Êtes-vous un étudiant de l'EPFL à la recherche d'un projet de semestre?

Travaillez avec nous sur des projets en science des données et en visualisation, et déployez votre projet sous forme d'application sur GraphSearch.

Découvrez-en plus sur les applications Graph.

Graph
Search

fr|en

Se Connecter

Concept

Finite field arithmetic

Résumé

In mathematics, finite field arithmetic is arithmetic in a finite field (a field containing a finite number of elements) contrary to arithmetic in a field with an infinite number of elements, like the field of rational numbers. There are infinitely many different finite fields. Their number of elements is necessarily of the form pn where p is a prime number and n is a positive integer, and two finite fields of the same size are isomorphic. The prime p is called the characteristic of the field, and the positive integer n is called the dimension of the field over its prime field. Finite fields are used in a variety of applications, including in classical coding theory in linear block codes such as BCH codes and Reed–Solomon error correction, in cryptography algorithms such as the Rijndael (AES) encryption algorithm, in tournament scheduling, and in the design of experiments. Effective polynomial representation The finite field with pn elements is denoted GF(pn) and is also c

Source officielle

https://en.wikipedia.org/wiki/Finite_field_arithmetic

À propos de ce résultat

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Publications associées (49)

Publications associées

Chargement

Personnes associées (7)

Personnes associées

Chargement

Unités associées (2)

Unités associées

Chargement

Concepts associés (3)

Concepts associés

Chargement

Cours associés (9)

Cours associés

Chargement

Séances de cours associées (15)

Séances de cours associées

Chargement

Faster Squaring in the Cyclotomic Subgroup of Sixth Degree Extensions

Robert Granger

This paper describes an extremely efficient squaring operation in the so-called ‘cyclotomic subgroup’ of

\mathbb{F}_{q^6}

, for

q \equiv 1 \bmod{6}

. Our result arises from considering the Weil restriction of scalars of this group from

\mathbb{F}_{q^6}

\mathbb{F}_{q^2}

, and provides efficiency improvements for both pairing-based and torus-based cryptographic protocols. In particular we argue that such fields are ideally suited for the latter when the field characteristic satisfies

p \equiv 1 \pmod{6}

, and since torus-based techniques can be applied to the former, we present a compelling argument for the adoption of a single approach to efficient field arithmetic for pairing-based cryptography.

Springer Berlin Heidelberg2010

Chargement

Computing Cyclic Isogenies between Principally Polarized Abelian Varieties over Finite Fields

Marius Lorenz Vuille

Abelian varieties are fascinating objects, combining the fields of geometry and arithmetic. While the interest in abelian varieties has long time been of purely theoretic nature, they saw their first real-world application in cryptography in the mid 1980's, and have ever since lead to broad research on the computational and the arithmetic side. The most instructive examples of abelian varieties are elliptic curves and Jacobian varieties of hyperelliptic curves, and they come naturally equipped with some additional structure, called a principal polarization. Morphisms between abelian varieties that respect both the geometric and the arithmetic structure are called isogenies. In this thesis we focus on the computation of isogenies with cyclic kernel between principally polarized abelian varieties over finite fields.

EPFL2020

Chargement

In this paper we propose a binary field variant of the Joux-Lercier medium-sized Function Field Sieve, which results not only in complexities as low as

L_{q^n}(1/3,(4/9)1/3)

for computing arbitrary logarithms, but also in an heuristic polynomial time algorithm for finding the discrete logarithms of degree one and two elements when the field has a subfield of an appropriate size. To illustrate the efficiency of the method, we have successfully solved the DLP in the finite fields with

2^{1971}

and

2^{3164}

elements, setting a record for binary fields.

Springer Berlin Heidelberg2013

Chargement

Faster Squaring in the Cyclotomic Subgroup of Sixth Degree Extensions

Robert Granger

This paper describes an extremely efficient squaring operation in the so-called ‘cyclotomic subgroup’ of

\mathbb{F}_{q^6}

, for

q \equiv 1 \bmod{6}

. Our result arises from considering the Weil restriction of scalars of this group from

\mathbb{F}_{q^6}

\mathbb{F}_{q^2}

p \equiv 1 \pmod{6}

, and since torus-based techniques can be applied to the former, we present a compelling argument for the adoption of a single approach to efficient field arithmetic for pairing-based cryptography.

Springer Berlin Heidelberg2010

In this paper we propose a binary field variant of the Joux-Lercier medium-sized Function Field Sieve, which results not only in complexities as low as

L_{q^n}(1/3,(4/9)1/3)

2^{1971}

and

2^{3164}

elements, setting a record for binary fields.

Springer Berlin Heidelberg2013