E. Rescorla
INTERNET-DRAFT RTFM Inc.
March 1999 (Expires September 1999)
Diffie-Hellman Key Agreement Method
Status of this Memo
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Abstract
This document standardizes one particular Diffie-Hellman variant,
based on the ANSI X9.42 draft, developed by the ANSI X9F1 working
group. Diffie-Hellman is a key agreement algorithm used by two par-
ties to agree on a shared secret. An algorithm for converting the
shared secret into an arbitrary amount of keying material is pro-
vided. The resulting keying material is used as a symmetric encryp-
tion key. The D-H variant described requires the recipient to have a
certificate, but the originator may have a static key pair (with the
public key placed in a certificate) or an ephemeral key pair.
1. Introduction
In [DH76] Diffie and Hellman describe a means for two parties to
agree upon a shared secret in such a way that the secret will be una-
vailable to eavesdroppers. This secret may then be converted into
cryptographic keying material for other (symmetric) algorithms. A
large number of minor variants of this process exist. This document
describes one such variant, based on the ANSI X9.42 specification.
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1.1. Discussion of this Draft
This draft is being discussed on the "ietf-smime" mailing list. To
join the list, send a message to with
the single word "subscribe" in the body of the message. Also, there
is a Web site for the mailing list at .
1.2. Requirements Terminology
Keywords "MUST", "MUST NOT", "REQUIRED", "SHOULD", "SHOULD NOT" and
"MAY" that appear in this document are to be interpreted as described
in [RFC2119].
2. Overview Of Method
Diffie-Hellman key agreement requires that both the sender and reci-
pient of a message have key pairs. By combining one's private key and
the other party's public key, both parties can compute the same
shared secret number. This number can then be converted into crypto-
graphic keying material. That keying material is typically used as a
key-encryption key (KEK) to encrypt (wrap) a content-encryption key
(CEK) which is in turn used to encrypt the message data.
2.1. Key Agreement
The first stage of the key agreement process is to compute a shared
secret number, called ZZ. When the same originator and recipient
public/private key pairs are used, the same ZZ value will result.
The ZZ value is then converted into a shared symmetric cryptographic
key. When the originator employs a static private/public key pair,
the introduction of public random values are used to ensure that the
resulting symmetric key will be different for each key agreement.
2.1.1. Generation of ZZ
X9.42 defines that the shared secret ZZ is generated as follows:
ZZ = g ^ (xb * xa) mod p
Note that the individual parties actually perform the computations:
ZZ = yb ^ xa (mod p) = ya ^ xb mod p
where ^ denotes exponentiation
ya is party a's public key; ya = g ^ xa mod p
yb is party b's public key; yb = g ^ xb mod p
xa is party a's private key
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xb is party b's private key
p is a large prime
q is a large prime
g = h^{(p-1)/q} mod p, where
h is any integer with 1 < h < p-1 such that h{(p-1)/q} mod p > 1
(g has order q mod p; i.e. g^q mod p = 1 if g!=1)
j a large integer such that p=qj + 1
(See Section 2.2 for criteria for keys and parameters)
In [CMS], the recipient's key is identified by the CMS RecipientIden-
tifier, which points to the recipient's certificate. The sender's
key is identified using the OriginatorIdentifierOrKey field, either
by reference to the sender's certificate or by inline inclusion of a
key.
2.1.2. Generation of Keying Material
X9.42 provides an algorithm for generating an essentially arbitrary
amount of keying material from ZZ. Our algorithm is derived from that
algorithm by mandating some optional fields and omitting others.
KM = H ( ZZ || OtherInfo)
H is the message digest function SHA-1 [FIPS-180]
ZZ is the shared secret value computed in Section 2.1.1. Leading zeros MUST
be preserved, so that ZZ occupies as many octets as p. For
instance, if p is 1024 bits, ZZ should be 128 bytes long.
OtherInfo is the DER encoding of the following structure:
OtherInfo ::= SEQUENCE {
keyInfo KeySpecificInfo,
partyAInfo [0] OCTET STRING OPTIONAL,
suppPubInfo [2] OCTET STRING
}
KeySpecificInfo ::= SEQUENCE {
algorithm OBJECT IDENTIFIER,
counter OCTET STRING SIZE (4..4) }
algorithm is the ASN.1 algorithm OID of the MEK wrapping algorithm with which
this KEK will be used. Note that this is NOT an AlgorithmIdentifier,
but simply the OBJECT IDENTIFIER. No parameters are used.
counter is a 32 bit number, represented in network byte order. Its
initial value is 1 for any ZZ, i.e. the byte sequence 00 00 00 01
(hex), and it is incremented by one every time the above key generation
function is run for a given KEK.
partyAInfo is a random string provided by the sender. In CMS, it is provided as
a parameter in the UserKeyingMaterial field (encoded as an OCTET STRING). If
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provided, partyAInfo MUST contain 512 bits.
suppPubInfo is the length of the generated KEK, in bits, represented
as a 32 bit number in network byte order. E.g. for 3DES it
would be the byte sequence 00 00 00 C0.
Note that these ASN.1 definitions use EXPLICIT tagging. (In ASN.1,
EXPLICIT tagging is implicit unless IMPLICIT is explicitly speci-
fied.)
To generate a KEK, one generates one or more KM blocks (incrementing
counter appropriately) until enough material has been generated. The
KM blocks are concatenated left to right in the obvious order. I.e.
KM(counter=1) || KM(counter=2)...
Note that the only source of secret entropy in this computation is
ZZ. Even if a string longer than ZZ is generated, the effective key
space of the KEK is limited by the size of ZZ, in addition to any
security level considerations imposed by the parameters p and
q.However, if partyAInfo is different for each message, a different
KEK will be generated for each message. Note that partyAInfo MUST be
used in Static-Static mode, but MAY appear in Ephemeral-Static mode.
2.1.3. KEK Computation
Each key encryption algorithm requires a specific size key (n). The
KEK is generated by mapping the left n-most bytes of KM onto the key.
For 3DES, which requires 192 bits of keying material, the algorithm
must be run twice, once with a counter value of 1 (to generate K1',
K2', and the first 32 bits of K3') and once with a counter value of 2
(to generate the last 32 bits of K3). K1',K2' and K3' are then parity
adjusted to generate the 3 DES keys K1,K2 and K3. For RC2-128, which
requires 128 bits of keying material, the algorithm is run once, with
a counter value of 1, and the left-most 128 bits are directly con-
verted to an RC2 key. Similarly, for RC2-40, which requires 40 bits
of keying material, the algorithm is run once, with a counter value
of 1, and the leftmost 40 bits are used as the key.
2.1.4. Keylengths for common algorithms
Some common key encryption algorithms have KEKs of the following
lengths.
3DES-EDE-ECB 192 bits
RC2-128 128 bits
RC2-40 40 bits
RC2 effective key lengths are equal to RC2 real key lengths.
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2.1.5. Public Key Validation
The following algorithm MAY be used to validate a received public key
y.
1. Verify that y lies within the interval [2,p-1]. If it does not,
the key is invalid.
2. Compute y^q mod p. If the result == 1, the key is valid.
Otherwise the key is invalid.
The primary purpose of public key validation is to prevent a small
subgroup attack [LAW98] on the sender's key pair. If Ephemeral-Static
mode is used, this check may not be necessary. See also [P1363] for
more information on Public Key validation.
Note that this procedure may be subject to pending patents.
2.1.6. Example 1
ZZ is the 20 bytes 00 01 02 03 04 05 06 07 08 09
0a 0b 0c 0d 0e 0f 10 11 12 13
The key wrap algorithm is 3DES-EDE wrap.
No partyAInfo is used
Consequently, the input to the first invocation of SHA-1 is:
00 01 02 03 04 05 06 07 08 09 0a 0b 0c 0d 0e 0f 10 11 12 13 ; ZZ
30 1d
30 13
06 0b 2a 86 48 86 f7 0d 01 09 10 03 06 ; 3DES wrap OID
04 04
00 00 00 01 ; Counter
a2 06
04 04
00 00 00 c0 ; key length
And the output is the 20 bytes:
a0 96 61 39 23 76 f7 04 4d 90 52 a3 97 88 32 46 b6 7f 5f 1e
The input to the second invocation of SHA-1 is:
00 01 02 03 04 05 06 07 08 09 0a 0b 0c 0d 0e 0f 10 11 12 13 ; ZZ
30 1d
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30 13
06 0b 2a 86 48 86 f7 0d 01 09 10 03 06 ; 3DES wrap OID
04 04
00 00 00 02 ; Counter
a2 06
04 04
00 00 00 c0 ; key length
And the output is the 20 bytes:
f6 3e b5 fb 5f 56 d9 b6 a8 34 03 91 c2 d3 45 34 93 2e 11 30
Consequently,
K1'=a0 96 61 39 23 76 f7 04
K2'=4d 90 52 a3 97 88 32 46
K3'=b6 7f 5f 1e f6 3e b5 fb
Note: These keys are not parity adjusted
2.1.7. Example 2
ZZ is the 20 bytes 00 01 02 03 04 05 06 07 08 09
0a 0b 0c 0d 0e 0f 10 11 12 13
The key wrap algorithm is RC2-128 key wrap, so we need 128 bits (16
bytes) of keying material.
The partyAInfo used is the 64 bytes
01 23 45 67 89 ab cd ef fe dc ba 98 76 54 32 01
01 23 45 67 89 ab cd ef fe dc ba 98 76 54 32 01
01 23 45 67 89 ab cd ef fe dc ba 98 76 54 32 01
01 23 45 67 89 ab cd ef fe dc ba 98 76 54 32 01
Consequently, the input to SHA-1 is:
00 01 02 03 04 05 06 07 08 09 0a 0b 0c 0d 0e 0f 10 11 12 13 ; ZZ
30 61
30 13
06 0b 2a 86 48 86 f7 0d 01 09 10 03 07 ; RC2 wrap OID
04 04
00 00 00 01 ; Counter
a0 42
04 40
01 23 45 67 89 ab cd ef fe dc ba 98 76 54 32 01 ; partyAInfo
01 23 45 67 89 ab cd ef fe dc ba 98 76 54 32 01
01 23 45 67 89 ab cd ef fe dc ba 98 76 54 32 01
01 23 45 67 89 ab cd ef fe dc ba 98 76 54 32 01
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a2 06
04 04
00 00 00 80 ; key length
And the output is the 20 bytes:
48 95 0c 46 e0 53 00 75 40 3c ce 72 88 96 04 e0 3e 7b 5d e9
Consequently,
K=48 95 0c 46 e0 53 00 75 40 3c ce 72 88 96 04 e0
2.2. Key and Parameter Requirements
X9.42 requires that the group parameters be of the form p=jq + 1
where q is a large prime of length m and j>=2. An algorithm for gen-
erating primes of this form (derived from the algorithms in FIPS PUB
186-1[FIPS-186] and [X942]can be found in appendix A.
X9.42 requires that the private key x be in the interval [2, (q -
2)]. x should be randomly generated in this interval. y is then com-
puted by calculating g^x mod p. To comply with this draft, m MUST be
>=160 bits in length, (consequently, q MUST be at least 160 bits
long). When symmetric ciphers stronger than DES are to be used, a
larger m may be advisable. p must be a minimum of 512 bits long.
2.2.1. Group Parameter Generation
Agents SHOULD generate domain parameters (g,p,q) using the following
algorithm, derived from [FIPS-186] and [X942]. When this algorithm is
used, the correctness of the generation procedure can be verified by
a third party by the algorithm of 2.2.2.
2.2.1.1. Generation of p, q
This algorithm generates a p, q pair where q is of length m and
p is of length L.
Let L - 1 = n*m + b where both b and n are integers and
0 <= b < 160.
1. Choose an arbitrary sequence of at least m bits and call it SEED.
Let g be the length of SEED in bits.
2. Set U = 0
3. Set m' = m / 160, where / represents integer division,
consequently, if m = 200, m' = 1.
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4. for i = 0 to m' - 1
U = U + SHA[SEED + i] XOR SHA[(SEED + m' + i) mod 2^160] * 2^(160 * i)
Note that for m=160, this reduces to the algorithm of [FIPS-186]
U = SHA[SEED] XOR SHA[(SEED+1) mod 2^160 ].
5. Form q from U by setting the most significant bit (the 2^(m-1) bit)
and the least significant bit to 1. In terms of boolean operations,
q = U OR 2^(m-1) OR 1. Note that 2^(m-1) < q < 2^m
6. Use a robust primality algorithm to test whether q is prime.
7. If q is not prime then go to 1.
8. Let counter = 0 and offset = 2
9. For k = 0 to n let
Vk = SHA[(SEED + offset + k) mod 2^160 ].
10. Let W be the integer
W = V0 + V1*2^160 + ... + Vn-1*2(n-1)*160 + (Vn mod 2^b)
* 2n*160
and let
X = W + 2^(L-1)
Note that 0 <= W < 2^(L-1) and hence 2^(L-1)
11. Let c = X mod (2 * q) and set p = X - (c-1). Note that p is congruent
to 1 mod (2 * q).
12. If p < 2^(L -1) then go to step 15.
13. Perform a robust primality test on p.
14. If p passes the test performed in step 13 go to step 17.
15. Let counter = counter + 1 and offset = offset + n + 1.
16. If counter >= 4096 go to step 1. Otherwise go to step 9.
17. Save the value of SEED and the value of counter for use
in certifying the proper generation of p and q.
Note: A robust primality test is one where the probability of
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a non-prime number passing the test is at most 2^-80. [FIPS-186]
provides a suitable algorithm, as does [X9.42].
2.2.1.2. Generation of g
This section gives an algorithm (derived from [FIPS-186]) for gen-
erating g.
1. Let j = (p - 1)/q.
2. Set h = any integer, where 1 < h < p - 1 and h differs
from any value previously tried.
3. Set g = h^j mod p
4. If g = 1 go to step 2
2.2.2. Group Parameter Validation
The ASN.1 for DH keys in [PKIX] includes elements j and validation-
Parms which MAY be used by recipients of a key to verify that the
group parameters were correctly generated. Two checks are possible:
1. Verify that p=qj + 1. This demonstrates that the parameters meet
the X9.42 parameter criteria.
2. Verify that when the p,q generation procedure of [FIPS-186]
Appendix 2 is followed with seed 'seed', that p is found when
This demonstrates that the parameters were randomly chosen and
do not have a special form.
Whether agents provide validation information in their certificates
is a local matter between the agents and their CA.
2.3. Ephemeral-Static Mode
In Ephemeral-Static mode, the recipient has a static (and certified)
key pair, but the sender generates a new key pair for each message
and sends it using the originatorKey production. If the sender's key
is freshly generated for each message, the shared secret ZZ will be
similarly different for each message and partyAInfo MAY be omitted,
since it serves merely to decouple multiple KEKs generated by the
same set of pairwise keys. If, however, the same ephemeral sender key
is used for multiple messages (e.g. it is cached as a performance
optimization) then a separate partyAInfo MUST be used for each mes-
sage. All implementations of this standard MUST implement Ephemeral-
Static mode.
In order to resist small subgroup attacks, the recipient SHOULD
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perform the check described in 2.1.5. If an opponent cannot determine
success or failure of a decryption operation by the recipient, the
recipient MAY choose to omit this check. See also [LL97] for a method
of generating keys which are not subject to small subgroup attack.
2.4. Static-Static Mode
In Static-Static mode, both the sender and the recipient have a
static (and certified) key pair. Since the sender's and recipient's
keys are therefore the same for each message, ZZ will be the same for
each message. Thus, partyAInfo MUST be used (and different for each
message) in order to ensure that different messages use different
KEKs. Implementations MAY implement Static-Static mode.
In order to prevent small subgroup attacks, both originator and reci-
pient SHOULD either perform the validation step described in Section
2.1.5 or verify that the CA has properly verified the validity of the
key. See also [LL97] for a method of generating keys which are not
subject to small subgroup attack.
Acknowledgements
The Key Agreement method described in this document is based on work
done by the ANSI X9F1 working group. The author wishes to extend his
thanks for their assistance.
The author also wishes to thank Burt Kaliski, Paul Hoffman, Stephen
Henson, Russ Housley, Brian Korver, John Linn, Jim Schaad, Mark
Schertler, Peter Yee, and Robert Zuccherato for their expert advice
and review.
References
[CMS] Housley, R., "Cryptographic Message Syntax", draft-ietf-smime-cms-07.txt.
[FIPS-46-1] Federal Information Processing Standards Publication (FIPS PUB)
46-1, Data Encryption Standard, Reaffirmed 1988 January 22
(supersedes FIPS PUB 46, 1977 January 15).
[FIPS-81] Federal Information Processing Standards Publication (FIPS PUB)
81, DES Modes of Operation, 1980 December 2.
[FIPS-180] Federal Information Processing Standards Publication (FIPS PUB)
180-1, "Secure Hash Standard", 1995 April 17.
[FIPS-186] Federal Information Processing Standards Publication (FIPS PUB)
186, "Digital Signature Standard", 1994 May 19.
[P1363] "Standard Specifications for Public Key Cryptography", IEEE P1363
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working group draft, 1998, Annex D.
[PKIX] Housley, R., Ford, W., Polk, W., Solo, D., "Internet X.509 Public
Key Infrastructure Certificate and CRL Profile", RFC-XXXX.
[LAW98] L. Law, A. Menezes, M. Qu, J. Solinas and S. Vanstone,
"An efficient protocol for authenticated key agreement",
Technical report CORR 98-05, University of Waterloo, 1998.
[LL97] C.H. Lim and P.J. Lee, "A key recovery attack on discrete log-based
schemes using a prime order subgroup", B.S. Kaliski, Jr., editor,
Advances in Cryptology - Crypto '97, Lecture Notes in Computer Science,
vol. 1295, 1997, Springer-Verlag, pp. 249-263.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate Requirement
Levels." RFC 2119. March 1997.
[X942] "Agreement Of Symmetric Keys Using Diffie-Hellman and MQV Algorithms",
ANSI draft, 1998.
Security Considerations
All the security in this system is provided by the secrecy of the
private keying material. If either sender or recipient private keys
are disclosed, all messages sent or received using that key are
compromised. Similarly, loss of the private key results in an inabil-
ity to read messages sent using that key.
Static Diffie-Hellman keys are vulnerable to a small subgroup attack
[LAW98]. In practice, this issue arises for both sides in Static-
Static mode and for the receiver during Ephemeral-Static mode. Sec-
tions 2.3 and 2.4 describe appropriate practices to protect against
this attack. Alternatively, it is possible to generate keys in such a
fashion that they are resistant to this attack. See [LL97]
The security level provided by these methods depends on several fac-
tors. It depends on the length of the symmetric key (typically, a 2^l
security level if the length is l bits); the size of the prime q (a
2^{m/2} security level); and the size of the prime p (where the secu-
rity level grows as a subexponential function of the size in bits).
A good design principle is to have a balanced system, where all three
security levels are approximately the same. If many keys are derived
from a given pair of primes p and q, it may be prudent to have higher
levels for the primes. In any case, the overall security is limited
by the lowest of the three levels.
Author's Address
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Eric Rescorla
RTFM Inc.
30 Newell Road, #16
East Palo Alto, CA 94303
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Table of Contents
1. Introduction ................................................... 1
1.1. Discussion of this Draft ..................................... 2
1.2. Requirements Terminology ..................................... 2
2. Overview Of Method ............................................. 2
2.1. Key Agreement ................................................ 2
2.1.1. Generation of ZZ ........................................... 2
2.1.2. Generation of Keying Material .............................. 3
2.1.3. KEK Computation ............................................ 4
2.1.4. Keylengths for common algorithms ........................... 4
2.1.5. Public Key Validation ...................................... 5
2.1.6. Example 1 .................................................. 5
2.1.7. Example 2 .................................................. 6
2.2. Key and Parameter Requirements ............................... 7
2.2.1. Group Parameter Generation ................................. 7
2.2.1.1. Generation of p, q ....................................... 7
2.2.1.2. Generation of g .......................................... 9
2.2.2. Group Parameter Validation ................................. 9
2.3. Ephemeral-Static Mode ........................................ 9
2.4. Static-Static Mode ........................................... 10
2.4. Acknowledgements ............................................. 10
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2.4. References ................................................... 10
Security Considerations ........................................... 11
Author's Address .................................................. 11