Conference Agenda

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Session Overview
Session
MS162, part 1: Applications of finite fields theory
Time:
Friday, 12/Jul/2019:
3:00pm - 5:00pm

Location: Unitobler, F-123
52 seats, 100m^2

Presentations
3:00pm - 5:00pm

Applications of finite fields theory

Chair(s): Antoine Joux (University of Sorbonne), Giacomo Micheli (EPFL), Violetta Weger (University of Zurich, Switzerland)

The theory of finite fields is one of the most important meeting points of Algebraic Geometry, Computer Science, and Number Theory. One of the most important challenges in the area is to develope the theory of finite fields in connection with useful applications, in particular in secure communication, coding theory, and pseudorandom number generation. In this minysimposium we plan to bring together experts from many different areas of the mathematics of communication who share the common interest towards the theory of finite fields. Our main purpose is to provide an overview of some of the cutting-edge research in the field, and to lay the fundations for new collaborations among researchers interested in applications of the theory of finite fields.
In the cryptographic setting, we focus on new post-quantum cryptographic schemes (Marco Baldi, Antoine Joux) and cryptanalysis (Gohar Kyureghyan, Yann Rotella). For pseudorandomness we propose construction of new pseudorandom generators (Federico Amadio Guidi, Laszlo Merai) and construction of polynomials over finite fields with given properties which are interesting for applications (Andrea Ferraguti).

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Introductory Talk

Giacomo Micheli
EPFL

This is an introductory talk to this session.

 

Using Mersenne and Fermat numbers in Cryptosystems

Antoine Joux
University of Sorbonne

Modern public-key cryptography is mostly based on the hardness of Factoring and computing discrete logarithms. However, due to Shor’s algorithm, large scale Quantum computer if and when they become available would put these systems at risk, with the danger of compromising the security of all computer applications. In this talk, we show the construction of new crypto algorithms based on arithmetic modulo Mersenne or Fermat numbers. We describe both a simple encryption algorithm and a fully homomorphic encryption scheme.

 

Cryptographic attacks against filter generator using monomial mapping

Yann Rotella
Inria

Filter generators are vulnerable to several attacks which have led to well-known design criteria on the Boolean filtering function (mainly Algebraic Immunity and Nonlinearity). However, Rønjom and Cid have observed that a change of the primitive root defining the LFSR leads to several equivalent generators. They usually offer different security levels since they involve filtering functions of the form F(xk) where k is coprime to (2n -1) and n denotes the LFSR length.

We prove that this monomial equivalence does not affect the resistance of the generator against algebraic attacks, but usually impacts the resistance to correlation attacks.

Most importantly, a more efficient attack can often be mounted by considering non-bijective monomial mappings. In this setting, a divide-and-conquer strategy applies based on a search within a multiplicative subgroup of F2n*. Moreover, if the LFSR length n is not a prime, a fast correlation attack involving a shorter LFSR can be performed.

This attack is generic and uses the decomposition in the multiplicative subgroups of F2n*, leading to new design criteria for Boolean functions used in Cryptography.

 

Permutation and complete rational functions via Chebotarev theorem for function fields

Andrea Ferraguti
Max Planck Institute for Mathmatics

Constructing permutation functions of finite fields is a task of great interest in coding theory and cryptography. Permutation polynomials over finite fields have been completely classified up to degree 6, with "ad hoc" methods for every degree. In this talk, we present a general approach for classifying permutation rational functions of any degree that exploits a refined version of Chebotarev density theorem for function fields due to Kosters. We will show how to use the method to completely classify permutation rational functions and complete rational functions of degree 3. This is joint work with Giacomo Micheli.