Hashing to Elliptic Curves
draft-irtf-cfrg-hash-to-curve-00

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Last updated 2018-04-08 (latest revision 2018-03-23)
Replaces draft-sullivan-cfrg-hash-to-curve
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Network Working Group                                        N. Sullivan
Internet-Draft                                                Cloudflare
Intended status: Informational                                   C. Wood
Expires: September 24, 2018                                   Apple Inc.
                                                          March 23, 2018

                       Hashing to Elliptic Curves
                    draft-irtf-cfrg-hash-to-curve-00

Abstract

   This document specifies a number of algorithms that may be used to
   hash arbitrary strings to Elliptic Curves.

Status of This Memo

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   provisions of BCP 78 and BCP 79.

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   This Internet-Draft will expire on September 24, 2018.

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   Copyright (c) 2018 IETF Trust and the persons identified as the
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Sullivan & Wood        Expires September 24, 2018               [Page 1]
Internet-Draft                hash-to-curve                   March 2018

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
     1.1.  Requirements  . . . . . . . . . . . . . . . . . . . . . .   3
   2.  Algorithm Recommendations . . . . . . . . . . . . . . . . . .   3
   3.  Generic Interface . . . . . . . . . . . . . . . . . . . . . .   4
     3.1.  Utility Functions . . . . . . . . . . . . . . . . . . . .   4
   4.  Hashing Variants  . . . . . . . . . . . . . . . . . . . . . .   5
     4.1.  Icart Method  . . . . . . . . . . . . . . . . . . . . . .   5
     4.2.  Shallue-Woestijne-Ulas Method . . . . . . . . . . . . . .   6
     4.3.  Simplified SWU Method . . . . . . . . . . . . . . . . . .   6
     4.4.  Elligator2 Method . . . . . . . . . . . . . . . . . . . .   8
   5.  Curve Transformations . . . . . . . . . . . . . . . . . . . .  10
   6.  Cost Comparison . . . . . . . . . . . . . . . . . . . . . . .  10
   7.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .  11
   8.  Security Considerations . . . . . . . . . . . . . . . . . . .  11
   9.  Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .  11
   10. Contributors  . . . . . . . . . . . . . . . . . . . . . . . .  11
   11. Normative References  . . . . . . . . . . . . . . . . . . . .  11
   Appendix A.  Try-and-Increment Method . . . . . . . . . . . . . .  13
   Appendix B.  Sample Code  . . . . . . . . . . . . . . . . . . . .  13
     B.1.  Icart Method  . . . . . . . . . . . . . . . . . . . . . .  13
     B.2.  Shallue-Woestijne-Ulas Method . . . . . . . . . . . . . .  14
     B.3.  Simplified SWU Method . . . . . . . . . . . . . . . . . .  15
     B.4.  Elligator2 Method . . . . . . . . . . . . . . . . . . . .  16
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  17

1.  Introduction

   Many cryptographic protocols require a procedure which maps arbitrary
   input, e.g., passwords, to points on an elliptic curve (EC).
   Prominent examples include Simple Password Exponential Key Exchange
   [Jablon96], Password Authenticated Key Exchange [BMP00], and Boneh-
   Lynn-Shacham signatures [BLS01].

   Let E be an elliptic curve over base field GF(p).  In practice,
   efficient (polynomial-time) functions that hash arbitrary input to E
   can be constructed by composing a cryptographically secure hash
   function F1 : {0,1}^* ->GF(p) and an injection F2 : GF(p) -> E, i.e.,
   Hash(m) = F2(F1(m)).  Probabilistic constructions of Hash, e.g., the
   MapToGroup function described by Boneh et al.  [BLS01].  Their
   algorithm fails with probability 2^I, where I is a tunable parameter
   that one can control.  Another variant, dubbed the "Try and
   Increment" approach, was described by Boneh et al.  [BLS01].  This
   function works by hashing input m using a standard hash function,
   e.g., SHA256, and then checking to see if the resulting point E(m,
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