Fast Algorithms for Towers of Finite Fields and Isogenies. (Algorithmes Rapides pour les Tours de Corps Finis et les Isogénies)
نویسنده
چکیده
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.
منابع مشابه
ON CURVES OVER FINITE FIELDS by
— In these notes we present some basic results of the Theory of Curves over Finite Fields. Assuming a famous theorem of A. Weil, which bounds the number of solutions in a finite field (i.e., number of rational points) in terms of the genus and the cardinality of the finite field, we then prove several other related bounds (bounds of Serre, Ihara, Stohr-Voloch, etc.). We then treat Maximal Curve...
متن کاملFast dot product over finite field
Finite fields have great applications in various areas as cryptography, that is why it is important to have fast ways of computation to manipulate them. A first approach developed in this report lies in representing integers of the field using floating-point numbers, which lead to efficient computations. Operations in our case are done by restricting the characteristic p of the field to a float...
متن کاملA Crt Algorithm for Constructing Genus 2 Curves over Finite Fields
— We present a new method for constructing genus 2 curves over a finite field Fn with a given number of points on its Jacobian. This method has important applications in cryptography, where groups of prime order are used as the basis for discrete-log based cryptosystems. Our algorithm provides an alternative to the traditional CM method for constructing genus 2 curves. For a quartic CM field K ...
متن کاملOn the Bilinear Complexity of the Multiplication in Finite Fields
— The aim of this paper is to introduce the bilinear complexity of the multiplication in finite fields and to give a brief exposition of the recent results obtained in this part of algebraic complexity theory. In particular we present the new results obtained using the Chudnovsky-Chudnovsky algorithm and its generalizations. Résumé (Sur la complexité bilinéaire de la multiplication dans les cor...
متن کاملOn an Application of the Definition Field Descent of a Tower of Function Fields
— Let us consider an algebraic function field defined over a finite Galois extension K of a perfect field k. We recall some elementary conditions allowing the descent of the definition field of the algebraic function field from K to k. We apply these results to the descent of the definition field of a tower of function fields. We give explicitly the equations of the intermediate steps of an Art...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2010