Elliptic Curves for Security
draft-irtf-cfrg-curves-06

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Last updated 2015-08-26
Replaces draft-agl-cfrgcurve
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CFRG                                                          A. Langley
Internet-Draft                                                    Google
Intended status: Informational                                M. Hamburg
Expires: February 27, 2016                  Rambus Cryptography Research
                                                               S. Turner
                                                              IECA, Inc.
                                                         August 26, 2015

                      Elliptic Curves for Security
                       draft-irtf-cfrg-curves-06

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

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   This Internet-Draft will expire on February 27, 2016.

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   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
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Langley, et al.         Expires February 27, 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  . . . . . . . . . . . . . . . . . . . . . . .  11
     6.1.  Curve25519  . . . . . . . . . . . . . . . . . . . . . . .  11
     6.2.  Curve448  . . . . . . . . . . . . . . . . . . . . . . . .  12
   7.  Deterministic Generation  . . . . . . . . . . . . . . . . . .  13
     7.1.  p = 1 mod 4 . . . . . . . . . . . . . . . . . . . . . . .  14
     7.2.  p = 3 mod 4 . . . . . . . . . . . . . . . . . . . . . . .  14
     7.3.  Base points . . . . . . . . . . . . . . . . . . . . . . .  15
   8.  Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .  15
   9.  References  . . . . . . . . . . . . . . . . . . . . . . . . .  16
     9.1.  Normative References  . . . . . . . . . . . . . . . . . .  16
     9.2.  Informative References  . . . . . . . . . . . . . . . . .  16
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  17

1.  Introduction

   Since the initial standardization of elliptic curve cryptography
   (ECC) 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 support constant-time, exception-free scalar multiplication that
   is resistant to a wide range of side-channel attacks, including
   timing and cache attacks.  They are Montgomery curves (where y^2 =
   x^3 + Ax^2 + x) and thus have birationally equivalent Edwards
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