Fundamental Elliptic Curve Cryptography Algorithms
RFC 6090
| Document | Type |
RFC - Informational
(February 2011; Errata)
Was draft-mcgrew-fundamental-ecc (individual in sec area)
|
|
|---|---|---|---|
| Last updated | 2020-01-21 | ||
| Stream | IETF | ||
| Formats | plain text html pdf htmlized with errata bibtex | ||
| Reviews | |||
| Stream | WG state | (None) | |
| Document shepherd | No shepherd assigned | ||
| IESG | IESG state | RFC 6090 (Informational) | |
| Consensus Boilerplate | Unknown | ||
| Telechat date | |||
| Responsible AD | Tim Polk | ||
| Send notices to | (None) | ||
Internet Engineering Task Force (IETF) D. McGrew
Request for Comments: 6090 Cisco Systems
Category: Informational K. Igoe
ISSN: 2070-1721 M. Salter
National Security Agency
February 2011
Fundamental Elliptic Curve Cryptography Algorithms
Abstract
This note describes the fundamental algorithms of Elliptic Curve
Cryptography (ECC) as they were defined in some seminal references
from 1994 and earlier. These descriptions may be useful for
implementing the fundamental algorithms without using any of the
specialized methods that were developed in following years. Only
elliptic curves defined over fields of characteristic greater than
three are in scope; these curves are those used in Suite B.
Status of This Memo
This document is not an Internet Standards Track specification; it is
published for informational purposes.
This document is a product of the Internet Engineering Task Force
(IETF). It represents the consensus of the IETF community. It has
received public review and has been approved for publication by the
Internet Engineering Steering Group (IESG). Not all documents
approved by the IESG are a candidate for any level of Internet
Standard; see Section 2 of RFC 5741.
Information about the current status of this document, any errata,
and how to provide feedback on it may be obtained at
http://www.rfc-editor.org/info/rfc6090.
McGrew, et al. Informational [Page 1]
RFC 6090 Fundamental ECC February 2011
Copyright Notice
Copyright (c) 2011 IETF Trust and the persons identified as the
document authors. All rights reserved.
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1. Conventions Used in This Document . . . . . . . . . . . . 4
2. Mathematical Background . . . . . . . . . . . . . . . . . . . 4
2.1. Modular Arithmetic . . . . . . . . . . . . . . . . . . . . 4
2.2. Group Operations . . . . . . . . . . . . . . . . . . . . . 5
2.3. The Finite Field Fp . . . . . . . . . . . . . . . . . . . 6
3. Elliptic Curve Groups . . . . . . . . . . . . . . . . . . . . 7
3.1. Homogeneous Coordinates . . . . . . . . . . . . . . . . . 8
3.2. Other Coordinates . . . . . . . . . . . . . . . . . . . . 9
3.3. ECC Parameters . . . . . . . . . . . . . . . . . . . . . . 9
3.3.1. Discriminant . . . . . . . . . . . . . . . . . . . . . 10
3.3.2. Security . . . . . . . . . . . . . . . . . . . . . . . 10
4. Elliptic Curve Diffie-Hellman (ECDH) . . . . . . . . . . . . . 10
4.1. Data Types . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2. Compact Representation . . . . . . . . . . . . . . . . . . 11
5. Elliptic Curve ElGamal Signatures . . . . . . . . . . . . . . 11
5.1. Background . . . . . . . . . . . . . . . . . . . . . . . . 11
5.2. Hash Functions . . . . . . . . . . . . . . . . . . . . . . 12
5.3. KT-IV Signatures . . . . . . . . . . . . . . . . . . . . . 12
5.3.1. Keypair Generation . . . . . . . . . . . . . . . . . . 12
5.3.2. Signature Creation . . . . . . . . . . . . . . . . . . 13
5.3.3. Signature Verification . . . . . . . . . . . . . . . . 13
5.4. KT-I Signatures . . . . . . . . . . . . . . . . . . . . . 14
5.4.1. Keypair Generation . . . . . . . . . . . . . . . . . . 14
5.4.2. Signature Creation . . . . . . . . . . . . . . . . . . 14
5.4.3. Signature Verification . . . . . . . . . . . . . . . . 14
5.5. Converting KT-IV Signatures to KT-I Signatures . . . . . . 15
5.6. Rationale . . . . . . . . . . . . . . . . . . . . . . . . 15
6. Converting between Integers and Octet Strings . . . . . . . . 16
6.1. Octet-String-to-Integer Conversion . . . . . . . . . . . . 17
6.2. Integer-to-Octet-String Conversion . . . . . . . . . . . . 17
McGrew, et al. Informational [Page 2]
RFC 6090 Fundamental ECC February 2011
7. Interoperability . . . . . . . . . . . . . . . . . . . . . . . 17
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