A comparison and a combination of SST and AGM algorithms for counting points of elliptic curves in characteristic 2

International audienceSince the first use of a p-adic method for counting points of elliptic curves, by Satoh in 1999, several variants of his algorithm have been proposed. In the current state, the AGM algorithm, proposed by Mestre is thought to be the fastest in practice, and the algorithm by Satoh­-Skjernaa­-Taguchi has the best asymptotic complexity but requires precomputations. We present an amelioration of the SST algorithm, borrowing ideas from the AGM. We make a precise comparison between this modified SST algorithm and the AGM, thus demonstrating that the former is faster by a significant factor, even for small cryptographic sizes

The mpFq library and implementing curve-based key exchanges

International audienceWe present a library for finite field arithmetic. The originality of this library lies in the fact that specialized code is automatically produced for the selected finite fields. The opportunity of compile-time optimizations yields substantial performance improvements compared to libraries which initialize the finite field at runtime. This library is used to present benchmarks on some curve-based public key cryptosystems

Algorithms for improved performance in cryptographic protocols.

Public key cryptographic algorithms provide data authentication and non-repudiation for electronic transmissions. The mathematical nature of the algorithms, however, means they require a significant amount of computation, and encrypted messages and digital signatures possess high bandwidth. Accordingly, there are many environments (e.g. wireless, ad-hoc, remote sensing networks) where public-key requirements are prohibitive and cannot be used. The use of elliptic curves in public-key computations has provided a means by which computations and bandwidth can be somewhat reduced. We report here on the research conducted in an LDRD aimed to find even more efficient algorithms and to make public-key cryptography available to a wider range of computing environments. We improved upon several algorithms, including one for which a patent has been applied. Further we discovered some new problems and relations on which future cryptographic algorithms may be based

Finding Secure Curves with the Satoh-FGH Algorithm and an Early-Abort Strategy

The use of elliptic curves in cryptography relies on the ability to count the number of points on a given curve. Before 1999, the SEA algorithm was the only ecient method known for random curves. Then Satoh proposed a new algorithm based on the canonical p-adic lift of the curve for p 5. In an earlier paper, the authors extended Satoh's method to the case of characteristics two and three. This paper presents an implementation of the Satoh-FGH algorithm and its application to the problem of nding curves suitable for cryptography. By combining SatohFGH and an early-abort strategy based on SEA, we are able to nd secure random curves in characteristic two in much less time than previously reported. In particular we can generate curves widely considered to be as secure as RSA-1024 in less than one minute each on a fast workstation