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A heuristic quasi-polynomial algorithm for discrete logarithm in finite fields of small characteristic. Phong Q. HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from(More)
We give details on solving the discrete logarithm problem in the 202-bit prime order subgroup of F × 2 809 using the Function Field Sieve algorithm (FFS). To our knowledge, this computation is the largest discrete logarithm computation so far in a binary field extension of prime degree. The Function Field Sieve is the traditional approach for solving these(More)
In this paper, we study the discrete logarithm problem in medium and high characteristic finite fields. We propose a variant of the Number Field Sieve (NFS) based on numerous number fields. Our improved algorithm computes discrete logarithms in F p n for the whole range of applicability of NFS and lowers the asymptotic complexity from L p n (1/3, (128/9)(More)
The difficulty of computing discrete logarithms in fields F q k depends on the relative sizes of k and q. Until recently all the cases had a sub-exponential complexity of type L(1/3), similar to the factorization problem. In 2013, Joux designed a new algorithm with a complexity of L(1/4 + ǫ) in small characteristic. In the same spirit, we propose in this(More)
The aim of this work is to investigate the hardness of the discrete logarithm problem in fields GF(p n) where n is a small integer greater than 1. Though less studied than the small characteristic case or the prime field case, the difficulty of this problem is at the heart of security evaluations for torus-based and pairing-based cryptography. The best(More)
The security of pairing-based crypto-systems relies on the difficulty to compute discrete logarithms in finite fields Fpn where n is a small integer larger than 1. The state-of-art algorithm is the number field sieve (NFS) together with its many variants. When p has a special form (SNFS), as in many pairings constructions, NFS has a faster variant due to(More)
We introduce a new variant of the number field sieve algorithm for discrete logarithms in Fpn called exTNFS. The most important modification is done in the polynomial selection step, which determines the cost of the whole algorithm: if one knows how to select good polynomials to tackle discrete logarithms in Fpκ , exTNFS allows to use this method when(More)
We describe a unified framework to search for optimal formulae evaluating bi-linear — or quadratic — maps. This framework applies to polynomial multiplication and squaring, finite field arithmetic, matrix multiplication, etc. We then propose a new algorithm to solve problems in this unified framework. With an implementation of this algorithm, we prove the(More)