draft-ietf-smime-x942-02.txt   draft-ietf-smime-x942-03.txt 
E. Rescorla E. Rescorla
INTERNET-DRAFT RTFM Inc. INTERNET-DRAFT RTFM Inc.
<draft-ietf-smime-x942-02.txt> November 1998 (Expires May 1999) <draft-ietf-smime-x942-03.txt> November 1998 (Expires May 1999)
Diffie-Hellman Key Agreement Method Diffie-Hellman Key Agreement Method
Status of this Memo Status of this Memo
This document is an Internet-Draft. Internet-Drafts are working This document is an Internet-Draft. Internet-Drafts are working
documents of the Internet Engineering Task Force (IETF), its areas, documents of the Internet Engineering Task Force (IETF), its areas,
and its working groups. Note that other groups may also distribute and its working groups. Note that other groups may also distribute
working documents as Internet-Drafts. working documents as Internet-Drafts.
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ftp.isi.edu (US West Coast). ftp.isi.edu (US West Coast).
Abstract Abstract
This document standardizes one particular Diffie-Hellman variant, This document standardizes one particular Diffie-Hellman variant,
based on the ANSI X9.42 standard, developed by the ANSI X9F1 working based on the ANSI X9.42 standard, developed by the ANSI X9F1 working
group. An algorithm for converting the shared secret into an arbi- group. Diffie-Hellam is a key agreement algorithm used by two parties
trary amount of keying material is provided. to agree on a shared secret. An algorithm for converting the shared
secret into an arbitrary amount of keying material is provided. The
resulting keying material is used as a symmetric encryption 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 1. Introduction
In [DH76] Diffie and Hellman describe a means for two parties to 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- 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 vailable to eavesdroppers. This secret may then be converted into
cryptographic keying material for other (symmetric) algorithms. A cryptographic keying material for other (symmetric) algorithms. A
large number of minor variants of this process exist. This document large number of minor variants of this process exist. This document
describes one such variant, based on the ANSI X9.42 specification. describes one such variant, based on the ANSI X9.42 specification.
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1.1. Discussion of this Draft 1.1. Discussion of this Draft
This draft is being discussed on the "ietf-smime" mailing list. To 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 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 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- is a Web site for the mailing list at <http://www.imc.org/ietf-
smime/>. smime/>.
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1.2. Requirements Terminology 1.2. Requirements Terminology
Keywords "MUST", "MUST NOT", "REQUIRED", "SHOULD", "SHOULD NOT" and Keywords "MUST", "MUST NOT", "REQUIRED", "SHOULD", "SHOULD NOT" and
"MAY" that appear in this document are to be interpreted as described "MAY" that appear in this document are to be interpreted as described
in [RFC2119]. in [RFC2119].
2. Overview Of Method 2. Overview Of Method
Diffie-Hellman key agreement requires that both the sender and reci- 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 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 the other party's public key, both parties can compute the same
shared secret number. This number can then be converted into crypto- shared secret number. This number can then be converted into crypto-
graphic keying material. That keying material is typically used as a graphic keying material. That keying material is typically used as a
key encryption key (KEK) to encrypt (wrap) a message encrytion key key encryption key (KEK) to encrypt (wrap) a content encryption key
(MEK) which is in turn used to encrypt the message data. (CEK) which is in turn used to encrypt the message data.
2.1. Key Agreement 2.1. Key Agreement
The first stage of the key agreement process is to compute a shared The first stage of the key agreement process is to compute a shared
secret number ZZ (which will be constant for any pair of Diffie- secret number, called ZZ. When the same originator and recipient
Hellman keys). ZZ is then converted into a shared symmetric key. Note public/private key pairs are used, the same ZZ value will result.
that the symmetric key will be different for each key agreement, due The ZZ value is then converted into a shared symmetric cryptographic
to the introduction of public random components. 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 2.1.1. Generation of ZZ
X9.42 defines that the shared secret ZZ is generated as follows: X9.42 defines that the shared secret ZZ is generated as follows:
ZZ = g ^ (xb * xa) (mod p) ZZ = g ^ (xb * xa) (mod p)
Note that the individual parties actually perform the computations: Note that the individual parties actually perform the computations:
ZZ = yb ^ xa (mod p) = ya ^ xb (mod p) ZZ = yb ^ xa (mod p) = ya ^ xb (mod p)
where ^ denotes exponentiation where ^ denotes exponentiation
ya is party a's public key; ya = g ^ xa (mod p) ya is party a's public key; ya = g ^ xa (mod p)
yb is party b's public key; yb = g ^ xb (mod p) yb is party b's public key; yb = g ^ xb (mod p)
xa is party a's private key xa is party a's private key
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xb is party b's private key xb is party b's private key
p is a large prime p is a large prime
g is a generator for the integer group specified by p. g is a generator for the integer group specified by p.
j a large integer such that
q is a large prime and p=qj + 1
(See Section 2.2 for criteria for keys and parameters) (See Section 2.2 for criteria for keys and parameters)
In CMS, the recipient's key is identified by the CMS RecipientIden- In CMS, the recipient's key is identified by the CMS RecipientIden-
tifier, which points to the recipient's certificate. The sender's tifier, which points to the recipient's certificate. The sender's
key is identified using the OriginatorIdentifierOrKey field, either key is identified using the OriginatorIdentifierOrKey field, either
by reference to the sender's certificate or by inline inclusion of a by reference to the sender's certificate or by inline inclusion of a
key. key.
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2.1.2. Generation of Keying Material 2.1.2. Generation of Keying Material
X9.42 provides an algorithm for generating an essentially arbitrary X9.42 provides an algorithm for generating an essentially arbitrary
amount of keying material from ZZ. Our algorithm is derived from that amount of keying material from ZZ. Our algorithm is derived from that
algorithm by mandating some optional fields and omitting others. algorithm by mandating some optional fields and omitting others.
KM = H ( ZZ || OtherInfo) KM = H ( ZZ || OtherInfo)
H is the message digest function SHA-1 [FIPS-180] H is the message digest function SHA-1 [FIPS-180]
ZZ is the shared key computed in Section 2.1.1 ZZ is the shared key computed in Section 2.1.1
skipping to change at line 129 skipping to change at line 139
KeySpecificInfo ::= SEQUENCE { KeySpecificInfo ::= SEQUENCE {
algorithm OBJECT IDENTIFIER, algorithm OBJECT IDENTIFIER,
counter OCTET STRING SIZE (4..4) } counter OCTET STRING SIZE (4..4) }
algorithm is the ASN.1 algorithm OID of the symmetric algorithm with which algorithm is the ASN.1 algorithm OID of the symmetric algorithm with which
this KEK will be used. this KEK will be used.
counter is a 32 bit number, represented in network byte order. counter is a 32 bit number, represented in network byte order.
Its initial value is 1, i.e. the byte sequence 00 00 00 01 (hex) Its initial value is 1, i.e. the byte sequence 00 00 00 01 (hex)
pubInfo is a random string provided by the sender. In CMS, it is provided pubInfo is a random string provided by the sender. In CMS, it is provided
as a parameter in the UserKeyingMaterial field (a 512 bit byte string). as a parameter in the UserKeyingMaterial field (a 512 bit value represented
as an OCTET STRING).
Note that the only source of secret entropy in this computation is Note that the only source of secret entropy in this computation is
ZZ, so the security of this data is limited to the size of ZZ, even ZZ, so the security of this data is limited to the size of ZZ, even
if more data than ZZ is generated. However, since pubInfo is dif- if more data than ZZ is generated. However, if pubInfo is different
ferent for each message, a different KEK will be generated for each for each message, a different KEK will be generated for each message.
message. Note that pubInfo is required in Static-Static mode, but MAY appear
in Ephemeral-Static mode.
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2.1.3. KEK Computation 2.1.3. KEK Computation
Each key encryption algorithm requires a specific size key (n). The 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. KEK is generated by mapping the left n-most bytes of KM onto the key.
Consequently, for a DES [FIPS-46-1] key, which requires 64 bits of For 3DES, which requires 192 bits of keying material, the algorithm
keying material, the algorithm is only run once, with a counter value must be run twice, once with a counter value of 1 (to generate K1',
of 1. The first 64 bits of the output are parity adjusted and con- K2', and the first 32 bits of K3') and once with a counter value of 2
verted into a DES key. For 3DES, which requires 192 bits of keying (to generate the last 32 bits of K3). K1',K2' and K3' are then parity
material, the algorithm must be run twice, once with a counter value adjusted to generate the 3 DES keys K1,K2 and K3. For RC2, which
of 1 (to generate K1', K2', and the first 32 bits of K3') and once requires 128 bits of keying material, the algorithm is run once, with
with a counter value of 2 (to generate the last 32 bits of K3). a counter value of 1, and the left-most 128 bits are directly con-
K1',K2' and K3' are then parity adjusted to generate the 3 DES keys verted to an RC2 key.
K1,K2 and K3.
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2.1.4. Keylengths for common algorithms 2.1.4. Keylengths for common algorithms
Some common key encryption algorithms have KEKs of the following Some common key encryption algorithms have KEKs of the following
lengths. lengths.
DES-ECB 64 bits
3DES-EDE-ECB 192 bits 3DES-EDE-ECB 192 bits
RC2 (all) 128 bits RC2 (all) 128 bits
2.1.5. Public Key Validation 2.1.5. Public Key Validation
The following algorithm MAY be used to validate received public keys. 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, 1. Verify that y lies within the interval [2,p-1]. If it does not,
the key is invalid. the key is invalid.
2. Compute y^q (mod p). If the result == 1, the key is valid. 2. Compute y^q (mod p). If the result == 1, the key is valid.
Otherwise the key is invalid. Otherwise the key is invalid.
The primary purpose of public key validation is to prevent a small The primary purpose of public key validation is to prevent a small
subgroup attack [SUBGROUP] on the sender's key pair. If Ephemeral- subgroup attack [LAW98] on the sender's key pair. If Ephemeral-Static
Static mode is used, this check is unnecessary. Note that this pro- mode is used, this check may not be necessary.
cedure may be subject to pending patents.
2.1.6. Example 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 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 3DES-EDE. The key encryption algorithm is 3DES-EDE.
No pubInfo is used No pubInfo is used
Consequently, the input to the first invocation of SHA-1 is: Consequently, the input to the first invocation of SHA-1 is:
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00 01 02 03 04 05 06 07 08 09 0a 0b 0c 0d 0e 0f ; ZZ 00 01 02 03 04 05 06 07 08 09 0a 0b 0c 0d 0e 0f ; ZZ
30 13 30 13
30 11 30 11
06 09 2a 86 48 86 f7 0d 03 06 00 ; 3DES-EDE OID 06 09 2a 86 48 86 f7 0d 03 06 00 ; 3DES-EDE OID
04 04 00 00 00 01 ; Counter 04 04 00 00 00 01 ; Counter
And the output is the 20 bytes: And the output is the 20 bytes:
a8 c6 4e 46 1a aa c2 36 45 c9 2e c6 0e 8a c1 96 8f fb 94 b3 a8 c6 4e 46 1a aa c2 36 45 c9 2e c6 0e 8a c1 96 8f fb 94 b3
The input to the second invocation of SHA-1 is: The input to the second invocation of SHA-1 is:
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00 01 02 03 04 05 06 07 08 09 0a 0b 0c 0d 0e 0f ; ZZ 00 01 02 03 04 05 06 07 08 09 0a 0b 0c 0d 0e 0f ; ZZ
30 13 30 13
30 11 30 11
06 09 2a 86 48 86 f7 0d 03 06 00 ; 3DES-EDE OID 06 09 2a 86 48 86 f7 0d 03 06 00 ; 3DES-EDE OID
04 04 00 00 00 01 ; Counter 04 04 00 00 00 01 ; Counter
And the output is the 20 bytes: And the output is the 20 bytes:
49 eb c8 09 27 77 19 c1 a3 0c cc 49 bd 0c 12 5e e0 f9 1a cc 49 eb c8 09 27 77 19 c1 a3 0c cc 49 bd 0c 12 5e e0 f9 1a cc
Consequently, Consequently,
K1=a8 c6 4e 46 1a aa c2 36 K1'=a8 c6 4e 46 1a aa c2 36
K2=45 c9 2e c6 0e 8a c1 96 K2'=45 c9 2e c6 0e 8a c1 96
K3=8f fb 94 b3 49 eb c8 09 K3'=8f fb 94 b3 49 eb c8 09
Note: These keys are not parity adjusted 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
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 56
30 10
06 08 2a 86 48 86 f7 0d 03 02 ; RC2 OID
04 04 00 00 00 02 ; 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
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01 23 45 67 89 ab cd ef 01 23 45 67 89 ab cd ef
And the output is the 20 bytes:
f3 91 81 0b 33 34 3e c5 48 e5 a5 49 51 83 c0 0a 99 7e b4 1e
Consequently,
K=f3 91 81 0b 33 34 3e c5 48 e5 a5 49 51 83 c0 0a
2.2. Key and Parameter Requirements 2.2. Key and Parameter Requirements
X9.42 requires that the group parameters be of the form p=jq + 1 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- where q is a large prime of length m and j>=2. An algorithm for gen-
erating primes of this form can be found in FIPS PUB 186-1[DSS] and erating primes of this form can be found in FIPS PUB 186-1[DSS] and
ANSI, X9.30-1 1996 [X930], as well as in [X942]. ANSI, X9.30-1 1996 [X930], as well as in [X942].
X9.42 requires that the private key x be in the interval [2^(m-1) + X9.42 requires that the private key x be in the interval [2, (q -
1, (q - 2)]. x should be randomly generated in this interval. y is 2)]. x should be randomly generated in this interval. y is then com-
then computed by calculating g^x (mod p). To comply with this draft, puted by calculating g^x (mod p). To comply with this draft, m MUST
m MUST be >=128, (consequently, q and x MUST each be at least 128 be >=128, (consequently, q and x MUST each be at least 128 bits
bits long). When symmetric ciphers stronger than DES are to be used, long). When symmetric ciphers stronger than DES are to be used, a
a larger m may be advisable. larger m may be advisable.
2.2.1. Group Parameter Generation 2.2.1. Group Parameter Generation
Agents SHOULD generate domain parameters (g,p,q) using the algorithms Agents SHOULD generate domain parameters (g,p,q) using the algorithms
specified in Appendixes 2 and 3 of [FIPS-186]. specified in Appendixes 2 and 3 of [FIPS-186].
2.2.2. Group Parameter Validation 2.2.2. Group Parameter Validation
The ASN.1 for DH keys in [PKIX] includes elements j and 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 Parms which MAY be used by recipients of a key to verify that the
group parameters were correctly generated. Two checks are possible: group parameters were correctly generated. Two checks are possible:
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1. Verify that p=qj + 1. This demonstrates that the parameters meet 1. Verify that p=qj + 1. This demonstrates that the parameters meet
the X9.42 parameter criteria. the X9.42 parameter criteria.
2. Verify that when the p,q generation procedure of [FIPS-186] Appendix 2 2. Verify that when the p,q generation procedure of [FIPS-186]
is followed with seed 'seed', that p is found when 'counter' = pgenCounter. Appendix 2 is followed with seed 'seed', that p is found when
This demonstrates that the parameters were randomly chosen and do not This demonstrates that the parameters were randomly chosen and
have a special form. do not have a special form.
Whether agents provide validation information in their certificates Whether agents provide validation information in their certificates
is a local matter between the agents and their CA. is a local matter between the agents and their CA.
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2.3. Ephemeral-Static Mode 2.3. Ephemeral-Static Mode
In Ephemeral-Static mode, the recipient has a static (and certified) In Ephemeral-Static mode, the recipient has a static (and certified)
key pair, but the sender generates a new key pair for each message 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 and sends it using the originatorKey production. If the sender's key
is freshly generated for each message, the shared secret ZZ will be is freshly generated for each message, the shared secret ZZ will be
similarly different for each message and pubInfo MAY be omitted, similarly different for each message and pubInfo MAY be omitted,
since it serves merely to decouple multiple KEKs generated by the since it serves merely to decouple multiple KEKs generated by the
same set of pairwise keys. If, however, the same ephemeral sender key 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 is used for multiple messages (e.g. it is cached as a performance
optimization) then a separate pubInfo MUST be used for each message. optimization) then a separate pubInfo MUST be used for each message.
All implementations of this standard MUST implement Ephemeral-Static All implementations of this standard MUST implement Ephemeral-Static
mode. mode.
Acknowledgements Acknowledgements
The Key Agreement method described in this document is based on work 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 done by the ANSI X9F1 working group. The author wishes to extend his
thanks for their assistance. thanks for their assistance.
The author also wishes to thank Paul Hoffman, Russ Housley, Brian The author also wishes to thank Paul Hoffman, Stephen Henson, Russ
Korver, Mark Schertler, and Peter Yee for their expert advice and Housley, Brian Korver, Mark Schertler, and Peter Yee for their expert
review. advice and review.
References References
[CMS] Housley, R., "Cryptographic Message Syntax", draft-ietf-smime-cms-05.txt. [CMS] Housley, R., "Cryptographic Message Syntax", draft-ietf-smime-cms-07.txt.
[FIPS-46-1] Federal Information Processing Standards Publication (FIPS PUB) [FIPS-46-1] Federal Information Processing Standards Publication (FIPS PUB)
46-1, Data Encryption Standard, Reaffirmed 1988 January 22 46-1, Data Encryption Standard, Reaffirmed 1988 January 22
(supersedes FIPS PUB 46, 1977 January 15). (supersedes FIPS PUB 46, 1977 January 15).
[FIPS-81] Federal Information Processing Standards Publication (FIPS PUB) [FIPS-81] Federal Information Processing Standards Publication (FIPS PUB)
81, DES Modes of Operation, 1980 December 2. 81, DES Modes of Operation, 1980 December 2.
[FIPS-180] Federal Information Processing Standards Publication (FIPS PUB) [FIPS-180] Federal Information Processing Standards Publication (FIPS PUB)
180-1, "Secure Hash Standard", 1995 April 17. 180-1, "Secure Hash Standard", 1995 April 17.
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[FIPS-186] Federal Information Processing Standards Publication (FIPS PUB) [FIPS-186] Federal Information Processing Standards Publication (FIPS PUB)
186, "Digital Signature Standard", 1994 May 19. 186, "Digital Signature Standard", 1994 May 19.
[PKIX] Housley, R., Ford, W., Polk, W., Solo, D., "Internet X.509 Public [PKIX] Housley, R., Ford, W., Polk, W., Solo, D., "Internet X.509 Public
Key Infrastructure Certificate and CRL Profile", RFC-XXXX. Key Infrastructure Certificate and CRL Profile", RFC-XXXX.
[SUBGROUP] I still need a reference for the Small Subgroup attack. [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.
[X942] "Agreement Of Symmetric Keys Using Diffie-Hellman and MQV Algorithms", [X942] "Agreement Of Symmetric Keys Using Diffie-Hellman and MQV Algorithms",
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ANSI draft, 1998. ANSI draft, 1998.
Security Considerations Security Considerations
All the security in this system is provided by the secrecy of the All the security in this system is provided by the secrecy of the
private keying material. If either sender or recipient private keys private keying material. If either sender or recipient private keys
are disclosed, all messages sent or received using that key are are disclosed, all messages sent or received using that key are
compromised. Similarly, loss of the private key results in an inabil- compromised. Similarly, loss of the private key results in an inabil-
ity to read messages sent using that key. 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. In
Static-Static mode, both originator and recipient SHOULD either per-
form the validation step described in S 2.1.5 or verify that the CA
has properly verified the validity of the key. In Ephemeral-Static
mode, the recipient SHOULD perform 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.
Author's Address Author's Address
Eric Rescorla <ekr@rtfm.com> Eric Rescorla <ekr@rtfm.com>
RTFM Inc. RTFM Inc.
30 Newell Road, #16 30 Newell Road, #16
East Palo Alto, CA 94303 East Palo Alto, CA 94303
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Table of Contents Table of Contents
1. Introduction ................................................... 1 1. Introduction ................................................... 1
1.1. Discussion of this Draft ..................................... 1 1.1. Discussion of this Draft ..................................... 2
1.2. Requirements Terminology ..................................... 2 1.2. Requirements Terminology ..................................... 2
2. Overview Of Method ............................................. 2 2. Overview Of Method ............................................. 2
2.1. Key Agreement ................................................ 2 2.1. Key Agreement ................................................ 2
2.1.1. Generation of ZZ ........................................... 2 2.1.1. Generation of ZZ ........................................... 2
2.1.2. Generation of Keying Material .............................. 3 2.1.2. Generation of Keying Material .............................. 3
2.1.3. KEK Computation ............................................ 3 2.1.3. KEK Computation ............................................ 4
2.1.4. Keylengths for common algorithms ........................... 4 2.1.4. Keylengths for common algorithms ........................... 4
2.1.5. Public Key Validation ...................................... 4 2.1.5. Public Key Validation ...................................... 4
2.1.6. Example .................................................... 4 2.1.6. Example 1 .................................................. 4
2.2. Key and Parameter Requirements ............................... 5 2.1.7. Example 2 .................................................. 5
2.2.1. Group Parameter Generation ................................. 5 2.2. Key and Parameter Requirements ............................... 6
2.2.2. Group Parameter Validation ................................. 5 2.2.1. Group Parameter Generation ................................. 6
2.3. Ephemeral-Static Mode ........................................ 6 2.2.2. Group Parameter Validation ................................. 6
2.3. Acknowledgements ............................................. 6 2.3. Ephemeral-Static Mode ........................................ 7
2.3. References ................................................... 6 2.3. Acknowledgements ............................................. 7
Security Considerations ........................................... 7 2.3. References ................................................... 7
Author's Address .................................................. 7 Security Considerations ........................................... 8
Author's Address .................................................. 8
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