Elliptic Curves for Security
draft-irtf-cfrg-curves-07

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CFRG                                                          A. Langley
Internet-Draft                                                    Google
Intended status: Informational                                M. Hamburg
Expires: March 1, 2016                      Rambus Cryptography Research
                                                               S. Turner
                                                              IECA, Inc.
                                                         August 29, 2015

                      Elliptic Curves for Security
                       draft-irtf-cfrg-curves-07

Abstract

   This memo specifies two elliptic curves over prime fields that offer
   high practical security in cryptographic applications, including
   Transport Layer Security (TLS).  These curves are intended to operate
   at the ~128-bit and ~224-bit security level, respectively, and are
   generated deterministically based on a list of required properties.

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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   Drafts is at http://datatracker.ietf.org/drafts/current/.

   Internet-Drafts are draft documents valid for a maximum of six months
   and may be updated, replaced, or obsoleted by other documents at any
   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 March 1, 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

Langley, et al.           Expires March 1, 2016                 [Page 1]
Internet-Draft                  cfrgcurve                    August 2015

   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.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
   2.  Requirements Language . . . . . . . . . . . . . . . . . . . .   3
   3.  Notation  . . . . . . . . . . . . . . . . . . . . . . . . . .   3
   4.  Recommended Curves  . . . . . . . . . . . . . . . . . . . . .   3
     4.1.  Curve25519  . . . . . . . . . . . . . . . . . . . . . . .   3
     4.2.  Curve448  . . . . . . . . . . . . . . . . . . . . . . . .   4
   5.  The X25519 and X448 functions . . . . . . . . . . . . . . . .   6
     5.1.  Side-channel considerations . . . . . . . . . . . . . . .   9
     5.2.  Test vectors  . . . . . . . . . . . . . . . . . . . . . .   9
   6.  Diffie-Hellman  . . . . . . . . . . . . . . . . . . . . . . .  12
     6.1.  Curve25519  . . . . . . . . . . . . . . . . . . . . . . .  12
     6.2.  Curve448  . . . . . . . . . . . . . . . . . . . . . . . .  13
   7.  Security Considerations . . . . . . . . . . . . . . . . . . .  14
   8.  Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .  14
   9.  References  . . . . . . . . . . . . . . . . . . . . . . . . .  15
     9.1.  Normative References  . . . . . . . . . . . . . . . . . .  15
     9.2.  Informative References  . . . . . . . . . . . . . . . . .  15
   Appendix A.  Deterministic Generation . . . . . . . . . . . . . .  16
     A.1.  p = 1 mod 4 . . . . . . . . . . . . . . . . . . . . . . .  17
     A.2.  p = 3 mod 4 . . . . . . . . . . . . . . . . . . . . . . .  18
     A.3.  Base points . . . . . . . . . . . . . . . . . . . . . . .  18
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  19

1.  Introduction

   Since the initial standardization of elliptic curve cryptography (ECC
   [RFC6090]) in [SEC1] there has been significant progress related to
   both efficiency and security of curves and implementations.  Notable
   examples are algorithms protected against certain side-channel
   attacks, various 'special' prime shapes that allow faster modular
   arithmetic, and a larger set of curve models from which to choose.
   There is also concern in the community regarding the generation and
   potential weaknesses of the curves defined by NIST [NIST].

   This memo specifies two elliptic curves ("curve25519" and "curve448")
   that lend themselves to constant-time implementation and an
   exception-free scalar multiplication that is resistant to a wide
   range of side-channel attacks, including timing and cache attacks.