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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                       December 28, 2015
Expires: June 30, 2016


                    CFRG ECDH and signatures in JOSE
                     draft-ietf-jose-cfrg-curves-00

Abstract

   This document defines how to use Diffie-Hellman algorithms "X25519"
   and "X448" as well as signature algorithms "Ed25519", "Ed25519ph",
   "Ed448" and "Ed448ph" from 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 June 30, 2016.

Copyright Notice

   Copyright (c) 2015 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.  Requirements Terminology  . . . . . . . . . . . . . . . .   3
   2.  Key type 'OKP'  . . . . . . . . . . . . . . . . . . . . . . .   3
   3.  Algorithms  . . . . . . . . . . . . . . . . . . . . . . . . .   3
     3.1.  Signatures  . . . . . . . . . . . . . . . . . . . . . . .   3
       3.1.1.  Algorithms  . . . . . . . . . . . . . . . . . . . . .   4
       3.1.2.  Signing . . . . . . . . . . . . . . . . . . . . . . .   4
       3.1.3.  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  . . . . . . . . . . . . . . . . . . . . . . . . .   8
     7.1.  Normative References  . . . . . . . . . . . . . . . . . .   8
     7.2.  Informative References  . . . . . . . . . . . . . . . . .   9
   Appendix A.  Examples . . . . . . . . . . . . . . . . . . . . . .   9
     A.1.  Ed25519 private key . . . . . . . . . . . . . . . . . . .   9
     A.2.  Ed25519 public key  . . . . . . . . . . . . . . . . . . .  10
     A.3.  JWK thumbprint canonicalization . . . . . . . . . . . . .  10
     A.4.  Ed25519 Signing . . . . . . . . . . . . . . . . . . . . .  10
     A.5.  Ed25519 Validation  . . . . . . . . . . . . . . . . . . .  11
     A.6.  ECDH-ES with X25519 . . . . . . . . . . . . . . . . . . .  12
     A.7.  ECDH-ES with X448 . . . . . . . . . . . . . . . . . . . .  13
   Author's Address  . . . . . . . . . . . . . . . . . . . . . . . .  14

1.  Introduction

   Internet Research Task Force (IRTF) Crypto Forum Research Group
   (CFRG) selected new Diffie-Hellman algorithms ("X25519" and "X448";
   [I-D.irtf-cfrg-curves]) and signature algorithms ("Ed25519",
   "Ed25519ph", "Ed448" and "Ed448ph"; [I-D.irtf-cfrg-eddsa]) for
   asymmetric key cryptography.  This document defines how those
   algorithms are to be used in JOSE in inter-operable manner.

   This document defines the conventions to be used in context of
   [RFC7517] and [RFC7518]

   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 (e.g. trying to apply
   ECDSA to those curves leads to nasty corner-cases and produces odd
   results).





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   All inputs to and outputs from the the ECDH and signature functions
   are defined to be octet strings, with the exception of output of
   verification function, which is a boolean.

1.1.  Requirements 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].

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 "JSON Web Elliptic curve" registry).

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

   o  The parameter "d" MUST be present for private keys, and contain
      the private key encoded using 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 thumbprints [RFC7638], the three public key fields
   are included in the hash.  That is, in lexicographic order: "crv",
   "kty" and "x".

   [TBD: Switch to "alg" parameter for subtyping?  But normally "alg" is
   not included in JWK thumbprints and there are multiple "ECDH-ES"
   algorithms already in JWA.]

3.  Algorithms

3.1.  Signatures







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3.1.1.  Algorithms

   The following signature algorithms are defined here (to be applied as
   values of "alg" parameter).  All these have keys with subtype ("crv")
   of the same name:

     "alg"/"crv"            The signature algorithm applied
     Ed25519                Ed25519
     Ed25519ph              Ed25519ph
     Ed448                  Ed448
     Ed448ph                Ed448ph

   The key type for these keys is "OKP" and key subtype for these
   algorithms MUST be the same as the algorithm name.

   The keys of these subtypes MUST NOT be used for ECDH-ES.

   [TBD: Merge the alg values into a single one that can perform signing
   with any signature-capable OKP subtype?  That would remove a source
   of possible errors, since then the message and key could not mismatch
   in algorithm.]

3.1.2.  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 value.  All inputs and
   outputs are octet strings.

3.1.3.  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 value (as
   signature).  All inputs are octet strings.  If the algorithm accepts,
   the signature is valid, otherwise signature is invalid.

3.2.  ECDH-ES

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

      "crv"             ECDH Function applied
      X25519            X25519
      X448              X448





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

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

3.2.1.  Performing the ECDH operation

   The "x" parameter of "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 output is the value for "x" parameter of "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 [I-D.irtf-cfrg-curves] and
   [I-D.irtf-cfrg-eddsa] apply here.

   Some algorithms interact in bad ways (e.g.  "Ed25519" and
   "Ed25519ph").  For this reason, those algorithms have different
   subtypes, so keys for each are not mixed up.

   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 system up to
   attacks via mixing up algorithms.  It is particularly dangerous to
   mix up signature and MAC algorithms.

   Do not assume that signature also binds the key used for signing, it
   does not (there are also other widespread signature algorithms where
   this binding fails, as such binding is not part of the definition of
   secure signature primitive).  As an example of such failure, the
   Ed25519ph signature of X under key (Ed25519ph,Y) is identical to
   Ed25519 signature of SHA512(X) under key (Ed25519,Y).  And often it
   takes only setting a few bits of message (easy to do by brute force)
   to make the message valid enough to be processed in some very
   surprising way.




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   If key generation or batch signature verification is performed, a
   well-seed 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 receiver public key has to be chosen maliciously or the
   sender has to be malicious in order to cause problems.  And in either
   case, all security evaporates anyway.

   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

   Mike Jones for comments on initial pre-draft.

6.  IANA considerations

   The following is added to 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 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



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   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 JSON Web Signature and Encryption
   Algorithms Registry:

   o  Algorithm Name: "Ed25519"
   o  Algorithm Description: Ed25519 signature algorithm
   o  Algorithm Usage Location(s): "alg"
   o  JOSE Implementation Requirements: Optional
   o  Change Controller: IESG
   o  Specification Document(s): Section 3.1 of [RFC-THIS]
   o  Algorithm Analysis Documents(s): [I-D.irtf-cfrg-eddsa]

   o  Algorithm Name: "Ed25519ph"
   o  Algorithm Description: Ed25519 signature algorithm with prehash
   o  Algorithm Usage Location(s): "alg"
   o  JOSE Implementation Requirements: Optional
   o  Change Controller: IESG
   o  Specification Document(s): Section 3.1 of [RFC-THIS]
   o  Algorithm Analysis Documents(s): [I-D.irtf-cfrg-eddsa]

   o  Algorithm Name: "Ed448"
   o  Algorithm Description: Ed448 signature algorithm
   o  Algorithm Usage Location(s): "alg"
   o  JOSE Implementation Requirements: Optional
   o  Change Controller: IESG
   o  Specification Document(s): Section 3.1 of [RFC-THIS]
   o  Algorithm Analysis Documents(s): [I-D.irtf-cfrg-eddsa]

   o  Algorithm Name: "Ed448ph"
   o  Algorithm Description: Ed448 signature algorithm with prehash
   o  Algorithm Usage Location(s): "alg"
   o  JOSE Implementation Requirements: Optional
   o  Change Controller: IESG
   o  Specification Document(s): Section 3.1 of [RFC-THIS]
   o  Algorithm Analysis Documents(s): [I-D.irtf-cfrg-eddsa]


   The following is added to 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



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   o  Specification Document(s): Section 3.1 of [RFC-THIS]

   o  Curve Name: "Ed25519ph"
   o  Curve Description: Ed25519 signature algorithm with prehash
      keypairs
   o  JOSE Implementation Requirements: Optional
   o  Change Controller: IESG
   o  Specification Document(s): Section 3.1 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 3.1 of [RFC-THIS]

   o  Curve Name: "Ed448ph"
   o  Curve Description: Ed448 signature algorithm with prehash keypairs
   o  JOSE Implementation Requirements: Optional
   o  Change Controller: IESG
   o  Specification Document(s): Section 3.1 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): [I-D.irtf-cfrg-curves]

   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): [I-D.irtf-cfrg-curves]

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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   [I-D.irtf-cfrg-curves]
              Langley, A. and M. Hamburg, "Elliptic Curves for
              Security", draft-irtf-cfrg-curves-11 (work in progress),
              October 2015.

   [I-D.irtf-cfrg-eddsa]
              Josefsson, S. and I. Liusvaara, "Edwards-curve Digital
              Signature Algorithm (EdDSA)", draft-irtf-cfrg-eddsa-01
              (work in progress), December 2015.

7.2.  Informative References

   [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>.

Appendix A.  Examples

   To the extent possible, the examples use material lifted from test
   vectors of [I-D.irtf-cfrg-curves] 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:

   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







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A.2.  Ed25519 public key

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

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

A.3.  JWK thumbprint canonicalization

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

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

   Which has the SHA-256 hash of:
   90facafea9b1556698540f70c0117a22ea37bd5cf3ed3c47093c1707282b4b89

A.4.  Ed25519 Signing

   The JWS protected header is:

   {"alg":"Ed25519"}

   This has base64url encoding of:

   eyJhbGciOiJFZDI1NTE5In0

   The payload is (text):

   Example of Ed25519 signing

   This has 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:

   eyJhbGciOiJFZDI1NTE5In0.RXhhbXBsZSBvZiBFZDI1NTE5IHNpZ25pbmc

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







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   53 18 48 60 b1 c6 83 7f 4d 54 22 e9 40 05 43 fd
   47 1f 3a 69 c6 48 2c cb 15 9a 17 62 42 e2 21 b1
   5c 72 63 9b fe a3 9b b2 08 f3 2c ab 1f 27 0f b8
   36 57 1c 52 0b d8 ac 41 eb 45 b3 55 d0 77 19 01

   Converting this to base64url yields:

   UxhIYLHGg39NVCLpQAVD_UcfOmnGSCzLFZoXYkLiIbFccmOb_qObsgjzLKsfJw-4NlccU
   gvYrEHrRbNV0HcZAQ

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

   eyJhbGciOiJFZDI1NTE5In0.RXhhbXBsZSBvZiBFZDI1NTE5IHNpZ25pbmc.UxhIYLHGg
   39NVCLpQAVD_UcfOmnGSCzLFZoXYkLiIbFccmOb_qObsgjzLKsfJw-4NlccUgvYrEHrRb
   NV0HcZAQ

A.5.  Ed25519 Validation

   The JWS from above example is:

   eyJhbGciOiJFZDI1NTE5In0.RXhhbXBsZSBvZiBFZDI1NTE5IHNpZ25pbmc.UxhIYLHGg
   39NVCLpQAVD_UcfOmnGSCzLFZoXYkLiIbFccmOb_qObsgjzLKsfJw-4NlccUgvYrEHrRb
   NV0HcZAQ

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

   {"alg":"Ed25519"}

   So this is Ed25519 signature.  Now the key has: "kty":"OKP" and
   "crv":"Ed25519", so the key is valid for the algorithm (if it had
   other values, the validation would have failed).

   The signing input is the part before second dot:

   eyJhbGciOiJFZDI1NTE5In0.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 base64 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 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 packed into 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 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 sender's value (the both sides run this through
   KDF before using as direct encryption key or AES128-KW key).




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A.7.  ECDH-ES with X448

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

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






Liusvaara                 Expires June 30, 2016                [Page 13]


Internet-Draft      CFRG ECDH and signatures in JOSE       December 2015


   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 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 June 30, 2016                [Page 14]


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