Fast Algorithms for Towers of Finite Fields and Isogenies. (Algorithmes Rapides pour les Tours de Corps Finis et les Isogénies)

In this thesis we apply techniques from computer algebra and languagetheory to speed up the elementary operations in some specific towers of finitefields. We apply our construction to the problem of computing isogeniesbetween elliptic curves and obtain faster (both asymptotically and in practice)variants of Couveignes’ algorithm.The document is divided in four parts. In Part I we recall some basicnotions from algebra and complexity theory. Part II deals with the transpositionprinciple: in it we generalize ideas of Bostan, Schost and Lecerf, and showthat it is possible to automatically transpose computer programs withoutlosses in time complexity and with a small loss in space complexity. Part IIIcombines the results on the transposition principle with classical techniquesfrom elimination theory; we apply these ideas to obtain asymptotically optimalalgorithms for the arithmetic of Artin-Schreier towers of finite fields. We alsodescribe an implementations of these algorithms. Finally, in Part IV we use theprevious results to speed up Couveignes’ algorithm and compare the resultwith the other state of the art algorithms for isogeny computation. We alsopresent a new generalization of Couveignes’ algorithm that computes isogeniesof unknown degree.