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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)
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)
The year 2013 has seen several major complexity advances for the discrete logarithm problem in multiplicative groups of small-characteristic finite fields. These outmatch, asymptotically, the Function Field Sieve (FFS) approach, which was so far the most efficient algorithm known for this task. Yet, on the practical side, it is not clear whether the new(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)
In this paper we prove some divisibility properties of the cardinal-ity of elliptic curves modulo primes. These proofs explain the good behavior of certain parameters when using Montgomery or Edwards curves in the setting of the elliptic curve method (ECM) for integer factorization. The ideas of the proofs help us to find new families of elliptic curves(More)