Alternative Elliptic Curve Representations
draft-ietf-lwig-curve-representations-01
lwig R. Struik
Internet-Draft Struik Security Consultancy
Intended status: Informational November 06, 2018
Expires: May 10, 2019
Alternative Elliptic Curve Representations
draft-ietf-lwig-curve-representations-01
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].
Status of This Memo
This Internet-Draft is submitted in full conformance with the
provisions of BCP 78 and BCP 79.
Internet-Drafts are working documents of the Internet Engineering
Task Force (IETF). Note that other groups may also distribute
working documents as Internet-Drafts. The list of current Internet-
Drafts is at https://datatracker.ietf.org/drafts/current/.
Internet-Drafts are draft documents valid for a maximum of six months
and may be updated, replaced, or obsoleted by other documents at any
time. It is inappropriate to use Internet-Drafts as reference
material or to cite them other than as "work in progress."
This Internet-Draft will expire on May 10, 2019.
Copyright Notice
Copyright (c) 2018 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
Provisions Relating to IETF Documents
Struik Expires May 10, 2019 [Page 1]
Internet-Draft lwig-curve-representations November 2018
(https://trustee.ietf.org/license-info) in effect on the date of
publication of this document. Please review these documents
carefully, as they describe your rights and restrictions with respect
to this document. Code Components extracted from this document must
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. Fostering Code Reuse with New Elliptic Curves . . . . . . . . 3
2. Specification of Wei25519 . . . . . . . . . . . . . . . . . . 3
3. Use of Representation Switches . . . . . . . . . . . . . . . 4
4. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.1. Implementation of X25519 . . . . . . . . . . . . . . . . 5
4.2. Implementation of Ed25519 . . . . . . . . . . . . . . . . 5
4.3. Specification of ECDSA-SHA256-Wei25519 . . . . . . . . . 5
4.4. Other Uses . . . . . . . . . . . . . . . . . . . . . . . 6
5. Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
6. Security Considerations . . . . . . . . . . . . . . . . . . . 7
7. Privacy Considerations . . . . . . . . . . . . . . . . . . . 8
8. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 8
9. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 8
10. References . . . . . . . . . . . . . . . . . . . . . . . . . 8
10.1. Normative References . . . . . . . . . . . . . . . . . . 8
10.2. Informative References . . . . . . . . . . . . . . . . . 9
Appendix A. Some (non-Binary) Elliptic Curves . . . . . . . . . 10
A.1. Curves in short-Weierstrass Form . . . . . . . . . . . . 10
A.2. Montgomery Curves . . . . . . . . . . . . . . . . . . . . 10
A.3. Twisted Edwards Curves . . . . . . . . . . . . . . . . . 10
Appendix B. Elliptic Curve Nomenclature . . . . . . . . . . . . 11
Appendix C. Elliptic Curve Group Operations . . . . . . . . . . 11
C.1. Group Law for Weierstrass Curves . . . . . . . . . . . . 11
C.2. Group Law for Montgomery Curves . . . . . . . . . . . . . 12
C.3. Group Law for Twisted Edwards Curves . . . . . . . . . . 13
Appendix D. Relationship Between Curve Models . . . . . . . . . 14
D.1. Mapping between twisted Edwards Curves and Montgomery
Curves . . . . . . . . . . . . . . . . . . . . . . . . . 14
D.2. Mapping between Montgomery Curves and Weierstrass Curves 14
D.3. Mapping between twisted Edwards Curves and Weierstrass
Curves . . . . . . . . . . . . . . . . . . . . . . . . . 15
Appendix E. Curve25519 and Cousins . . . . . . . . . . . . . . . 15
E.1. Curve Definition and Alternative Representations . . . . 15
E.2. Switching between Alternative Representations . . . . . . 16
E.3. Domain Parameters . . . . . . . . . . . . . . . . . . . . 17
Show full document text