E. Rescorla
INTERNET-DRAFT                                                RTFM Inc.
<draft-ietf-smime-x942-04.txt>         December 1998 (Expires June 1999)

                  Diffie-Hellman Key Agreement Method

Status of this Memo

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   This document standardizes one particular Diffie-Hellman variant,
   based on the ANSI X9.42 standard, 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 <ietf-smime-request@imc.org> with
   the single word "subscribe" in the body of the message.  Also, there
   is a Web site for the mailing list at <http://www.imc.org/ietf-

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
         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)
         q is a large prime
         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

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 key computed in Section 2.1.1
   OtherInfo is the DER encoding of the following structure:

           OtherInfo ::= SEQUENCE {
             keyInfo KeySpecificInfo,
             pubInfo [2] OCTET STRING OPTIONAL,

           KeySpecificInfo ::= SEQUENCE {
             algorithm OBJECT IDENTIFIER,
             counter OCTET STRING SIZE (4..4) }

   algorithm is the ASN.1 algorithm OID of the symmetric algorithm with which
     this KEK will be used. Note that this is NOT an AlgorithmIdentifier,
     but simply the OBJECT IDENTIFIER. No paramters 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.
   pubInfo 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
     provided this pubInfo MUST contain 512 bits.

   Note that the only source of secret entropy in this computation is

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   ZZ, so the security of this data is limited to the size of ZZ, even
   if more data than ZZ is generated. However, if pubInfo is different
   for each message, a different KEK will be generated for each message.
   Note that pubInfo 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, 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.

2.1.4.  Keylengths for common algorithms

   Some common key encryption algorithms have KEKs of the following

           3DES-EDE-ECB    192 bits
           RC2 (all)       128 bits

2.1.5.  Public Key Validation

   The following algorithm MAY be used to validate received public keys.

           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.

   Note that this procedure may be subject to pending patents.

2.1.6.  Example 1

   ZZ is the 16 bytes 00 01 02 03 04 05 06 07 08 09 0a 0b 0c 0d 0e 0f

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   The key encryption algorithm is 3DES-EDE.

   No pubInfo 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       ; ZZ
   30 12
      30 10
         06 08 2a 86 48 86 f7 0d 03 07                   ; 3DES-EDE OID
         04 04 00 00 00 01                               ; Counter

   And the output is the 20 bytes:

   20 be 23 e3 3b 72 ef 16 8e e3 ae 18 5a 00 93 b0 d6 49 56 22

   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       ; ZZ
   30 12
      30 10
         06 08 2a 86 48 86 f7 0d 03 07                   ; 3DES-EDE OID
         04 04 00 00 00 02                               ; Counter

   And the output is the 20 bytes:

   3b 4e fd d4 6e ff 6b 6d 35 a9 cd e3 e3 e7 05 39 e0 31 53 de

   K1'=20 be 23 e3 3b 72 ef 16
   K2'=8e e3 ae 18 5a 00 93 b0
   K3'=d6 49 56 22 3b 4e fd d4

   Note: These keys are not parity adjusted

2.1.7.  Example 2

   ZZ is the 16 bytes 00 01 02 03 04 05 06 07 08 09 0a 0b 0c 0d 0e 0f

   The key encryption algorithm is RC2

   The pubInfo used is the 64 bytes 01 23 45 67 89 ab cd ef 01 23 45 67
   89 ab cd ef 01 23 45 67 89 ab cd ef 01 23 45 67 89 ab cd ef 01 23 45
   67 89 ab cd ef 01 23 45 67 89 ab cd ef 01 23 45 67 89 ab cd ef 01 23
   45 67 89 ab cd ef

   Consequently, the input to SHA-1 is:

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   00 01 02 03 04 05 06 07 08 09 0a 0b 0c 0d 0e 0f       ; ZZ
   30 56
      30 10
         06 08 2a 86 48 86 f7 0d 03 02                   ; RC2 OID
         04 04 00 00 00 01                               ; Counter
      a2 42
         04 40
         01 23 45 67 89 ab cd ef 01 23 45 67 89 ab cd ef ; PubInfo
         01 23 45 67 89 ab cd ef 01 23 45 67 89 ab cd ef
         01 23 45 67 89 ab cd ef 01 23 45 67 89 ab cd ef
         01 23 45 67 89 ab cd ef 01 23 45 67 89 ab cd ef

   And the output is the 20 bytes:

   c7 4e c5 80 68 7d 70 aa 06 38 d3 e3 c7 2a da 5a 67 4e cf 06

   K=c7 4e c5 80 68 7d 70 aa 06 38 d3 e3 c7 2a da 5a

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[DSS], 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, (consequently, q and x MUST each be at least 128 bits long).
   When symmetric ciphers stronger than DES are to be used, a larger m
   may be advisable.

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.  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.

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   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.

   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 [FIPS186]

           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.

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   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
   a non-prime number passing the test is at most 2^-80.  Generation of g

   This section gives an algorithm (derived from [FIPS186]) for generat-
   ing 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 pubInfo MAY be omitted,
   since it serves merely to decouple multiple KEKs generated by the

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   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 pubInfo MUST be used for each message.
   All implementations of this standard MUST implement Ephemeral-Static

   In order to resist small subgroup attacks, the recipient SHOULD per-
   form the check described in 2.1.5. If the sender cannot determine
   success or failure of decryption, the recipient MAY choose to omit
   this check.

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, pubInfo MUST be used (and different for each mes-
   sage) 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 S 2.1.5
   or verify that the CA has properly verified the validity of the key.


   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 Paul Hoffman, Stephen Henson, Russ
   Housley, Brian Korver, Jim Schaad, Mark Schertler, and Peter Yee for
   their expert advice and review.

   [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)

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       186, "Digital Signature Standard", 1994 May 19.

   [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.

   [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.  S 2.3
   and 2.4 describe appropriate practices to protect against this

   In order to securely generate a symmetric key of length l, m (the
   length in bits of q, and hence x) should be at least 2l. m MUST
   always be >= 160. Consequently, for RC2, m should be >=256 and for
   3DES, >=336 (the effective length of 3DES keys is 168 bits).

Author's Address

Eric Rescorla <ekr@rtfm.com>
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 ......................................    4

2.1.6. Example 1 ..................................................    4

2.1.7. Example 2 ..................................................    5

2.2. Key and Parameter Requirements ...............................    6

2.2.1. Group Parameter Generation .................................    6 Generation of p, q .......................................    6 Generation of g ..........................................    8

2.2.2. Group Parameter Validation .................................    8

2.3. Ephemeral-Static Mode ........................................    8

2.4. Static-Static Mode ...........................................    9

2.4. Acknowledgements .............................................    9

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2.4. References ...................................................    9

Security Considerations ...........................................   10

Author's Address ..................................................   10