Pairing-Friendly Curves
draft-irtf-cfrg-pairing-friendly-curves-00
CFRG Y. Sakemi
Internet-Draft Lepidum
Intended status: Experimental T. Kobayashi
Expires: May 4, 2020 T. Saito
NTT
November 1, 2019
Pairing-Friendly Curves
draft-irtf-cfrg-pairing-friendly-curves-00
Abstract
This memo introduces pairing-friendly curves used for constructing
pairing-based cryptography. It describes recommended parameters for
each security level and recent implementations of pairing-friendly
curves.
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1. Pairing-Based Cryptography . . . . . . . . . . . . . . . 2
1.2. Applications of Pairing-Based Cryptography . . . . . . . 3
1.3. Goal . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4. Requirements Terminology . . . . . . . . . . . . . . . . 4
2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1. Elliptic Curve . . . . . . . . . . . . . . . . . . . . . 4
2.2. Pairing . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3. Barreto-Naehrig Curve . . . . . . . . . . . . . . . . . . 6
2.4. Barreto-Lynn-Scott Curve . . . . . . . . . . . . . . . . 6
2.5. Representation Convention for an Extension Field . . . . 7
3. Security of Pairing-Friendly Curves . . . . . . . . . . . . . 8
3.1. Evaluating the Security of Pairing-Friendly Curves . . . 8
3.2. Impact of the Recent Attack . . . . . . . . . . . . . . . 9
4. Security Evaluation of Pairing-Friendly Curves . . . . . . . 9
4.1. For 100 Bits of Security . . . . . . . . . . . . . . . . 9
4.2. For 128 Bits of Security . . . . . . . . . . . . . . . . 10
4.2.1. BN Curves . . . . . . . . . . . . . . . . . . . . . . 10
4.2.2. BLS Curves . . . . . . . . . . . . . . . . . . . . . 12
4.3. For 192 Bits of Security . . . . . . . . . . . . . . . . 14
4.4. For 256 Bits of Security . . . . . . . . . . . . . . . . 15
5. Implementations of Pairing-Friendly Curves . . . . . . . . . 19
6. Security Considerations . . . . . . . . . . . . . . . . . . . 21
7. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 21
8. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 21
9. References . . . . . . . . . . . . . . . . . . . . . . . . . 21
9.1. Normative References . . . . . . . . . . . . . . . . . . 21
9.2. Informative References . . . . . . . . . . . . . . . . . 22
Appendix A. Computing Optimal Ate Pairing . . . . . . . . . . . 26
A.1. Optimal Ate Pairings over Barreto-Naehrig Curves . . . . 27
A.2. Optimal Ate Pairings over Barreto-Lynn-Scott Curves . . . 27
Appendix B. Test Vectors of Optimal Ate Pairing . . . . . . . . 28
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 35
1. Introduction
1.1. Pairing-Based Cryptography
Elliptic curve cryptography is one of the important areas in recent
cryptography. The cryptographic algorithms based on elliptic curve
cryptography, such as ECDSA (Elliptic Curve Digital Signature
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