properly behaved and isn't forcing the local system into 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.
Traditional Finite Field Diffie-Hellman has each peer choose their secret exponent from the range [2,p-2]. Using exponentiation by squaring, this means each peer must do roughly 2*log_2(p) multiplications, twice (once for the generator and once for the peer's public key).
Peers concerned with performance may also prefer to choose their secret exponent from a smaller range, doing fewer multiplications, while retaining the same level of overall security. Each named group indicates its approximate security level, and provides a lower-bound on the range of secret exponents that should preserve it. For example, rather than doing 2*2*3072 multiplications for a ffdhe3072 handshake, each peer can choose to do 2*2*250 multiplications by choosing their secret exponent from the range [2^249,2^250] (that is, a m-bit integer where m is at least 224) and still keep the approximate 125-bit security level.
A similar short-exponent approach is suggested in SSH's Diffie-Hellman key exchange (See section 6.2 of [RFC4419]).
Peers wishing to further accelerate FFDHE key exchange can also pre-compute a table of powers of the generator of a known group. This is a memory vs. time tradeoff, and it only accelerates the first exponentiation of the ephemeral DH exchange (the fixed-base exponentiation). The variable-base exponentiation (using the peer's public exponent as a base) still needs to be calculated as normal.
The ordering of named groups in the Supported Groups extension may contain some ECDHE groups and some FFDHE groups. These SHOULD be ranked in the order preferred by the client.
However, the ClientHello also contains list of desired ciphersuites, also ranked in preference order. This presents the possibility of conflicted preferences. For example, if the ClientHello contains a CipherSuite with two choices in order <TLS_DHE_RSA_WITH_AES_128_CBC_SHA, TLS_ECDHE_RSA_WITH_AES_128_CBC_SHA> and the Supported Groups Extension contains two choices in order <secp256r1,ffdhe3072> then there is a clear contradiction. Clients SHOULD NOT present such a contradiction since it does not represent a sensible ordering. A server that encounters such an contradiction when selecting between an ECDHE or FFDHE key exchange mechanism while trying to respect client preferences SHOULD give priority to the Supported Groups extension (in the example case, it should select TLS_ECDHE_RSA_WITH_AES_128_CBC_SHA with secp256r1), but MAY resolve the contradiction any way it sees fit.
More subtly, clients MAY interleave preferences between ECDHE and FFDHE groups, for example if stronger groups are preferred regardless of cost, but weaker groups are acceptable, the Supported Groups extension could consist of: <ffdhe8192,secp384p1,ffdhe3072,secp256r1>. In this example, with the same CipherSuite offered as the previous example, a server configured to respect client preferences and with support for all listed groups SHOULD select TLS_DHE_RSA_WITH_AES_128_CBC_SHA with ffdhe8192. A server configured to respect client preferences and with support for only secp384p1 and ffdhe3072 SHOULD select TLS_ECDHE_RSA_WITH_AES_128_CBC_SHA with secp384p1.
Thanks to Fedor Brunner, Dave Fergemann, Sandy Harris, Watson Ladd, Nikos Mavrogiannopolous, Niels Möller, Bodo Moeller, Kenny Paterson, Eric Rescorla, Tom Ritter, Rene Struik, Martin Thomson, Sean Turner, and other members of the TLS Working Group for their comments and suggestions on this draft. Any mistakes here are not theirs.
IANA maintains the registry currently known as EC Named Curves (originally defined in [RFC4492] and updated by [RFC7027]) at https://www.iana.org/assignments/tls-parameters/tls-parameters.xhtml#tls-parameters-8.
This document expands the semantics of this registry slightly, to include groups based on finite fields in addition to groups based on elliptic curves. It should add a range designation to that registry, indicating that values from 256-511 (inclusive) are set aside for "Finite Field Diffie-Hellman groups", and that all other entries in the registry are "Elliptic curve groups".
This document allocates five codepoints in the registry, as follows:
| Value | Description | DTLS-OK | Reference |
|---|---|---|---|
| 256 | ffdhe2048 | Y | [this document] |
| 257 | ffdhe3072 | Y | [this document] |
| 258 | ffdhe4096 | Y | [this document] |
| 259 | ffdhe8192 | Y | [this document] |
Because the contents of the Supported Groups 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:
An attacker who impersonates the client can try to do the following:
TLS implementations using FFDHE key exchange should consider the strength of the group they negotiate. The strength of the selected group is one of the factors which defines the connection's resiliance against attacks on the session's confidentiality and integrity, since the session keys are derived from the DHE handshake.
While attacks on integrity must generally happen while the session is in progress, attacks against session confidentiality can happen significantly later, if the entire TLS session is stored for offline analysis. Therefore, FFDHE groups should be selected by clients and servers based on confidentiality guarantees they need. Sessions which need extremely long-term confidentiality should prefer stronger groups.
[ENISA] provides rough estimates of group resistance to attack, and recommends that forward-looking implementations ("future systems") should use FFDHE group sizes of at least 3072 bits. ffdhe3072 is intended for use in these implementations.
Note that this document 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 address elliptic curve DHE (ECDHE) key exchange, which is defined in [RFC4492].
Measured by computational cost to the TLS peers, ECDHE appears today to offer much a stronger key exchange than FFDHE.
Advances in hardware or in finite field cryptanalysis may cause some of the negotiated groups to not provide the desired security margins, as indicated by the estimated work factor of an adversary to discover the premaster secret (and may therefore compromise the confidentiality and integrity of the TLS session).
Revisions of this document should mark known-weak groups as explicitly deprecated for use in TLS, and should update the estimated work factor needed to break the group, if the cryptanalysis has changed. Implementations that require strong confidentiality and integrity guarantees should avoid using deprecated groups and should be updated when the estimated security margins are updated.
Other lists of named finite field Diffie-Hellman groups [STRONGSWAN-IKE] exist. This draft chooses to not reuse them for several reasons:
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).
[SECURE-RESUMPTION], [CROSS-PROTOCOL], and [SSL3-ANALYSIS] all show a malicious peer using a bad FFDHE 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-FFDHE 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 FFDHE 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 FFDHE 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.
One of the main reasons to prefer FFDHE ciphersuites is Forward Secrecy, the ability to resist decryption even if when the endpoint's long-term secret key (usually RSA) is revealed in the future.
This property depends on both sides of the connection discarding their ephemeral keys promptly. Implementations should wipe their FFDHE secret key material from memory as soon as it is no longer needed, and should never store it in persistent storage.
Forward secrecy also depends on the strength of the Diffie-Hellman group; using a very strong symmetric cipher like AES256 with a forward-secret ciphersuite, but generating the keys with a much weaker group like dhe2048 simply moves the adversary's cost from attacking the symmetric cipher to attacking the dh_Ys or dh_Yc ephemeral keyshares.
If the goal is to provide forward secrecy, attention should be paid to all parts of the ciphersuite selection process, both key exchange and symmetric cipher choice.
Clients capable of TLS False Start [FALSE-START] may receive a proposed FFDHE group from a server that is attacker-controlled. In particular, the attacker can modify the ClientHello to strip the proposed FFDHE groups, which may cause the server to offer a weaker FFDHE group than it should, and this will not be detected until receipt of the server's Finished message. This could cause the a client using the False Start protocol modification to send data encrypted under a weak key agreement.
Clients should have their own classification of FFDHE groups that are "cryptographically strong" in the same sense described in the description of symmetric ciphers in [FALSE-START], and MUST offer at least one of these in the initial handshake if they contemplate using the False Start protocol modification.
Compatible clients performing a full handshake MUST NOT use the False Start protocol modification if the server selects an FFDHE ciphersuite but sends a group that is not cryptographically strong from the client's perspective.
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.
| [FALSE-START] | Langley, A., Modadugu, N. and B. Moeller, "Transport Layer Security (TLS) False Start", Work in Progress, draft-bmoeller-tls-falsestart-01, November 2014. |
| [RFC2119] | Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, March 1997. |
| [RFC4492] | Blake-Wilson, S., Bolyard, N., Gupta, V., Hawk, C. and B. Moeller, "Elliptic Curve Cryptography (ECC) Cipher Suites for Transport Layer Security (TLS)", RFC 4492, May 2006. |
| [RFC5246] | Dierks, T. and E. Rescorla, "The Transport Layer Security (TLS) Protocol Version 1.2", RFC 5246, August 2008. |
Each description below indicates the group itself, its derivation, its expected strength (estimated roughly from guidelines in [ECRYPTII]), and whether it is recommended for use in TLS key exchange at the given security level. It is not recommended to add further finite field groups to the NamedCurves registry; any attempt to do so should consider Section 10.1.
The primes in these finite field groups are all safe primes, that is, a prime p is a safe prime when q = (p-1)/2 is also prime. Where e is the base of the natural logarithm, and square brackets denote the floor operation, the groups 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 of FFDHE groups 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.
New additions of FFDHE groups MUST use a safe prime as the modulus to enable the inexpensive peer verification described in Section 5.1.
The 2048-bit group has registry value 256, and is calcluated from the following formula:
The modulus is: p = 2^2048 - 2^1984 + {[2^1918 * e] + 560315 } * 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 61285C97 FFFFFFFF FFFFFFFF
The generator is: g = 2
The group size is: q = (p-1)/2
The hexadecimal representation of q is:
7FFFFFFF FFFFFFFF D6FC2A2C 515DA54D 57EE2B10 139E9E78 EC5CE2C1 E7169B4A D4F09B20 8A3219FD E649CEE7 124D9F7C BE97F1B1 B1863AEC 7B40D901 576230BD 69EF8F6A EAFEB2B0 9219FA8F AF833768 42B1B2AA 9EF68D79 DAAB89AF 3FABE49A CC278638 707345BB F15344ED 79F7F439 0EF8AC50 9B56F39A 98566527 A41D3CBD 5E0558C1 59927DB0 E88454A5 D96471FD DCB56D5B B06BFA34 0EA7A151 EF1CA6FA 572B76F3 B1B95D8C 8583D3E4 770536B8 4F017E70 E6FBF176 601A0266 941A17B0 C8B97F4E 74C2C1FF C7278919 777940C1 E1FF1D8D A637D6B9 9DDAFE5E 17611002 E2C778C1 BE8B41D9 6379A513 60D977FD 4435A11C 30942E4B FFFFFFFF FFFFFFFF
The estimated symmetric-equivalent strength of this group is 103 bits.
Peers using ffdhe2048 that want to optimize their key exchange with a short exponent [short_exponents] should choose a secret key of at least 206 bits.
The 3072-bit prime has registry value 257, and is calcluated from the following formula:
The modulus is: p = 2^3072 - 2^3008 + {[2^2942 * e] + 2625351} * 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 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 [short_exponents] should choose a secret key of at least 250 bits.
The 4096-bit group has registry value 258, 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:
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 [short_exponents] should choose a secret key of at least 300 bits.
The 8192-bit group has registry value 259, and is calcluated from the following formula:
The modulus is: p = 2^8192 - 2^8128 + {[2^8062 * e] + 10965728} * 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 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
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
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 [short_exponents] should choose a secret key of at least 384 bits.