Alternative Elliptic Curve Representations
draft-ietf-lwig-curve-representations-04
lwig R. Struik
Internet-Draft Struik Security Consultancy
Intended status: Informational April 19, 2019
Expires: October 21, 2019
Alternative Elliptic Curve Representations
draft-ietf-lwig-curve-representations-04
Abstract
This document specifies how to represent Montgomery curves and
(twisted) Edwards curves as curves in short-Weierstrass form and
illustrates how this can be used to carry out elliptic curve
computations using existing implementations of, e.g., ECDSA and ECDH
using NIST prime curves.
Requirements Language
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
"OPTIONAL" in this document are to be interpreted as described in RFC
2119 [RFC2119].
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provisions of BCP 78 and BCP 79.
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Table of Contents
1. Fostering Code Reuse with New Elliptic Curves . . . . . . . . 4
2. Specification of Wei25519 . . . . . . . . . . . . . . . . . . 4
3. Use of Representation Switches . . . . . . . . . . . . . . . 4
4. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.1. Implementation of X25519 . . . . . . . . . . . . . . . . 5
4.2. Implementation of Ed25519 . . . . . . . . . . . . . . . . 6
4.3. Specification of ECDSA25519 . . . . . . . . . . . . . . . 6
4.4. Other Uses . . . . . . . . . . . . . . . . . . . . . . . 7
5. Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
6. Security Considerations . . . . . . . . . . . . . . . . . . . 9
7. Privacy Considerations . . . . . . . . . . . . . . . . . . . 9
8. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 10
8.1. COSE Elliptic Curves Registration . . . . . . . . . . . . 10
8.2. COSE Algorithms Registration (1/2) . . . . . . . . . . . 10
8.3. COSE Algorithms Registration (2/2) . . . . . . . . . . . 11
8.4. JOSE Elliptic Curves Registration . . . . . . . . . . . . 11
8.5. JOSE Algorithms Registration (1/2) . . . . . . . . . . . 11
8.6. JOSE Algorithms Registration (2/2) . . . . . . . . . . . 12
9. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 12
10. References . . . . . . . . . . . . . . . . . . . . . . . . . 12
10.1. Normative References . . . . . . . . . . . . . . . . . . 12
10.2. Informative References . . . . . . . . . . . . . . . . . 13
Appendix A. Some (non-Binary) Elliptic Curves . . . . . . . . . 15
A.1. Curves in short-Weierstrass Form . . . . . . . . . . . . 15
A.2. Montgomery Curves . . . . . . . . . . . . . . . . . . . . 15
A.3. Twisted Edwards Curves . . . . . . . . . . . . . . . . . 15
Appendix B. Elliptic Curve Nomenclature and Finite Fields . . . 16
B.1. Elliptic Curve Nomenclature . . . . . . . . . . . . . . . 16
B.2. Finite Fields . . . . . . . . . . . . . . . . . . . . . . 17
Appendix C. Elliptic Curve Group Operations . . . . . . . . . . 18
C.1. Group Law for Weierstrass Curves . . . . . . . . . . . . 18
C.2. Group Law for Montgomery Curves . . . . . . . . . . . . . 19
C.3. Group Law for Twisted Edwards Curves . . . . . . . . . . 20
Appendix D. Relationship Between Curve Models . . . . . . . . . 21
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