Alternative Elliptic Curve Representations
draft-ietf-lwig-curve-representations-01

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Last updated 2018-11-06
Replaces draft-struik-lwig-curve-representations
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lwig                                                           R. Struik
Internet-Draft                               Struik Security Consultancy
Intended status: Informational                         November 06, 2018
Expires: May 10, 2019

               Alternative Elliptic Curve Representations
                draft-ietf-lwig-curve-representations-01

Abstract

   This document specifies how to represent Montgomery curves and
   (twisted) Edwards curves as curves in short-Weierstrass form and
   illustrates how this can be used to carry out elliptic curve
   computations using existing implementations of, e.g., ECDSA and ECDH
   using NIST prime curves.

Requirements Language

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
   "OPTIONAL" in this document are to be interpreted as described in RFC
   2119 [RFC2119].

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
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   This Internet-Draft will expire on May 10, 2019.

Copyright Notice

   Copyright (c) 2018 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

Struik                    Expires May 10, 2019                  [Page 1]
Internet-Draft         lwig-curve-representations          November 2018

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Table of Contents

   1.  Fostering Code Reuse with New Elliptic Curves . . . . . . . .   3
   2.  Specification of Wei25519 . . . . . . . . . . . . . . . . . .   3
   3.  Use of Representation Switches  . . . . . . . . . . . . . . .   4
   4.  Examples  . . . . . . . . . . . . . . . . . . . . . . . . . .   5
     4.1.  Implementation of X25519  . . . . . . . . . . . . . . . .   5
     4.2.  Implementation of Ed25519 . . . . . . . . . . . . . . . .   5
     4.3.  Specification of ECDSA-SHA256-Wei25519  . . . . . . . . .   5
     4.4.  Other Uses  . . . . . . . . . . . . . . . . . . . . . . .   6
   5.  Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . .   6
   6.  Security Considerations . . . . . . . . . . . . . . . . . . .   7
   7.  Privacy Considerations  . . . . . . . . . . . . . . . . . . .   8
   8.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .   8
   9.  Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .   8
   10. References  . . . . . . . . . . . . . . . . . . . . . . . . .   8
     10.1.  Normative References . . . . . . . . . . . . . . . . . .   8
     10.2.  Informative References . . . . . . . . . . . . . . . . .   9
   Appendix A.  Some (non-Binary) Elliptic Curves  . . . . . . . . .  10
     A.1.  Curves in short-Weierstrass Form  . . . . . . . . . . . .  10
     A.2.  Montgomery Curves . . . . . . . . . . . . . . . . . . . .  10
     A.3.  Twisted Edwards Curves  . . . . . . . . . . . . . . . . .  10
   Appendix B.  Elliptic Curve Nomenclature  . . . . . . . . . . . .  11
   Appendix C.  Elliptic Curve Group Operations  . . . . . . . . . .  11
     C.1.  Group Law for Weierstrass Curves  . . . . . . . . . . . .  11
     C.2.  Group Law for Montgomery Curves . . . . . . . . . . . . .  12
     C.3.  Group Law for Twisted Edwards Curves  . . . . . . . . . .  13
   Appendix D.  Relationship Between Curve Models  . . . . . . . . .  14
     D.1.  Mapping between twisted Edwards Curves and Montgomery
           Curves  . . . . . . . . . . . . . . . . . . . . . . . . .  14
     D.2.  Mapping between Montgomery Curves and Weierstrass Curves   14
     D.3.  Mapping between twisted Edwards Curves and Weierstrass
           Curves  . . . . . . . . . . . . . . . . . . . . . . . . .  15
   Appendix E.  Curve25519 and Cousins . . . . . . . . . . . . . . .  15
     E.1.  Curve Definition and Alternative Representations  . . . .  15
     E.2.  Switching between Alternative Representations . . . . . .  16
     E.3.  Domain Parameters . . . . . . . . . . . . . . . . . . . .  17
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