SCIENTIFIC PROGRAMS AND ACTIVITIES

Number Theory Seminar Series
July 19, 2006-December 2006
Fields Institute

1) Number Theory and Cryptography Research Seminar
Tuesdays 2-3 p.m., held at the Fields Institute, Room 230

Tuesday, December 12, 2006 -- 3:30 PM	Roger Oyono Fast arithmetic in the Jacobian of non-hyperelliptic curves of genus 3 In this talk, I will present a fast addition algorithm in the Jacobian of non-hyperelliptic curves of genus 3. The presented algorithm has a nice geometric interpretation, comparable to the classic chord and tangent law for the elliptic curves.
Tuesday, December 5 2:00 PM	Adrian Tang Low-Cost RFID, Private Key Authentication and Abstractions of Integer Arithmetic With Radio Frequency Identification (RFID) tags posed to replaced Universal Product Codes, serious security and information privacy issues have arisen. Attempts have been made to provide a means of authentication between tags and readers, and simultaneously meet computing and memory specifications for the lowest ranges of RFID tags. This talk will discuss a proposed mutual authentication scheme that requires 32 bits of read/write memory, 62 bits of read-only memory and can be deployed using as few as 224 logic gates. We also propose a stream cipher with the same memory constraints and magnitude of logic gates. The underlying idea behind these schemes is a notion that we call the abstractions of integer arithmetic.
Tuesday, November 21, 2006 2:00 PM	Reinier Broker p-adic class invariants The theory of complex multiplication provides us with a means of computing a generating polynomial for the Hilbert class field of a given imaginary quadratic number field. The classical approach of using the modular j-function yields polynomials with huge coefficients, and as was disovered by Weber already, we can do better by using `smaller' functions. In this talk we focus on new p-adic algorithms to compute such generating polynomials. For the j-function this is based on a paper of Couveignes and Henocq, and we explain how to generalize their approach to cope with smaller functions over p-adic fields by using modular curves.
Tuesday, November 14, 2006 2:00 PM	Andy Klapper, University of Kentucky Function Field and Number Field Generalizations of Linear Feedback Shift Registers Linear feedback shift registers are very fast generators of statistically random sequences. They are used in a vast array of applications, including cryptographic stream ciphers, error correcting codes, code division multiple access, radar ranging, and quasi-Monte Carlo integration. From a mathematical point of view, they are based on the algebra of polynomials and power series over finite fields. In recent years we have generalized this construction to build sequence generators based first on the algebra of N-adic numbers (the case N = 2 has been used in random number generators for quasi-Monte Carlo and as building blocks for stream ciphers), and more recently on more general completions of algebraic rings. The resulting generators are called algebraic feedback shift registers (AFSRs). In this talk we will review the basic definitions and properties of algebraic feedback shift registers. We will then examine the case when the underlying ring is an function field in some detail. In particular we will see how these sequence generators relate to an old conjecture of Golomb's. If time permits, we will touch on various other topics concerning AFSRs.
Tuesday, November 7, 2006 2:00 pm	Lennart Berg and Jerome Grand'Maison When Games Meet Secret Sharings Part 1: Game theoretic analysis of cryptographic protocols Speaker 1: Lennart Berg Abstract 1: Some cryptographic protocols can be seen as strategies, and self enforcing protocols can be formally defined in terms of a Nash equilibrium. This will help us to create protocols that works also with strictly self-interested participants. Part 2: Cryptography to the rescue of game theory Speaker 2: Jerome Grand'Maison Abstract 2: The use of a trusted mediator often increase the expected payoffs of players, but is not practical. We will use cryptography and so called "cheap talks" to remove this mediator without changing the outcome of the game.
Friday October 27 2:00	Jens Zumbraegel Public-Key Cryptography using semigroup actions and semirings The classical Diffie Hellman key exchange protocol can be generalized by using an action of an Abelian semigroup on a set. Its security is based on the assumed hardness of the analog to the Discrete Log Problem, which we call Semigroup Action Problem (SAP). I will present various examples of such semigroup actions and discuss their security. Also I will introduce an important ingredient to build some promising semigroup actions, the so-called congruence-free (or simple) semirings.
Tuesday Oct 24, 2006 2 PM	Nicolas Theriault Factoring polynomials of small degree over fields of characteristic 2 We present a new algorithm to factor polynomials of very small degree defined over fields of characteristic 2. An interesting application of our algorithm can be found in discrete logartihm problems for hyperelliptic curves. We also discuss what happens when the algorithm is adated to asymptotic situations.
Tuesday, October 10, 2006 2:00 PM	Andreas Stein Approximating Euler Products and an Algorithm for Computing the Class Number of an Algebraic Function Field A fundamental problem in the theory of function fields and curves over finite fields is the effective computation of the class number h and thus the order of the Jacobian of an algebraic function field. If the characteristic of the finite field is small, various recent algorithms solve this problem. Our main focus will be algebraic function fields of large characteristic, in which case not much is known about effective computation of the order of the Jacobian. However, our methods are very general for any genus and any characteristic. In our talk, we will first discuss how to perform arithmetic in an algebraic function fields based on recent results. Then we will provide tight estimates for the class number via truncated Euler products, and show how these estimates can be used to develop an effective method of computing h.
Thursday, October 5, 2006 2:00 - 3:00	Felix Fontain (University of Zurich) Computing in Divisor Class Groups of Global Function Fields In this talk, I will present an algorithm from Florian Hess to effectively compute Riemann-Roch spaces of divisors in global function fields. Then I will describe a unique representation of divisor classes and an algorithm to effectively compute in the divisor class group.

---

2) Number Theory Seminar on Modular Curves

Wednesday, October 25, 2006 10:00-11:00	Dr. Reiner Broker Modular curves as moduli spaces
Monday, October 23, 2006 1:00-2:30	Dr. Reiner Broker and Alina Cojocaru Modular curves as moduli spaces and diophantine applications
Monday, October 16, 2006 1:30-3:00	Roger Oyono Introduction to modular curves
Thursday, October 5, 2006 11:00 - 12:00	Elisa Gorla An introduction to schemes and group schemes
Friday, October 6, 2006 1:00 - 3:00	Elisa Gorla The Neron model of an abelian variety

We are starting a working seminar on rational points of modular curves. We plan to cover the following topics:

1. Introduction to schemes and group schemes
2. The Neron model of an abelian variety
3. Introduction to modular curves
4. Characterization of modular curves as moduli spaces of elliptic curves with additional structure, their relation with Serre's Open Image Theorem and with solving diophantine equations
5. Rational points on X_0(N): Mazur's work
6. Mestre's graph method
7. The winding quotient of an abelian variety: Merel's work
8. Heegner points
9. Rational points on X_split(N): work of Momose and Parent
10. Special values of L-functions: work of Gross and Gross-Kudla

If you are interested in giving a talk, please contact Prof. A.C. Cojocaru at: acojocar@fields.utoronto.ca

---

3) OPEN IMAGE THEOREMS FOR l-ADIC REPRESENTATIONS ASSOCIATED TO ELLIPTIC CURVES

This is a working seminar focusing on l-adic representations associated to elliptic curves. Elliptic curves (smooth curves of genus 1 with a fixed point) are fundamental objects in today's number theory. They posses very rich arithmetic and complex structure, and are subject to major open questions such as the Birch and Swinnerton-Dyer Conjecture and the Lang-Trotter Conjectures. Moreover, they have been used in cryptography starting with the 1980s and (most importantly) they played a crucial role in Wiles' celebrated proof of Fermat's Last Theorem from the mid 1990s.

Associated to an elliptic curve defined over the field of rational numbers (say) there is a family of l-adic Galois representations. In 1972, Serre proved that if the elliptic curve is without complex multiplication (the "generic" case), then each l-adic representation has image as large as possible provided that l is sufficiently large. The focus in this seminar is to understand what "sufficiently large" means in Serre's result, and in generalizations of Serre's result (due to Ribet) to l-adic representations associated to modular forms. We will attempt to survey works of Mazur, Serre, Kraus/Cojocaru, Masser and Wustholz, Duke, Cojocaru and Hall, Imin Chen, Darmon and Merel, Merel and Rebolledo-Hochart (based on Gross-Kudla).

Fri. Sep 22, 1 p.m. Fields Library	Igor Shparlisnki (Macquarie University, Australia) Sato-Tate, cyclicity and divisibility statistics for elliptic curves: vertically, horizontally and diagonally
Wed. Aug. 23, 1 p.m. Fields Library	Alina Cojocaru Uniform results related to Serre's Theorem for elliptic curves
Wed. Aug. 16, 1 p.m. **BA 6183	Alina Cojocaru Uniform versions of Serre's Theorem for elliptic curves
Wed. Aug. 9, 1 p.m. Fields Library	Alina Cojocaru More on effective versions of Serre's Theorem for elliptic curves I will continue the survey on ways of making Serre's Theorem effective.
Wed. July 26, 1 p.m. Room 210	Alina Cojocaru An effective version of Serre's Theorem for elliptic curve A celebrated result of Serre from 1972 asserts that if E/Q is an elliptic curve over Q without complex multiplication, then its associated mod l representation is surjective for any sufficiently large prime l. We will discuss how "sufficiently large" can be made effective in terms of the conductor of E. More precisely, we will explain the conditional (upon Riemann Hypothesis) approach given by Serre in 1981 ("Quelques applications du theoreme de densite de Chebotarev") and the uncoditional approach given by Kraus/Cojocaru. No knowledge from the first lecture is assumed, as the techniques to be discussed are now analytic.
Fri. July 21, 1 p.m. Fields Library	Liangyi Zhao , University of Toronto Large Sieve fo Square Moduli and Primes in Quadratic Progressions In recent joint works with Stephan Baier, we were able to improve the large sieve inequalities for square moduli. The result is better, in certain ranges, than all previously known results which were obtained both jointly and independently by Baier and myself. I shall speak about the history, heuristics and conjectures about this problem and the techniques that enabled us to obtain the new result. I will also talk about application which is an approximation to the n^2+1 problem. In that direction, I will also talk about a recent result of ours regarding primes in quadratic progressions on average.
Wed. July 19, 1 p.m Stewart Library	Alina Cojocaru Serre's Open Image Theorem for elliptic curves (a sketch of the proof)

Everybody is welcome to attend!

If you are intersted in speaking at the Seminar please contact: acojocar@fields.utoronto.ca

Back to top