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- Antoine Joux, Reynald Lercier, Nigel P. Smart, Frederik Vercauteren
- CRYPTO
- 2006

In this paper, we study several variations of the number field sieve to compute discrete logarithms in finite fields of the form Fpn , with p a medium to large prime. We show that when n is not too large, this yields a Lpn(1/3) algorithm with efficiency similar to that of the regular number field sieve over prime fields. This approach complements the recentâ€¦ (More)

- Antoine Joux, Reynald Lercier
- EUROCRYPT
- 2006

In this paper, we study the application of the function field sieve algorithm for computing discrete logarithms over finite fields of the form Fqn when q is a medium-sized prime power. This approach is an alternative to a recent paper of Granger and Vercauteren for computing discrete logarithms in tori, using efficient torus representations. We show thatâ€¦ (More)

- Antoine Joux, Reynald Lercier
- Math. Comput.
- 2003

In this paper, we describe many improvements to the number field sieve. Our main contribution consists of a new way to compute individual logarithms with the number field sieve without solving a very large linear system for each logarithm. We show that, with these improvements, the number field sieve outperforms the gaussian integer method in the hundredâ€¦ (More)

- Antoine Joux, Reynald Lercier
- ANTS
- 2002

In this paper, we describe improvements to the function field sieve (FFS) for the discrete logarithm problem in Fpn , when p is small. Our main contribution is a new way to build the algebraic function fields needed in the algorithm. With this new construction, the heuristic complexity is as good as the complexity of the construction proposed by Adleman andâ€¦ (More)

- Reynald Lercier, Frann Cois, Morain Yzx
- 1995

Cryptographic schemes using elliptic curves over nite elds require the computation of the cardinality of the curves. Dramatic progress have been achieved recently in that eld by various authors. The aim of this article is to highlight part of these improvements and to describe an eecient implementation of them in the particular case of the elds GF (2 n),â€¦ (More)

- Ã‰liane Jaulmes, Reynald Lercier
- IACR Cryptology ePrint Archive
- 2004

We revisit the randomized approach followed in the design of the RMAC message authentication code in order to construct a MAC with similar properties, but based on WegmanCarterâ€™s Îµ-universal hash families instead of a classical CBC chain. This yields a new message authentication code called FRMAC whose security bounds are, as in RMAC, beyond the birthdayâ€¦ (More)

- Reynald Lercier
- ANTS
- 1996

Contrary to what happens over prime elds of large characteristic , the main cost when counting the number of points of an elliptic curve E over F2n is the computation of isogenies of prime degreÃ¨. The best method so far is due to Couveignes and needs asymptotically O(` 3) eld operations. We outline in this article some nice properties satissed by theseâ€¦ (More)

- Pierre-Alain Fouque, Reynald Lercier, Denis RÃ©al, FrÃ©dÃ©ric Valette
- 2008 5th Workshop on Fault Diagnosis andâ€¦
- 2008

In this paper, we present a new fault attack on elliptic curve scalar product algorithms. This attack is tailored to work on the classical Montgomery ladder method when the y-coordinate is not used. No weakness has been reported so far on such implementations, which are very efficient and were promoted by several authors. But taking into account the twistâ€¦ (More)

- Antoine Joux, Reynald Lercier, David Naccache, Emmanuel ThomÃ©
- IACR Cryptology ePrint Archive
- 2008

This paper extends Joux-Naccache-ThomÃ©â€™s e-th root algorithm to the static Diffie-Hellman problem (sdhp). The new algorithm can be adapted to diverse finite fields by customizing it with an nfs-like core or an ffs-like core. In both cases, after a number of non-adaptive sdhp oracle queries, the attacker buildsup the ability to solve new sdhp instancesâ€¦ (More)

- Jean Marc Couveignes, Reynald Lercier
- Finite Fields and Their Applications
- 2009

We construct two new families of basis for finite field extensions. Bases in the first family, the so-called elliptic bases, are not quite normal bases, but they allow very fast Frobenius exponentiation while preserving sparse multiplication formulas. Bases in the second family, the so-called normal elliptic bases are normal bases and allow fastâ€¦ (More)