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source: Centre International de Rencontres Mathématiques 2015年6月8日

Jean-Morlet Chair - Research Talks - Shparlinski/Kohel

Semester on 'Number Theory and its Applications to Cryptography'

February - July 2014

General themes

This chair was linked in parts to the thematic month on 'Arithmetics' which took part in February 2014 at CIRM. Igor Shparlinski has a career in Number theory and its applications to cryptography, with significant overlap with the research interests of the groups Dynamique Arithmétique, Combinatoire (DAC) and Arithmétique et Théorie de l'Information (ATI) in Marseille. The idea was to start the month with a week on 'Unlikely Intersections' followed by a workshop organized by members of the DAC research group. Weeks 3 and 4 were on 'Frobenius distributions' and were co-organized with the ATI group. The focus was to introduce and explore new directions of research around the proof of the Sato-Tate conjecture, its generalizations, and the related Lang-Trotter conjecture. Continuing the progression to the interactions of arithmetics with geometry, the thematic month closed with a week on the topic 'On the Conjectures of Lang and Volta'.

The project was concentrated around several areas of number theory and its applications to quasi-Monte Carlo methods and cryptography. For both applications, the notion of pseudorandomness plays a very crucial role and thus they both require high quality pseudorandom number generators and randomness extractors. In turn, these applications lead to several subtle questions of analytic and combinatorial number theory, which are of intrinsic mathematical interest and involve the study of distribution of integers with prescribed arithmetic or combinatorial structure (e.g primes or smooth numbers and numbers with prescribed digit expansions). One of the new directions envisaged was to obtain polynomial analogues of several important results and conjectures which are known in the number case.

Furthermore, driven by applications to elliptic curve cryptography, the project also addressed several theoritic and algorithmic questions related to elliptic and higher genus curves. The above applications were used on a combination of advanced number theory methods such as a) bounds of exponential and character sums; b) sieve methods and c) Subspace theorem and other Diophantine methods, which are developed by the members of DAC as well as the methods of algebraic geometry and commutative algebra such as d) effective forms of Hilbert's Nullstellensatz; e) Newton polytopes and f) Hilbert's Irreducibility theorem, which are developed by the members of ATI. The potential applications to pseudorandomness are of main interest to the members of DAC, while the applications to elliptic curve cryptography are one of the main directions of ATI.

More specifically, the project consisted of the following closely related and cross-fertilising areas:

1. Pseudorandom number generators

2. Integers of cryptographic interest

3. Distribution of points in small boxes on curves over finite fields

4. Arithmetic and group theoretic properties of elliptic curves over finite fields.

http://shparlinskikohel.weebly.com

Igor Shparlinski: Group structures of elliptic curves #1 59:06

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We give a survey of results which address the following generic question: How does a random elliptic curve over a finite field look like. This question has a rich variety of specfic interpretations, which depend on how one defines a random curve and what properties which are of interest. The former may include randomisation of the coefficients of the Weierstrass equation or the prime power defining the field, or both. The latter may include studying the group structure, arithmetic structure of the number of points (primality, smoothness, etc.) and certain divisibility conditions. These questions are related to such celebrated problems as Lang-Trotter and Sato-Tate conjectures. More recently the interest to these questions was re-fueled by the needs of pairing based cryptography. In a series of talks we will describe the state of art in some of these directions, demonstrate the richness of underlying mathematics and pose some open questions.

Recording during the thematic meeting: "Frobenius distribution on curves" the February 18, 2014 at the Centre International de Rencontres Mathématiques (Marseille, France)

Igor Shparlinski: Group structures of elliptic curves #2 59:44

Igor Shparlinski: Group structures of elliptic curves #3 59:41

Chantal David: Distributions of Frobenius of elliptic curves #1 56:29

Chantal David: Distributions of Frobenius of elliptic curves #2 1:00:47

Nathan Jones: Distributions of Frobenius of elliptic curves #3 58:25

Nathan jones: Distributions of Frobenius of elliptic curves #4 1:04:13

Chantal David: Distributions of Frobenius of elliptic curves #5 1:02:11

Nathan Jones : Distributions of Frobenius of elliptic curves #6 45:16

Christian Elsholtz: Hilbert cubes in arithmetic sets 31:47

Francesc Fité: The Galois type of an Abelian surface 1:04:05

Francesc Fité: Sato-Tate axioms 53:53

Francesc Fité: The generalized Sato-Tate conjecture 58:52

Peter Stevenhagen: The Chebotarev density theorem 1:02:29

Peter Stevenhagen: Character sums for primitive root densities 1:05:46

Andrew Sutherland: Introduction to Sato-Tate distributions 1:03:38

Andrew Sutherland: Computing Sato-Tate statistics 1:11:19

Andrew Sutherland: Moment sequences of Sato-Tate groups 58:06

Jean Pierre Serre: Distributions des valeurs propres des Frobenius des variétés abéliennes ... 1:00:04

Mike Zieve: Unlikely intersections of polynomial orbits 40:05

Gilles Lachaud: Formulas for the limiting distribution of traces of Frobenius 48:59

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