Twisting commutative algebraic groups

If $V$ is a commutative algebraic group over a field $k$, $O$ is a commutative ring that acts on $V$, and $I$ is a finitely generated free $O$-module with a right action of the absolute Galois group of $k$, then there is a commutative algebraic group $I \otimes_O V$ over $k$, which is a twist of a power of $V$. These group varieties have applications to cryptography (in the cases of abelian varieties and algebraic tori over finite fields) and to the arithmetic of abelian varieties over number fields. For purposes of such applications we devote this article to making explicit this tensor product construction and its basic properties.Comment: To appear in Journal of Algebra. Minor changes from original versio

07381 Executive Summary - Cryptography

The topics covered in the seminar spanned most areas of cryptography, in one way or another, both in terms of the types of schemes (public-key cryptography, symmetric cryptography, hash functions and other cryptographic functions, multi-party protocols, etc.) and in terms of the mathematical methods and techniques used (algebra, number theory, elliptic curves, probability theory, information theory, combinatorics, quantum theory, etc.). The range of applications addressed in the various talks was broad, ranging from secure communication, key management, authentication, digital signatures and payment systems to e-voting and Internet security. While the initial plan had been to focus more exclusively on public-key cryptography, it turned out that this sub-topic branches out into many other areas of cryptography and therefore the organizers decided to expand the scope, emphasizing quality rather than close adherence to public-key cryptography. This decision turned out to be a wise one. What was common to almost all the talks is that rigorous mathematical proofs for the security of the presented schemes were given. In fact, a central topic of many of the talks were proof methodologies for various contexts

Group theory in cryptography

This paper is a guide for the pure mathematician who would like to know more about cryptography based on group theory. The paper gives a brief overview of the subject, and provides pointers to good textbooks, key research papers and recent survey papers in the area.Comment: 25 pages References updated, and a few extra references added. Minor typographical changes. To appear in Proceedings of Groups St Andrews 2009 in Bath, U

Solving multivariate polynomial systems and an invariant from commutative algebra

The complexity of computing the solutions of a system of multivariate polynomial equations by means of Gr\"obner bases computations is upper bounded by a function of the solving degree. In this paper, we discuss how to rigorously estimate the solving degree of a system, focusing on systems arising within public-key cryptography. In particular, we show that it is upper bounded by, and often equal to, the Castelnuovo Mumford regularity of the ideal generated by the homogenization of the equations of the system, or by the equations themselves in case they are homogeneous. We discuss the underlying commutative algebra and clarify under which assumptions the commonly used results hold. In particular, we discuss the assumption of being in generic coordinates (often required for bounds obtained following this type of approach) and prove that systems that contain the field equations or their fake Weil descent are in generic coordinates. We also compare the notion of solving degree with that of degree of regularity, which is commonly used in the literature. We complement the paper with some examples of bounds obtained following the strategy that we describe