Cryptology ePrint Archive: Report 2014/130

Selecting Elliptic Curves for Cryptography: An Efficiency and Security Analysis

Joppe W. Bos and Craig Costello and Patrick Longa and Michael Naehrig

Abstract: We select a set of elliptic curves for cryptography and analyze our selection from a performance and security perspective. This analysis complements recent curve proposals that suggest (twisted) Edwards curves by also considering the Weierstrass model. Working with both Montgomery-friendly and pseudo-Mersenne primes allows us to consider more possibilities which improves the overall efficiency of base field arithmetic. Our Weierstrass curves are backwards compatible with current implementations of prime order NIST curves, while providing improved efficiency and stronger security properties. We choose algorithms and explicit formulas to demonstrate that our curves support constant-time, exception-free scalar multiplications, thereby offering high practical security in cryptographic applications. Our implementation shows that variable-base scalar multiplication on the new Weierstrass curves at the 128-bit security level is about 1.4 times faster than the recent implementation record on the corresponding NIST curve. For practitioners who are willing to use a different curve model and sacrifice a few bits of security, we present a collection of twisted Edwards curves with particularly efficient arithmetic that are up to 1.43, 1.26 and 1.24 times faster than the new Weierstrass curves at the 128-, 192- and 256-bit security levels, respectively. Finally, we discuss how these curves behave in a real world protocol by considering different scalar multiplication scenarios in the transport layer security (TLS) protocol.

Category / Keywords: public-key cryptography /

Date: received 19 Feb 2014, last revised 30 Jun 2014

Contact author: plonga at microsoft com

Available format(s): PDF | BibTeX Citation

Note: Updates to curve selection, including new twisted Edwards curves with small "d" and isogenous Montgomery form with small "A" (i.e., optimal parameters for both curve forms). Includes link to the open-source library that supports a subset of the selected curves. Includes improved results and analysis.

Version: 20140630:154727 (All versions of this report)

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