Pairing-Friendly Curves
draft-yonezawa-pairing-friendly-curves-02

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Last updated 2019-07-08
Replaces draft-kato-threat-pairing
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Network Working Group                                        S. Yonezawa
Internet-Draft                                                   Lepidum
Intended status: Experimental                               T. Kobayashi
Expires: January 9, 2020                                        T. Saito
                                                                     NTT
                                                           July 08, 2019

                        Pairing-Friendly Curves
               draft-yonezawa-pairing-friendly-curves-02

Abstract

   This memo introduces pairing-friendly curves used for constructing
   pairing-based cryptography.  It describes recommended parameters for
   each security level and recent implementations of pairing-friendly
   curves.

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   Copyright (c) 2019 IETF Trust and the persons identified as the
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Yonezawa, et al.         Expires January 9, 2020                [Page 1]
Internet-Draft           Pairing-Friendly Curves               July 2019

   the Trust Legal Provisions and are provided without warranty as
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Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
     1.1.  Pairing-Based Cryptography  . . . . . . . . . . . . . . .   2
     1.2.  Applications of Pairing-Based Cryptography  . . . . . . .   3
     1.3.  Goal  . . . . . . . . . . . . . . . . . . . . . . . . . .   4
     1.4.  Requirements Terminology  . . . . . . . . . . . . . . . .   4
   2.  Preliminaries . . . . . . . . . . . . . . . . . . . . . . . .   4
     2.1.  Elliptic Curve  . . . . . . . . . . . . . . . . . . . . .   4
     2.2.  Pairing . . . . . . . . . . . . . . . . . . . . . . . . .   5
     2.3.  Barreto-Naehrig Curve . . . . . . . . . . . . . . . . . .   6
     2.4.  Barreto-Lynn-Scott Curve  . . . . . . . . . . . . . . . .   6
     2.5.  Representation Convention for an Extension Field  . . . .   7
   3.  Security of Pairing-Friendly Curves . . . . . . . . . . . . .   8
     3.1.  Evaluating the Security of Pairing-Friendly Curves  . . .   8
     3.2.  Impact of the Recent Attack . . . . . . . . . . . . . . .   9
   4.  Security Evaluation of Pairing-Friendly Curves  . . . . . . .   9
     4.1.  For 100 Bits of Security  . . . . . . . . . . . . . . . .   9
     4.2.  For 128 Bits of Security  . . . . . . . . . . . . . . . .  10
       4.2.1.  BN Curves . . . . . . . . . . . . . . . . . . . . . .  10
       4.2.2.  BLS Curves  . . . . . . . . . . . . . . . . . . . . .  12
     4.3.  For 192 Bits of Security  . . . . . . . . . . . . . . . .  14
     4.4.  For 256 Bits of Security  . . . . . . . . . . . . . . . .  15
   5.  Implementations of Pairing-Friendly Curves  . . . . . . . . .  19
   6.  Security Considerations . . . . . . . . . . . . . . . . . . .  21
   7.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .  21
   8.  Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .  21
   9.  References  . . . . . . . . . . . . . . . . . . . . . . . . .  21
     9.1.  Normative References  . . . . . . . . . . . . . . . . . .  21
     9.2.  Informative References  . . . . . . . . . . . . . . . . .  22
   Appendix A.  Computing Optimal Ate Pairing  . . . . . . . . . . .  26
     A.1.  Optimal Ate Pairings over Barreto-Naehrig Curves  . . . .  27
     A.2.  Optimal Ate Pairings over Barreto-Lynn-Scott Curves . . .  27
   Appendix B.  Test Vectors of Optimal Ate Pairing  . . . . . . . .  28
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  35

1.  Introduction

1.1.  Pairing-Based Cryptography

   Elliptic curve cryptography is one of the important areas in recent
   cryptography.  The cryptographic algorithms based on elliptic curve
   cryptography, such as ECDSA (Elliptic Curve Digital Signature
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