Elliptic curve 2y^2=x^3+x over field size 8^91+5
draft-brown-ec-2y2-x3-x-mod-8-to-91-plus-5-05

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Internet-Draft                                                D. Brown
Intended status: Experimental                               BlackBerry
Expires: 2020-10-05                                         2020-04-03
 
 
          Elliptic curve 2y^2=x^3+x over field size 8^91+5
         <draft-brown-ec-2y2-x3-x-mod-8-to-91-plus-5-05.txt>

Abstract

  Multi-curve elliptic curve cryptography with 2y^2=x^3+x/GF(8^91+5)
  hedges a risk of new curve-specific attacks.  The curve features:
  isomorphism to Miller's curve from 1985; low Kolmogorov complexity
  (little room for embedded weaknesses of Gordon, Young--Yung, or
  Teske); prime field; Montgomery ladder or Edwards unified arithmetic
  (Hisil--Carter--Dawson--Wong); complex multiplication by i
  (Gallant--Lambert--Vanstone); 34-byte keys; five 64-bit-word field
  arithmetic; easy reduction, inversion, Legendre symbol, and square
  root; similarity to a Bitcoin curve; and string-as-point encoding.

Status of This Memo
 
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Brown                2y^2=x^3+x over 8^91+5                 [Page 1]
Internet-Draft                                             2020-04-03

Table of contents
  1.  Introduction
  2.  Requirements Language (RFC 2119)
  3.  Overview
  3.1. Not for single-curve ECC
  3.2.  Risks of new curve-specific attacks
  3.3.  Multi-curve ECC
  3.3.1.  Multi-curve ECC is a redundancy strategy
  3.3.2.  Whether to use multi-ECC
  3.3.2.1.  Benefits of multi-curve ECC
  3.3.2.2.  Costs of multi-curve ECC
  3.3.3.  Applying multi-curve ECC
  3.4.  Curve features
  3.4.1.  Field features
  3.4.3.  Equation features
  3.4.4.  Finite curve feature
  3.4.4.1.  Curve size and cofactor
  3.4.4.2.  Pollard rho security
  3.4.4.3.  Pohlig--Hellman security
  3.4.4.2.  Menezes--Okamoto--Vanstone security
  3.4.4.3.  Semaev--Araki--Satoh--Smart security
  3.4.4.4.  Edwards and Hessian form
  3.4.4.5.  Bleichenbacher security
  3.4.4.6.  Bernstein's "twist" security
  3.4.4.7.  Cheon security
  4.  Encoding points
  4.1.  Point encoding process
  4.1.1.  Summary
  4.1.2.  Details
  4.2.  Point decoding process
  4.2.1.  Summary
  4.2.2.  Detail
  5.  Point validation
  5.1.  When to validate
  5.1.1.  Mandatory validation
  5.1.2.  Simplified validation
  5.1.4.  Minimal validation
  5.2.  Point validation process
  6.  OPTIONAL encodings
  6.1.  Encoding scalars
  6.2.  Encoding strings as points
  7.  IANA Considerations
  8.  Security considerations
  8.1.  Field choice
  8.2.  Curve choice
  8.3.  Encoding choices
  8.4.  General subversion concerns
  8.5.  Concerns about 'aegis'
  9.  References

Brown                2y^2=x^3+x over 8^91+5                 [Page 2]
Internet-Draft                                             2020-04-03

  9.1.  Normative References
  9.2.  Informative References
  Appendix A.  Test vectors
  Appendix B.  Minimizing trapdoors and backdoors
  Appendix C.  Pseudocode
  C.1.  Scalar multiplication of 34-byte strings
  C.1.1.  Field arithmetic for GF(8^91+5)
  C.1.2.  Montgomery ladder scalar multiplication
  C.1.3.  Bernstein's 2-dimensional Montgomery ladder
  C.1.4.  GLV in Edwards coordinates (Hisil--Carter--Dawson--Wong)
  C.2  Pseudocode for test vectors
  C.3.  Pseudocode for a command-line demo of Diffie--Hellman
  C.4  Pseudocode for public-key validation and twist insecurity
  C.5.  Elligator i
  D. Primality proofs and certificates
  D.1.  Pratt certificate for the field size 8^91+5
  D.2.  Pratt certificate for subgroup order

1.  Introduction
 
  Elliptic curve cryptography (ECC) is now part of several IETF
  protocols.

  Multi-curve ECC mitigates the risk of new curve-specific attacks on
  ECC.  This document aims to contribute to multi-curve ECC by
  describing how to use the curve
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