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Versions: (draft-liusvaara-jose-cfrg-curves) 00 01 02 03 04 05 06 RFC 8037

Network Working Group                                       I. Liusvaara
Internet-Draft                                               Independent
Intended status: Standards Track                         August 18, 2016
Expires: February 19, 2017


                    CFRG ECDH and signatures in JOSE
                     draft-ietf-jose-cfrg-curves-06

Abstract

   This document defines how to use the Diffie-Hellman algorithms
   "X25519" and "X448" as well as the signature algorithms "Ed25519" and
   "Ed448" from the IRTF CFRG elliptic curves work in JOSE.

Status of This Memo

   This Internet-Draft is submitted in full conformance with the
   provisions of BCP 78 and BCP 79.

   Internet-Drafts are working documents of the Internet Engineering
   Task Force (IETF).  Note that other groups may also distribute
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   Internet-Drafts are draft documents valid for a maximum of six months
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   time.  It is inappropriate to use Internet-Drafts as reference
   material or to cite them other than as "work in progress."

   This Internet-Draft will expire on February 19, 2017.

Copyright Notice

   Copyright (c) 2016 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents
   (http://trustee.ietf.org/license-info) in effect on the date of
   publication of this document.  Please review these documents
   carefully, as they describe your rights and restrictions with respect
   to this document.  Code Components extracted from this document must
   include Simplified BSD License text as described in Section 4.e of
   the Trust Legal Provisions and are provided without warranty as
   described in the Simplified BSD License.





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Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
     1.1.  Terminology . . . . . . . . . . . . . . . . . . . . . . .   3
   2.  Key type "OKP"  . . . . . . . . . . . . . . . . . . . . . . .   3
   3.  Algorithms  . . . . . . . . . . . . . . . . . . . . . . . . .   3
     3.1.  Signatures  . . . . . . . . . . . . . . . . . . . . . . .   4
       3.1.1.  Signing . . . . . . . . . . . . . . . . . . . . . . .   4
       3.1.2.  Verification  . . . . . . . . . . . . . . . . . . . .   4
     3.2.  ECDH-ES . . . . . . . . . . . . . . . . . . . . . . . . .   4
       3.2.1.  Performing the ECDH Operation . . . . . . . . . . . .   5
   4.  Security considerations . . . . . . . . . . . . . . . . . . .   5
   5.  Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .   6
   6.  IANA considerations . . . . . . . . . . . . . . . . . . . . .   6
   7.  References  . . . . . . . . . . . . . . . . . . . . . . . . .   7
     7.1.  Normative References  . . . . . . . . . . . . . . . . . .   7
     7.2.  Informative References  . . . . . . . . . . . . . . . . .   8
   Appendix A.  Examples . . . . . . . . . . . . . . . . . . . . . .   8
     A.1.  Ed25519 Private Key . . . . . . . . . . . . . . . . . . .   8
     A.2.  Ed25519 Public Key  . . . . . . . . . . . . . . . . . . .   9
     A.3.  JWK Thumbprint Canonicalization . . . . . . . . . . . . .   9
     A.4.  Ed25519 Signing . . . . . . . . . . . . . . . . . . . . .   9
     A.5.  Ed25519 Validation  . . . . . . . . . . . . . . . . . . .  10
     A.6.  ECDH-ES with X25519 . . . . . . . . . . . . . . . . . . .  11
     A.7.  ECDH-ES with X448 . . . . . . . . . . . . . . . . . . . .  12
   Author's Address  . . . . . . . . . . . . . . . . . . . . . . . .  13

1.  Introduction

   Internet Research Task Force (IRTF) Crypto Forum Research Group
   (CFRG) selected new Diffie-Hellman algorithms ("X25519" and "X448";
   [RFC7748]) and signature algorithms ("Ed25519" and "Ed448";
   [I-D.irtf-cfrg-eddsa]) for asymmetric key cryptography.  This
   document defines how to use those algorithms in JOSE in interoperable
   manner.

   This document defines the conventions to use in the context of
   [RFC7515], [RFC7516], and [RFC7517].

   While the CFRG also defined two pairs of isogenous elliptic curves
   that underlie these algorithms, these curves are not directly
   exposed, as the algorithms laid on top are sufficient for the
   purposes of JOSE and are much easier to use.

   All inputs to and outputs from the ECDH and signature functions are
   defined to be octet strings, with the exception of outputs of
   verification functions, which are booleans.




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1.1.  Terminology

   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].

   "JWS Signing Input" and "JWS Signature" are defined by [RFC7515].

   "Key Agreement with Elliptic Curve Diffie-Hellman Ephemeral Static"
   is defined by [RFC7518], section 4.6.

   The JOSE key format ("JSON Web Key (JWK)") is defined by [RFC7517]
   and thumbprints for it ("JSON Web Key (JWK) Thumbprint") in
   [RFC7638].

2.  Key type "OKP"

   A new key type (kty) value "OKP" (Octet Key Pair) is defined for
   public key algorithms that use octet strings as private and public
   keys.  It has the following parameters:

   o  The parameter "kty" MUST be "OKP".

   o  The parameter "crv" MUST be present and contain the subtype of the
      key (from the "JSON Web Elliptic Curve" registry).

   o  The parameter "x" MUST be present and contain the public key
      encoded using the base64url [RFC4648] encoding.

   o  The parameter "d" MUST be present for private keys and contain the
      private key encoded using the base64url encoding.  This parameter
      MUST NOT be present for public keys.

   Note: Do not assume that there is an underlying elliptic curve,
   despite the existence of the "crv" and "x" parameters.  (For
   instance, this key type could be extended to represent DH algorithms
   based on hyperelliptic surfaces.)

   When calculating JWK Thumbprints [RFC7638], the three public key
   fields are included in the hash input in lexicographic order: "crv",
   "kty", and "x".

3.  Algorithms








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3.1.  Signatures

   For purpose of using EdDSA for signing data using "JSON Web Signature
   (JWS)" ([RFC7515]), algorithm "EdDSA" is defined here, to be applied
   as the value of the "alg" parameter.

   The following key subtypes are defined here for use with EdDSA:

      "crv"             EdDSA Variant
      Ed25519           Ed25519
      Ed448             Ed448

   The key type used with these keys is "OKP" and the algorithm used for
   signing is "EdDSA".  These subtypes MUST NOT be used for ECDH-ES.

   The EdDSA variant used is determined by the subtype of the key
   (Ed25519 for "Ed25519" and Ed448 for "Ed448").

3.1.1.  Signing

   Signing for these is preformed by applying the signing algorithm
   defined in [I-D.irtf-cfrg-eddsa] to the private key (as private key),
   public key (as public key) and the JWS Signing Input (as message).
   The resulting signature is the JWS Signature.  All inputs and outputs
   are octet strings.

3.1.2.  Verification

   Verification is performed by applying the verification algorithm
   defined in [I-D.irtf-cfrg-eddsa] to the public key (as public key),
   the JWS Signing Input (as message) and the JWS Signature (as
   signature).  All inputs are octet strings.  If the algorithm accepts,
   the signature is valid; otherwise, the signature is invalid.

3.2.  ECDH-ES

   The following key subtypes are defined here for purpose of "Key
   Agreement with Elliptic Curve Diffie-Hellman Ephemeral Static" (ECDH-
   ES):

      "crv"             ECDH Function Applied
      X25519            X25519
      X448              X448

   The key type used with these keys is "OKP".  These subtypes MUST NOT
   be used for signing.





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   [RFC7518] Section 4.6 defines the ECDH-ES algorithms "ECDH-
   ES+A128KW", "ECDH-ES+A192KW", "ECDH-ES+A256KW" and "ECDH-ES".

3.2.1.  Performing the ECDH Operation

   The "x" parameter of the "epk" field is set as follows:

   Apply the appropriate ECDH function to the ephemeral private key (as
   scalar input) and the standard basepoint (as u-coordinate input).
   The base64url encoding of the output is the value for the "x"
   parameter of the "epk" field.  All inputs and outputs are octet
   strings.

   The Z value (raw key agreement output) for key agreement (to be used
   in subsequent KDF as per [RFC7518] section 4.6.2) is determined as
   follows:

   Apply the appropriate ECDH function to the ephemeral private key (as
   scalar input) and receiver public key (as u-coordinate input).  The
   output is the Z value.  All inputs and outputs are octet strings.

4.  Security considerations

   Security considerations from [RFC7748] and [I-D.irtf-cfrg-eddsa]
   apply here.

   Do not separate key material from information about what key subtype
   it is for.  When using keys, check that the algorithm is compatible
   with the key subtype for the key.  To do otherwise opens the system
   up to attacks via mixing up algorithms.  It is particularly dangerous
   to mix up signature and MAC algorithms.

   Although for Ed25519 and Ed448, the signature binds the key used for
   signing, do not assume this, as there are many signature algorithms
   that fail to make such a binding.  If key-binding is desired, include
   the key used for signing either inside the JWS protected header or
   the data to sign.

   If key generation or batch signature verification is performed, a
   well-seeded cryptographic random number generator is REQUIRED.
   Signing and non-batch signature verification are deterministic
   operations and do not need random numbers of any kind.

   The JWA ECDH-ES KDF construction does not mix keys into the final
   shared secret.  While in key exchange such could be a bad mistake,
   here either the receiver public key has to be chosen maliciously or
   the sender has to be malicious in order to cause problems.  In either
   case, all security evaporates.



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   The nominal security strengths of X25519 and X448 are ~126 and ~223
   bits.  Therefore, using 256-bit symmetric encryption (especially key
   wrapping and encryption) with X448 is RECOMMENDED.

5.  Acknowledgements

   Thanks to Michael B.  Jones for his comments on an initial pre-draft
   and editorial help.

   Thanks to Matt Miller for some editorial help.

6.  IANA considerations

   The following is added to the "JSON Web Key Types" registry:

   o  "kty" Parameter Value: "OKP"
   o  Key Type Description: Octet string key pairs
   o  JOSE Implementation Requirements: Optional
   o  Change Controller: IESG
   o  Specification Document(s): Section 2 of [RFC-THIS]


   The following is added to the "JSON Web Key Parameters" registry:

   o  Parameter Name: "crv"
   o  Parameter Description: The subtype of keypair
   o  Parameter Information Class: Public
   o  Used with "kty" Value(s): "OKP"
   o  Change Controller: IESG
   o  Specification Document(s): Section 2 of [RFC-THIS]

   o  Parameter Name: "d"
   o  Parameter Description: The private key
   o  Parameter Information Class: Private
   o  Used with "kty" Value(s): "OKP"
   o  Change Controller: IESG
   o  Specification Document(s): Section 2 of [RFC-THIS]

   o  Parameter Name: "x"
   o  Parameter Description: The public key
   o  Parameter Information Class: Public
   o  Used with "kty" Value(s): "OKP"
   o  Change Controller: IESG
   o  Specification Document(s): Section 2 of [RFC-THIS]


   The following is added to the "JSON Web Signature and Encryption
   Algorithms" registry:



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   o  Algorithm Name: "EdDSA"
   o  Algorithm Description: EdDSA signature algorithms
   o  Algorithm Usage Location(s): "alg"
   o  JOSE Implementation Requirements: Optional
   o  Change Controller: IESG
   o  Specification Document(s): Section 3 of [RFC-THIS]
   o  Algorithm Analysis Documents(s): [I-D.irtf-cfrg-eddsa]

   The following is added to the "JSON Web Key Elliptic Curve" registry:

   o  Curve Name: "Ed25519"
   o  Curve Description: Ed25519 signature algorithm keypairs
   o  JOSE Implementation Requirements: Optional
   o  Change Controller: IESG
   o  Specification Document(s): Section 3 of [RFC-THIS]

   o  Curve Name: "Ed448"
   o  Curve Description: Ed448 signature algorithm keypairs
   o  JOSE Implementation Requirements: Optional
   o  Change Controller: IESG
   o  Specification Document(s): Section 3 of [RFC-THIS]

   o  Curve name: "X25519"
   o  Curve Description: X25519 function keypairs
   o  JOSE Implementation Requirements: Optional
   o  Change Controller: IESG
   o  Specification Document(s): Section 3.2 of [RFC-THIS]
   o  Analysis Documents(s): [RFC7748]

   o  Curve Name: "X448"
   o  Curve Description: X448 function keypairs
   o  JOSE Implementation Requirements: Optional
   o  Change Controller: IESG
   o  Specification Document(s): Section 3.2 of [RFC-THIS]
   o  Analysis Documents(s): [RFC7748]

7.  References

7.1.  Normative References

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997,
              <http://www.rfc-editor.org/info/rfc2119>.

   [RFC4648]  Josefsson, S., "The Base16, Base32, and Base64 Data
              Encodings", RFC 4648, DOI 10.17487/RFC4648, October 2006,
              <http://www.rfc-editor.org/info/rfc4648>.



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   [RFC7515]  Jones, M., Bradley, J., and N. Sakimura, "JSON Web
              Signature (JWS)", RFC 7515, DOI 10.17487/RFC7515, May
              2015, <http://www.rfc-editor.org/info/rfc7515>.

   [RFC7517]  Jones, M., "JSON Web Key (JWK)", RFC 7517,
              DOI 10.17487/RFC7517, May 2015,
              <http://www.rfc-editor.org/info/rfc7517>.

   [RFC7518]  Jones, M., "JSON Web Algorithms (JWA)", RFC 7518,
              DOI 10.17487/RFC7518, May 2015,
              <http://www.rfc-editor.org/info/rfc7518>.

   [RFC7638]  Jones, M. and N. Sakimura, "JSON Web Key (JWK)
              Thumbprint", RFC 7638, DOI 10.17487/RFC7638, September
              2015, <http://www.rfc-editor.org/info/rfc7638>.

   [RFC7748]  Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves
              for Security", RFC 7748, DOI 10.17487/RFC7748, January
              2016, <http://www.rfc-editor.org/info/rfc7748>.

   [I-D.irtf-cfrg-eddsa]
              Josefsson, S. and I. Liusvaara, "Edwards-curve Digital
              Signature Algorithm (EdDSA)", draft-irtf-cfrg-eddsa-06
              (work in progress), August 2016.

7.2.  Informative References

   [RFC7516]  Jones, M. and J. Hildebrand, "JSON Web Encryption (JWE)",
              RFC 7516, DOI 10.17487/RFC7516, May 2015,
              <http://www.rfc-editor.org/info/rfc7516>.

Appendix A.  Examples

   To the extent possible, the examples use material taken from test
   vectors of [RFC7748] and [I-D.irtf-cfrg-eddsa].

A.1.  Ed25519 Private Key

   {"kty":"OKP","crv":"Ed25519",
   "d":"nWGxne_9WmC6hEr0kuwsxERJxWl7MmkZcDusAxyuf2A"
   "x":"11qYAYKxCrfVS_7TyWQHOg7hcvPapiMlrwIaaPcHURo"}

   The hexadecimal dump of private key is:

   9d 61 b1 9d ef fd 5a 60 ba 84 4a f4 92 ec 2c c4
   44 49 c5 69 7b 32 69 19 70 3b ac 03 1c ae 7f 60

   And of the public key is:



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   d7 5a 98 01 82 b1 0a b7 d5 4b fe d3 c9 64 07 3a
   0e e1 72 f3 da a6 23 25 af 02 1a 68 f7 07 51 1a

A.2.  Ed25519 Public Key

   This is the public parts of the previous private key (which just
   omits "d"):

   {"kty":"OKP","crv":"Ed25519",
   "x":"11qYAYKxCrfVS_7TyWQHOg7hcvPapiMlrwIaaPcHURo"}

A.3.  JWK Thumbprint Canonicalization

   The JWK Thumbprint canonicalization of the two above examples (with
   linebreak inserted for formatting reasons) is:

   {"crv":"Ed25519","kty":"OKP","x":"11qYAYKxCrfVS_7TyWQHOg7hcvPapiMlrwI
   aaPcHURo"}

   Which has the SHA-256 hash (in hexadecimal) of
   90facafea9b1556698540f70c0117a22ea37bd5cf3ed3c47093c1707282b4b89,
   which results in the base64url encoded JWK Thumbprint representation
   of "kPrK_qmxVWaYVA9wwBF6Iuo3vVzz7TxHCTwXBygrS4k".

A.4.  Ed25519 Signing

   The JWS protected header is:

   {"alg":"EdDSA"}

   This has the base64url encoding of:

   eyJhbGciOiJFZERTQSJ9

   The payload is (text):

   Example of Ed25519 signing

   This has the base64url encoding of:

   RXhhbXBsZSBvZiBFZDI1NTE5IHNpZ25pbmc

   The JWS signing input is (concatenation of base64url encoding of the
   (protected) header, a dot and base64url encoding of the payload) is:

   eyJhbGciOiJFZERTQSJ9.RXhhbXBsZSBvZiBFZDI1NTE5IHNpZ25pbmc





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   Applying the Ed25519 signing algorithm using the private key, public
   key, and the JWS signing input yields the signature (hex):

   86 0c 98 d2 29 7f 30 60 a3 3f 42 73 96 72 d6 1b
   53 cf 3a de fe d3 d3 c6 72 f3 20 dc 02 1b 41 1e
   9d 59 b8 62 8d c3 51 e2 48 b8 8b 29 46 8e 0e 41
   85 5b 0f b7 d8 3b b1 5b e9 02 bf cc b8 cd 0a 02

   Converting this to base64url yields:

   hgyY0il_MGCjP0JzlnLWG1PPOt7-09PGcvMg3AIbQR6dWbhijcNR4ki4iylGjg5BhVsPt
   9g7sVvpAr_MuM0KAg

   So the compact serialization of the JWS is (concatenation of signing
   input, a dot, and base64url encoding of the signature):

   eyJhbGciOiJFZERTQSJ9.RXhhbXBsZSBvZiBFZDI1NTE5IHNpZ25pbmc.hgyY0il_MGCj
   P0JzlnLWG1PPOt7-09PGcvMg3AIbQR6dWbhijcNR4ki4iylGjg5BhVsPt9g7sVvpAr_Mu
   M0KAg

A.5.  Ed25519 Validation

   The JWS from above example is:

   eyJhbGciOiJFZERTQSJ9.RXhhbXBsZSBvZiBFZDI1NTE5IHNpZ25pbmc.hgyY0il_MGCj
   P0JzlnLWG1PPOt7-09PGcvMg3AIbQR6dWbhijcNR4ki4iylGjg5BhVsPt9g7sVvpAr_Mu
   M0KAg

   This has 2 dots in it, so it might be valid a JWS.  Base64url
   decoding the protected header yields:

   {"alg":"EdDSA"}

   So this is an EdDSA signature.  Now the key has: "kty":"OKP" and
   "crv":"Ed25519", so the signature is Ed25519 signature.

   The signing input is the part before second dot:

   eyJhbGciOiJFZERTQSJ9.RXhhbXBsZSBvZiBFZDI1NTE5IHNpZ25pbmc

   Applying Ed25519 verification algorithm to the public key, JWS
   signing input and the signature yields true.  So the signature is
   valid.  The message is the base64url decoding of the part between the
   dots:

   Example of Ed25519 Signing





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A.6.  ECDH-ES with X25519

   The public key to encrypt to is:

   {"kty":"OKP","crv":"X25519","kid":"Bob"
   "x":"3p7bfXt9wbTTW2HC7OQ1Nz-DQ8hbeGdNrfx-FG-IK08"}

   The public key from the target key is (hex):

   de 9e db 7d 7b 7d c1 b4 d3 5b 61 c2 ec e4 35 37
   3f 83 43 c8 5b 78 67 4d ad fc 7e 14 6f 88 2b 4f

   The ephemeral secret happens to be (hex):

   77 07 6d 0a 73 18 a5 7d 3c 16 c1 72 51 b2 66 45
   df 4c 2f 87 eb c0 99 2a b1 77 fb a5 1d b9 2c 2a

   So the ephemeral public key is X25519(ephkey,G) (hex):

   85 20 f0 09 89 30 a7 54 74 8b 7d dc b4 3e f7 5a
   0d bf 3a 0d 26 38 1a f4 eb a4 a9 8e aa 9b 4e 6a

   This is represented as the ephemeral public key value:

   {"kty":"OKP","crv":"X25519",
   "x":"hSDwCYkwp1R0i33ctD73Wg2_Og0mOBr066SpjqqbTmo"}

   So the protected header could, for example, be:

   {"alg":"ECDH-ES+A128KW","epk":{"kty":"OKP","crv":"X25519",
   "x":"hSDwCYkwp1R0i33ctD73Wg2_Og0mOBr066SpjqqbTmo"},
   "enc":"A128GCM","kid":"Bob"}

   And the sender computes as the DH Z value as X25519(ephkey,recv_pub)
   (hex):

   4a 5d 9d 5b a4 ce 2d e1 72 8e 3b f4 80 35 0f 25
   e0 7e 21 c9 47 d1 9e 33 76 f0 9b 3c 1e 16 17 42

   The receiver computes as the DH Z value as X25519(seckey,ephkey_pub)
   (hex):

   4a 5d 9d 5b a4 ce 2d e1 72 8e 3b f4 80 35 0f 25
   e0 7e 21 c9 47 d1 9e 33 76 f0 9b 3c 1e 16 17 42

   Which is the same as the sender's value (the both sides run this
   through the KDF before using it as a direct encryption key or
   AES128-KW key).



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A.7.  ECDH-ES with X448

   The public key to encrypt to (with linebreak inserted for formatting
   reasons) is:

   {"kty":"OKP","crv":"X448","kid":"Dave",
   "x":"PreoKbDNIPW8_AtZm2_sz22kYnEHvbDU80W0MCfYuXL8PjT7QjKhPKcG3LV67D2
   uB73BxnvzNgk"}

   The public key from target key is (hex):

   3e b7 a8 29 b0 cd 20 f5 bc fc 0b 59 9b 6f ec cf
   6d a4 62 71 07 bd b0 d4 f3 45 b4 30 27 d8 b9 72
   fc 3e 34 fb 42 32 a1 3c a7 06 dc b5 7a ec 3d ae
   07 bd c1 c6 7b f3 36 09

   The ephemeral secret happens to be (hex):

   9a 8f 49 25 d1 51 9f 57 75 cf 46 b0 4b 58 00 d4
   ee 9e e8 ba e8 bc 55 65 d4 98 c2 8d d9 c9 ba f5
   74 a9 41 97 44 89 73 91 00 63 82 a6 f1 27 ab 1d
   9a c2 d8 c0 a5 98 72 6b

   So the ephemeral public key is X448(ephkey,G) (hex):

   9b 08 f7 cc 31 b7 e3 e6 7d 22 d5 ae a1 21 07 4a
   27 3b d2 b8 3d e0 9c 63 fa a7 3d 2c 22 c5 d9 bb
   c8 36 64 72 41 d9 53 d4 0c 5b 12 da 88 12 0d 53
   17 7f 80 e5 32 c4 1f a0

   This is packed into ephemeral public key value (linebreak inserted
   for formatting purposes):

   {"kty":"OKP","crv":"X448",
   "x":"mwj3zDG34-Z9ItWuoSEHSic70rg94Jxj-qc9LCLF2bvINmRyQdlT1AxbEtqIEg1
   TF3-A5TLEH6A"}

   So the protected header could for example be (linebreak inserted for
   formatting purposes):

   {"alg":"ECDH-ES+A256KW","epk":{"kty":"OKP","crv":"X448",
   "x":"mwj3zDG34-Z9ItWuoSEHSic70rg94Jxj-qc9LCLF2bvINmRyQdlT1AxbEtqIEg1
   TF3-A5TLEH6A"},"enc":"A256GCM","kid":"Dave"}

   And the sender computes as the DH Z value as X448(ephkey,recv_pub)
   (hex):





Liusvaara               Expires February 19, 2017              [Page 12]


Internet-Draft      CFRG ECDH and signatures in JOSE         August 2016


   07 ff f4 18 1a c6 cc 95 ec 1c 16 a9 4a 0f 74 d1
   2d a2 32 ce 40 a7 75 52 28 1d 28 2b b6 0c 0b 56
   fd 24 64 c3 35 54 39 36 52 1c 24 40 30 85 d5 9a
   44 9a 50 37 51 4a 87 9d

   The receiver computes as the DH Z value as X448(seckey,ephkey_pub)
   (hex):

   07 ff f4 18 1a c6 cc 95 ec 1c 16 a9 4a 0f 74 d1
   2d a2 32 ce 40 a7 75 52 28 1d 28 2b b6 0c 0b 56
   fd 24 64 c3 35 54 39 36 52 1c 24 40 30 85 d5 9a
   44 9a 50 37 51 4a 87 9d

   Which is the same as the sender's value (the both sides run this
   through KDF before using as direct encryption key or AES256-KW key).

Author's Address

   Ilari Liusvaara
   Independent

   Email: ilariliusvaara@welho.com





























Liusvaara               Expires February 19, 2017              [Page 13]


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