https://hal.uvsq.fr/hal-02177225Joux, A.A.JouxPRISM - Parallélisme, Réseaux, Systèmes, Modélisation - UVSQ - Université de Versailles Saint-Quentin-en-Yvelines - CNRS - Centre National de la Recherche ScientifiqueA new index calculus algorithm with complexity L(1/4 + o(1)) in small characteristicHAL CCSD2014CalculationsCryptographyDiscrete logarithmsFinite field elementsFinite fieldsIndex calculusAlgorithms[MATH] Mathematics [math]HAL UVSQ, Équipe2019-07-08 17:12:332022-06-25 21:12:452019-07-08 17:12:33enConference papers10.1007/978-3-662-43414-7_181In this paper, we describe a new algorithm for discrete logarithms in small characteristic. This algorithm is based on index calculus and includes two new contributions. The first is a new method for generating multiplicative relations among elements of a small smoothness basis. The second is a new descent strategy that allows us to express the logarithm of an arbitrary finite field element in terms of the logarithm of elements from the smoothness basis. For a small characteristic finite field of size Q = pn, this algorithm achieves heuristic complexity LQ(1/4 + o(1)). For technical reasons, unless is already a composite with factors of the right size, this is done by embedding double-struck FQ in a small extension with double-struck FQe with e ≤ 2⌈logpn⌉. © 2014 Springer-Verlag.