# 2012 Fall

## November 01, Sadoughian, 13:00 @HU1012

 Reza Sadoughian (McMaster University) GANITA Seminar Thursday November 01 13:00 HU1012 Title: Rankin L-functions and the twisted Birch and Swinnerton-Dyer Conjecture Abstract: We use Rankin's method to evaluate the central critical value of the L-series attached to an elliptic curve E over Q and certain odd irreducible 2-dimensional Artin representations. As an application we use the result to study the twisted Birch and Swinnerton-Dyer conjecture. [ arXiv] 2012_11_01_Sadoughian_Notes 2012_11_01

## October 25, George, 14:00 @HU1012

 William George (University of Toronto) GANITA Seminar Thursday October 25 14:00 HU1012 Title: A Surface of Qi Cheng and Ming-Deh Huang With Applications to the ECDLP Abstract: I will investigate properties of an elliptic surface for which, by work of Qi Cheng and Ming-Deh Huang, if a positive density of the fibers have rank one then one can relate the ECDLP to certain lifting problems. I will show how by Rosen and Silverman's proof of certain cases of Nagao's Conjecture one can compute the generic rank of this surface and then I will look at some work of Helfgott towards computing the average root number of this surface. [ arXiv] 2012_10_25_George_Notes 2012_10_25

## September 27, Hanson, 14:00 @HU1012

 Brandon Hanson (University of Toronto) GANITA Seminar Thursday September 27 14:00 HU1012 Title: Monochromatic Sums and Products in Finite Fields Abstract: [ arXiv] 2012_09_27_Hanson_Notes 2012_09_27

## September 27, Mourtada, 13:00 @HU1012

 Mariam Mourtada (University of Toronto) GANITA Seminar Thursday September 27 13:00 HU1012 Title: TBA Abstract: [ arXiv] 2012_09_27_Mourtada_Notes 2012_09_27

## September 20, Burko, 14:00 @HU1012

 Robby Burko (University of Toronto) GANITA Seminar Thursday September 20 14:00 HU1012 Title: Precision Estimates for Kedlaya's Point-Counting Algorithm Applied to Certain Singular Curves over Finite Fields Abstract: [ arXiv] 2012_09_20_Burko_Notes 2012_09_20

## September 13, Seraj, 13:00 @HU1012

 Samer Seraj (University of Toronto) GANITA Seminar Thursday September 13 13:00 HU1012 Title: Sum of Cubes is Square of Sum Abstract: An intriguing mathematical fact about the first n natural numbers is that the sum of their cubes is the same as the square of their sum. We explore other sets with the same property by establishing definite answers to the question of finitude vs. infinitude of solutions for each n, and end by asking open problems. The material is from a paper written by Edward Barbeau and Samer Seraj. [ arXiv] 2012_09_13_Seraj_Notes 2012_09_13

## September 06, Lim, 14:00 @HU1012

 Meng Fai Lim (University of Toronto) GANITA Seminar Thursday September 06 14:00 HU1012 Title: Some remarks on the BSD conjecture and (p-)parity conjecture Abstract: We present some (minor) improvement to results on BSD conjecture and (p-)parity conjecture. Namely, we improve certain relativity results of Dokchitser-Dokchitser on the p-parity conjecture. We will also present a statement on the BSD conjecture which is a mild improvement of results of Kolyvagin, Kato and Abkary-Murty. The method used in is mainly group-theoretical. [ arXiv] 2012_09_06_Lim_Notes 2012_09_06

# 2012 Summer

## August 30, Zong, 14:00 @HU1012

 Ying Zong (University of Toronto) GANITA Seminar Thursday August 30 14:00 HU1012 Title: p-adic simpson correspondence, d'apres Faltings Abstract: I give an expose of Faltings' paper "a p-adic simpson correspondence". [ arXiv] 2012_08_30_Zong_Notes 2012_08_30

## August 23, Eskandari, 14:00 @HU1012

 Payman Eskandari (University of Toronto) GANITA Seminar Thursday August 23 14:00 HU1012 Title: A deformation of the free associative algebra on a set Abstract: [ arXiv] 2012_08_23_Eskandari_Notes 2012_08_23

## July 26, George, 14:00 @HU1012

 William George (University of Toronto) GANITA Seminar Thursday July 26 14:00 HU1012 Title: Root Numbers of Fibers of Elliptic Surface Abstract: I will look at some work of David Rohrlich concerning the behavior of the root number of fibers of certain elliptic surfaces. For certain surfaces, he shows that the variation of the root number is related to topological properties of the surface. The talk will focus on gaining a sense of for what one requires on a surface to have this kind of behavior. [ arXiv] 2012_07_26_George_Notes 2012_07_26

## July 19, Sastry, 15:00 @HU1012

 Pramathanath Sastry (Chennai Mathematical Institute) GANITA Seminar Thursday July 19 15:00 HU1012 Title: Abelian varieties and cryptogrpahy Abstract: [ arXiv] 2012_07_19_Sastry_Notes 2012_07_19

## July 19, Anghel, 14:00 @HU1012

 Catalina Anghel (University of Toronto) GANITA Seminar Thursday July 19 14:00 HU1012 Title: Results related to the self-power map Abstract: [ arXiv] 2012_07_19_Anghel_Notes 2012_07_19

## July 12, Sastry, 15:00 @HU1012

 Pramathanath Sastry (Chennai Mathematical Institute) GANITA Seminar Thursday July 12 15:00 HU1012 Title: Abelian varieties and cryptogrpahy Abstract: [ arXiv] 2012_07_12_Sastry_Notes 2012_07_12

## July 12, Burko, 14:00 @HU1012

 Robby Burko (University of Toronto) GANITA Seminar Thursday July 12 14:00 HU1012 Title: p-adic methods in computing zeta functions of singular curves over finite fields Abstract: [ arXiv] 2012_07_12_Burko_Notes 2012_07_12

## July 05, Marie, 15:00 @HU1012

 Vincent Marie (ENSICAEN) GANITA Seminar Thursday July 05 15:00 HU1012 Title: Implementation of a payment protocol preserving privacy Abstract: [ arXiv] 2012_07_05_Marie_Notes 2012_07_05

## July 05, Khachlouf, 14:00 @HU1012

 Mejdi Khachlouf (ENSICAEN) GANITA Seminar Thursday July 05 14:00 HU1012 Title: Enhancing privacy in the e-health information system Abstract: Numerous country-specific infrastructures for medical records are emerging. The critical impact of these sensitive data on the privacy of patients gives rise to many standardization activities. Security of such medical information systems has attracted a major attention due to the large amount of data shared by several actors, such as doctors or nurses in different institutions. However, the privacy preserving issue is at most partially treated. Concepts of data minimisation and data sovereignty for medical data are often neglected. The proposed architecture presents an e-health infrastructure intended to minimise personal data disclosure and data aggregation, whereas data sovereignty principle is realized in accordance with medical constraints. [ arXiv] 2012_07_05_Khachlouf_Notes 2012_07_05

## June 28, Murty, 15:00 @HU1012

 Kumar Murty (University of Toronto) GANITA Seminar Thursday June 28 15:00 HU1012 Title: On Bounded Generation Abstract: [ arXiv] 2012_06_28_Murty_Notes 2012_06_28

## June 28, Lim, 14:00 @HU1012

 Meng Fai Lim (University of Toronto) GANITA Seminar Thursday June 28 14:00 HU1012 Title: Growth of p-rank of Selmer groups of elliptic curves with split multiplicative reduction Abstract: This is a continuation of the talk given in Ganita seminar on 2012.04.26. Let p be a fixed prime. Suppose that E is an elliptic curve with split multiplicative reduction at a prime, and that the Tamagawa factor at the split multiplicative prime is divisible by p. In this talk, we present some results on the growth of Selmer p-rank of the above class of elliptic curves in certain p-adic Lie extension. This is a joint work with Prof. V. Kumar Murty. [ arXiv] 2012_06_28_Lim_Notes 2012_06_28

## June 21, Mitra, 15:00 @HU1012

 Mahan Mitra (Ramakrishna Mission Vivekananda University) GANITA Seminar Thursday June 21 15:00 HU1012 Title: Unbounded quasi-morphisms and bounded generation (part 2) Abstract: [ arXiv] 2012_06_21_Mitra_Notes 2012_06_21

## June 21, Murty, 14:00 @HU1012

 Kumar Murty (University of Toronto) GANITA Seminar Thursday June 21 14:00 HU1012 Title: The error term in the Sato-Tate conjecture (part 2) Abstract: [ arXiv] 2012_06_21_Murty_Notes 2012_06_21

## June 14, Eskandari, 16:00 @HU1012

 Payman Eskandari (University of Toronto) GANITA Seminar Thursday June 14 16:00 HU1012 Title: TBA Abstract: [ arXiv] 2012_06_14_Eskandari_Notes 2012_06_14

## June 14, Murty, 15:20 @HU1012

 Kumar Murty (University of Toronto) GANITA Seminar Thursday June 14 15:20 HU1012 Title: The error term in the Sato-Tate conjecture Abstract: [ arXiv] 2012_06_14_Murty_Notes 2012_06_14

## June 14, Mitra, 14:40 @HU1012

 Mahan Mitra (Ramakrishna Mission Vivekananda University) GANITA Seminar Thursday June 14 14:40 HU1012 Title: Unbounded quasi-morphisms and bounded generation Abstract: [ arXiv] 2012_06_14_Mitra_Notes 2012_06_14

## May 03, Lim, 14:00 @HU1012

 Meng Fai Lim (University of Toronto) GANITA Seminar Thursday May 03 14:00 HU1012 Title: Growth of p-rank of Selmer groups of elliptic curves with split multiplicative reduction Abstract: This is a joint work with Prof. V. Kumar Murty. We present a result on the growth of Selmer p-rank of elliptic curves with split multiplicative reduction in an infinite Hilbert p-class tower. We are namely interested in the case when the Tamagawa factor at the split multiplicativ prime is divisible by p. We will show that under certain assumptions, the p-rank is unbounded. This result will complement the case for CM-abelian varieties considered by us, and also the result of Matar. The idea of the proof revolves around the techniques occuring in the theme of the so-called "Mazur Control Theorem". We will use a slight variant of this as considered by Matsuno. [ arXiv] 2012_05_03_Lim_Notes 2012_05_03

## April 26, George, 14:00 @HU1012

 William George (University of Toronto) GANITA Seminar Thursday April 26 14:00 HU1012 Title: Finding Quadratic Residue Fibres as Outputs in an Algorithm Abstract: [ arXiv] 2012_04_26_George_Notes 2012_04_26

# 2012 Winter

## March 08, Burko, 14:00 @HU1012

 Robby Burko (University of Toronto) GANITA Seminar Thursday March 08 14:00 HU1012 Title: Vanishing of Log-Cohomology Abstract: [ arXiv] 2012_03_08_Burko_Notes 2012_03_08

## March 01, Zaytseva, 14:00 @HU1012

 Anastasia Zaytseva (University of Toronto) GANITA Seminar Thursday March 01 14:00 HU1012 Title: Generalized Gamma function Abstract: "The Bohr-Mollerup theorem states that $\log{\Gamma(x)}$" is the only eventially convex function which satisfies the functional equation $f(x+1)-f(x)=\log{x}$ for $x>0$ with the additional condition $f(1)=0$. It is natural to consider higher analogues of this function, R_k(x), defined as the eventially convex solution of the equation $f(x+1)-f(x)=\log^k{x}$, for $x>0$, $k\in\mathbb{Z}^{+}$ with some initial condition. The case $k=2$ was studied by Deninger in 1983 in his paper "On the analogue of the formula of Chowla and Selberg". In particular, he showed connection between $R_2(x)$ and the Hurwitz zeta function and used it to obtain several relations for $R_2(x)$. In my talk I will discuss Deninger's work and further properties of the functions $R_k(x)$. [ arXiv] 2012_03_01_Zaytseva_Notes 2012_03_01

## February 16, Eskandari, 14:00 @HU1012

 Payman Eskandari (University of Toronto) GANITA Seminar Thursday February 16 14:00 HU1012 Title: Motives as systems of realizations Abstract: [ arXiv] 2012_02_16_Eskandari_Notes 2012_02_16

## February 09, George, 14:00 @HU1012

 William George (University of Toronto) GANITA Seminar Thursday February 09 14:00 HU1012 Title: Finding Rank One Lifts and the Elliptic Curve Discrete Log Problem Abstract: I will discuss my attempts to find desirable lifts of elliptic curves with the aim of fulfilling a hypothesis of Miri and Murty that would show the equivalence of ECDLP to a lifting problem. I will then compare this to related work done by Cheng and Huang in 2006. [ arXiv] 2012_02_09_George_Notes 2012_02_09

## January 26, Chow, 15:00 @HU1012

 Aaron Chow (University of Toronto) GANITA Seminar Thursday January 26 15:00 HU1012 Title: Primality Testing via Modular Forms Abstract: This is an informal discussion about work in progress on primality testing using Fourier coefficients of modular forms. [ arXiv] 2012_01_26_Chow_Notes 2012_01_26

## January 26, Burko, 14:00 @HU1012

 Robby Burko (University of Toronto) GANITA Seminar Thursday January 26 14:00 HU1012 Title: TBA Abstract: [ arXiv] 2012_01_26_Burko_Notes 2012_01_26

## January 19, Zaytseva, 14:00 @HU1012

 Anastasia Zaytseva (Queen's University) GANITA Seminar Thursday January 19 14:00 HU1012 Title: On the analogue of a Kronecker Limit Formula Abstract: Let $Q(x,y)=ax^2+bxy+cy^2$ be a definite quadratic form such that $a,c>0, b^2-4ac=1$. The Epstein zeta-function is defined as $K(Q,s)=\sum_{m,n}\frac{1}{Q(m,n)^s}$ for $Re(s)>1$. The summation here is over all integers $(m,n)\neq (0,0)$. Kronecker showed that $K(Q,s)$ can be extended meromorphically to the neighborhood $s=1$ with the only pole at $s=1$. The Kronecker's limit formula gives the first two coefficients in the Laurent expansion of $K(Q,s)$ at $s=1$. This important formula can applied to study the properties of the Dedekind zeta function of quadratic fields. In 1975, Zagier proved the analogue of the Kronecker's Theorem for the function $Z(Q,s)=\sum_{m>0,n>-1}\frac{1}{Q(m,n)^s},$ where $Q(x,y)=ax^2+bxy+cy^2$ is indefinite quadratic form with conditions $a,b,c>0 , b^2-4ac=1$. In this talk we will discuss the theorem's of Kronecker and Zagier and their connection with the Dedekind zeta function. [ arXiv] 2012_01_19_Zaytseva_Notes 2012_01_19

# 2011 Fall

## November 24, Shahrokhi Tehrani, 14:00 @HU1012

 Shervin Shahrokhi Tehrani (University of Toronto) GANITA Seminar Thursday November 24 14:00 HU1012 Title: The Geometry of Siegel Modular Forms Abstract: Local system, on the moduli space of abelian surfaces, are locally constant sheaves that obtained by action of GSp(4) on a finite dimensional vector spaces. We will consider the cohomology of these kinds of bundles and make connection to theory of Siegel automorphic forms of genus two AND theta correspondence. [ arXiv] 2011_11_24_Shahrokhi Tehrani_Notes 2011_11_24

## November 10, George, 15:00 @HU1012

 William George (University of Toronto) GANITA Seminar Thursday November 10 15:00 HU1012 Title: Primes in Elliptic Divisibility Sequences Abstract: I will look at some results on the primes that appear in the denominator of the multiples of a generator for a rank one curve that Poonen uses to show topological properties of certain sets of S-integers points on the curve. I will then look at a refinement of some of these ideas developed by Eisentrager and Everest. [ arXiv] 2011_11_10_George_Notes 2011_11_10

## November 10, Anghel, 14:00 @HU1012

 Catalina Anghel (University of Toronto) GANITA Seminar Thursday November 10 14:00 HU1012 Title: Lower Bounds for Discrete Logarithms and Related Problems Abstract: [ arXiv] 2011_11_10_Anghel_Notes 2011_11_10

## November 03, Chow, 15:00 @HU1012

 Aaron Chow (University of Toronto) GANITA Seminar Thursday November 03 15:00 HU1012 Title: The Rank One Lifting Problem and the AKS Algorithm Abstract: This will be a survey of two papers. The first one is by Miri and Murty on the equivalence of the lifting problem for elliptic curves of rank one and the discrete logarithm problem. The second paper is by Agrawal, Kayal, and Saxena on their primality testing algorithm. [ arXiv] 2011_11_03_Chow_Notes 2011_11_03

## November 03, Eskandari, 14:00 @HU1012

 Payman Eskandari (University of Toronto) GANITA Seminar Thursday November 03 14:00 HU1012 Title: The algebra of multiple zeta values Abstract: This will be an introductory talk on the algebra of multiple zeta values. The first part of the talk will be devoted to defining multiple zeta values and their algebra and some elementary observations such as writing them as integrals. In the second part I will give a conjectural description of the algebra of multiple zeta values. [ arXiv] 2011_11_03_Eskandari_Notes 2011_11_03

## October 27, Venkatesan, 14:00 @HU1012

 Ramarathnam Venkatesan (Microsoft Research) GANITA Seminar Thursday October 27 14:00 HU1012 Title: Non Abelian Analogues of Lattice Rounding Abstract: [ arXiv] 2011_10_27_Venkatesan_Notes 2011_10_27

## October 13, Burko, 15:00 @HU1012

 Robby Burko (University of Toronto) GANITA Seminar Thursday October 13 15:00 HU1012 Title: Limiting Frobenius Structures and the Legendre family of Elliptic curves Abstract: [ arXiv] 2011_10_13_Burko_Notes 2011_10_13

## October 13, Zaytseva, 14:00 @HU1012

 Anastasia Zaytseva (Queen's University) GANITA Seminar Thursday October 13 14:00 HU1012 Title: On the arithmetic properties of the incomplete Gamma functions Abstract: The arithmetic properties of the Gamma function at rational points is an open problem in Number Theory. Apart from very few classical results, we still don't know whether \Gamma(p/q) is transcendental or even irrational. I will show how the theory of E-functions may be used to study the arithmetic nature of the incomplete Gamma functions, which are functions closely related to the Gamma function. This is a joint work with Ram Murty. [ arXiv] 2011_10_13_Zaytseva_Notes 2011_10_13

## October 06, George, 15:00 @HU1012

 William George (University of Toronto) GANITA Seminar Thursday October 06 15:00 HU1012 Title: A Conjecture of Mazur for Elliptic Surfaces and a Decision Problem Abstract: I will look at a conjecture of Mazur regarding, for any given elliptic surface, the topology properties of the set of fibers with positive rank. In particular, I will look at some known special cases and the implications of the conjecture to relating certain decision problems. [ arXiv] 2011_10_06_George_Notes 2011_10_06

## October 06, Tehrani, 14:00 @HU1012

 Shervin Shahrokhi Tehrani (University of Toronto) GANITA Seminar Thursday October 06 14:00 HU1012 Title: Local Systems Abstract: I am going to talk about two different kind of bundles on the moduli spaces of abelian varieties. The Local systems have important applications in arithmetic geometry. [ arXiv] 2011_10_06_Tehrani_Notes 2011_10_06

## September 29, Anghel, 15:00 @HU1012

 Catalina Anghel (University of Toronto) GANITA Seminar Thursday September 29 15:00 HU1012 Title: ' Abstract: [ arXiv] 2011_09_29_Anghel_Notes 2011_09_29

## September 29, Burko, 14:00 @HU1012

 Robby Burko (University of Toronto) GANITA Seminar Thursday September 29 14:00 HU1012 Title: ' Abstract: [ arXiv] 2011_09_29_Burko_Notes 2011_09_29

## September 22, Lim, 14:00 @HU1012

 Meng Fai Lim (University of Toronto) GANITA Seminar Thursday September 22 14:00 HU1012 Title: ' Abstract: [ arXiv] 2011_09_22_Lim_Notes 2011_09_22

## September 15, Virag, 13:00 @HU1012

 Balint Virag (University of Toronto) GANITA Seminar Thursday September 15 13:00 HU1012 Title: Limits of random matrices and the Brownian Carousel Abstract: The Montgomery conjecture relates the eigenvalues of random matrices to zeros of the Riemann zeta function. I will describe a new representation of limits of random matrices in terms of a random self-adjoint operator (called the Brownian carousel) based on Brownian motion on $\text{SL}_2(\mathbb{R})$. This can be used to make conjectures about what the Riemann zeta function itself looks like near a random location on the critical line. [ arXiv] 2011_09_15_Virag_Notes 2011_09_15

# 2011 Summer

## August 23, Chow, 10:00 @HU1012

 Aaron Chow (University of Toronto) GANITA Seminar Tuesday August 23 10:00 HU1012 Title: eHMAC: A side-channel resistant HMAC proposal Abstract: A keyed message authentication code (MAC) protects a messages integrity and authenticity with a secret key. The standard way of constructing a MAC from a cryptographic (block-based) hash function is the HMAC construction (RFC 2104). However, there are side-channel attacks that allow forgery of HMAC's based on SHA-1 and SHA-2, and more recently SHA-3 candidates. In this talk, HMAC is extended to resist these side-channel attacks, and its security is discussed. [ arXiv] 2011_08_23_Chow_Notes 2011_08_23

## August 16, Milcak, 11:00 @HU1012

 Juraj Milcak (University of Toronto) GANITA Seminar Tuesday August 16 11:00 HU1012 Title: On the Lonely Runner Conjecture Abstract: The Lonely Runner Conjecture was first stated by J. M. Wills in 1967 and independently by T. W. Cusick in 1974. The conjecture is the very simple assertion that for any set $V$ of $k-1$ positive integers there exists some $\alpha \in \mathbb{R}$ such that for all $v \in V$, $\Vert \alpha v \Vert \geq \dfrac{1}{k}$, where $\Vert \cdot \Vert$ denotes the distance to the nearest integer. The conjecture has been proved for $k \le 7$, using a wide variety of methods. In this talk we will discuss some of the approaches taken to prove the above cases. The case $k=2$ is trivial; $k=3$ has been settled by Betke and Wills using Diophantine approximation methods, as well as by Cusick by solving a related "view-obstruction" problem. We explore a new constructive solution for this case. The proofs for $k=4,5,6$ follow a case by case approach and will not be discussed. We further investigate the most recent proof for $k=7$. Finally, we will show that the proposed bound is always tight for any $k$, and prove the conjecture when the set of integers $V$ forms some particular types of sequences. [ arXiv] 2011_08_16_Milcak_Notes 2011_08_16

## August 09, David, 10:00 @HU1012

 Chantal David http://www.mathstat.concordia.ca/faculty/cdavid/ (Concordia University) GANITA Seminar Tuesday August 09 10:00 HU1012 Title: Fluctuations in the number of points of curves over finite fields Abstract: We study in this talk the distribution of the number of points for two families of curves over a finite field with q elements: cyclic covers of $P^1$ and smooth plane curves. The Katz-Sarnak philosophy makes predictions about the statistics for such families in the large q limit when the genus is fixed. We are looking at the complementary statistics, when the genus varies, but the field of definition is fixed. In that case, one can obtain statistics for the distribution of the number of points by sieving the families of curves.This is joint work with A. Bucur, B. Feigon and M. Lalin. [ arXiv] 2011_08_09_David_Notes 2011_08_09

## July 19, Shakoori, 10:00 @BA6180

 Azar Shakoori http://azarshakoori.com/ (University of Cantabria) GANITA Seminar Tuesday July 19 10:00 BA6180 Title: Polynomial Algebra by Values and Its Applications Abstract: The polynomial root-finding problem has a long and rich history in computational math- ematics, mainly because of its frequent applications in engineering. It may be the case that the polynomial system as it appears naturally in the application formulation is not pre- sented in the familiar monomial basis. In this talk, I will present our root-finding algorithm, a hybrid of a resultant-type al- gorithm and eigenvalue techniques. Here, we use some new formulations for the so-called B ´ezout matrix and the companion matrix pencil associated with a pair of bivariate poly- nomial equations described by their values (i.e., in the Lagrange polynomial basis). The approach we take avoids conversion between bases. This is important because converting from the basis in which the polynomials are given to the monomial basis (or any other basis) is likely to degrade the conditioning of the problem. I will also describe some of the applications of our algorithm; one such application in Computer-Aided-Geometric-Design is to compute the topology of an algebraic curve whose parametric or implicit equations have rather high degrees, numerous terms and very large coefficients. Applying the currently available methods for computing the topology of the curve, with such polynomial equations, requires a big amount of memory space and a significant amount of computing time. This is one of the main reasons why, in this work, we assume that neither the implicit equation, nor the parametrization of the given curve are known. Instead, sufficient number of points on the curve and the corresponding values of the equations and derivatives are known, and the degree is also given or can be deduced. [ arXiv] 2011_07_19_Shakoori_Notes 2011_07_19

## July 12, Kahrobaei, 10:00 @HU1012

 Delaram Kahrobaei (City University of New York) GANITA Seminar Tuesday July 12 10:00 HU1012 Title: A decade of using non-commutative groups in cryptography Abstract: [ arXiv] 2011_07_12_Kahrobaei_Notes 2011_07_12

## July 05, Guerreiro, 10:00 @HU1012

 Marinês Guerreiro [1] (Universidade Federal de Vicosa (Brasil)) GANITA Seminar Tuesday July 05 10:00 HU1012 Title: Computing weights in abelian group codes Abstract: In this talk we propose a technique to compute idempotents that generate minimal abelian codes in group algebras of abelian groups of exponent pq, for p and q distinct odd prime numbers over a field of characteristic two, under some restrictions. We also present a way to compute the weight of the minimal codes. The main results extend previous work done by C. Polcino Milies and R.A. Ferraz. This is a joint work with R.A. Ferraz, C. P. Milies and G. Chalom from IME-USP (São Paulo). [ arXiv] 2011_07_05_Guerreiro_Notes 2011_07_05

## June 23, Anghel, 11:00 @HU1012

 Catalina Anghel [2] (University of Toronto) GANITA Seminar Thursday June 23 11:00 HU1012 Title: On the number of solutions of exponential congruences Abstract: [ arXiv] 2011_06_23_Anghel_Notes 2011_06_23

## June 23, Migonney, 10:00 @HU1012

 Coline Migonney [3] (Ensicaen) GANITA Seminar Thursday June 23 10:00 HU1012 Title: Concepts for the development of a secure & privacy e-health platform Abstract: [ arXiv] 2011_06_23_Migonney_Notes 2011_06_23

## June 21, Plateaux, 13:00 @HU1012

 Aude Plateaux [4] (Ensicaen) GANITA Seminar Tuesday June 21 13:00 HU1012 Title: Notion of privacy and security within the secure electronic transactions Abstract: [ arXiv] 2011_06_21_Plateaux_Notes 2011_06_21

## June 16, Burko, 11:00 @HU1012

 Robby Burko [5] (University of Toronto) GANITA Seminar Thursday June 16 11:00 HU1012 Title: A Method for Computing Zeta Functions of Varieties over Finite Fields using Fibrations Abstract: In 2006 Lauder extended his p-adic deformation method to include fibrations of certain varieties. The idea is to treat an n-dimensional variety as a parametrized family of (n-1)-dimensional subvarieties, and compute the action of Frobenius on this family. This reduces complexity of the original deformation method, since it only deals with the (n-1) level cohomology, however there are some restrictions on the types of varieties which can be used in this method. These ideas were generalized in G.M. Walker's Thesis in 2009. [ arXiv] 2011_06_16_Burko_Notes 2011_06_16

## June 16, Stein, 10:00 @HU1012

 Andreas Stein [6] (University of Oldenburg) GANITA Seminar Thursday June 16 10:00 HU1012 Title: Ellipic Curves Public-Key Cryptography - Applications and Attacks Abstract: In recent years, elliptic curves have become objects of intense investigation because of their significance to public-key cryptography. The major advantage of ECC is that the cryptographic security is in general believed to grow exponentially with the length of the input parameters. This implies short parameters, short digital signatures, and fast computations. We provide a survey of elliptic curves over finite fields and their interactions with algorithmic number theory. Our main focus will be the discussion of various interesting attacks to the so-called elliptic curve discrete logarithm problem (ECDLP) and their mathematical background as well as their important impact on public-key cryptography. For several attacks, results on algebraic curves, especially hyperelliptic curves, are needed. [ arXiv] 2011_06_16_Stein_Notes 2011_06_16

## June 09, Mourtada, 10:00 @HU1012

 Mariam Mourtada [7] (University of Toronto) GANITA Seminar Thursday June 09 10:00 HU1012 Title: Average over fundamental discriminants of $L'/L(\sigma,\chi_D), 1/2 < \sigma < 1$. Abstract: This is a subsequent work following the average over fundamental discriminants of $L'/L(1,\chi_D)$. We will show under GRH that this average is a constant for all $\sigma$ . Then we will look at a specific set of $\sigma$ that has a positive measure, and show that the average is still a constant, unconditionally. [ arXiv] 2011_06_09_Mourtada_Notes 2011_06_09

## June 02, George, 13:00 @HU1012

 William George [8] (University of Toronto) GANITA Seminar Thursday June 02 13:00 HU1012 Title: Some Consequences of Mazur's Conjecture Abstract: I will explore some of the ideas related to Mazur's Conjecture on the topological closure (in the real topology) of the rational points on varieties. Particularly, I will look at its implications for Diophantine decidability and how it affects the specialized ranks on elliptic surfaces. [ arXiv] 2011_06_02_George_Notes 2011_06_02

## May 26, Mameri, 13:00 @HU1012

 Sabrina Mameri [9] (Ensicaen) GANITA Seminar Thursday May 26 13:00 HU1012 Title: Zero-knowledge identification protocols Abstract: [ arXiv] 2011_05_26_Mameri_Notes 2011_05_26

# 2011 Winter

## April 21, Volkovs, 13:00 @HU1012

 Nikolajs Volkovs (PRATA) GANITA Seminar Thursday April 21 13:00 HU1012 Title: Digital Signature Scheme with a generator kept secret Abstract: We present the digital signature algorithm where a generator should not be used for verifying a signature, and therefore it can be kept secret. Furthermore, we show that the problem of extracting a private key from a public key is a new problem, different from the DLP. [ arXiv] 2011_04_21_Volkovs_Notes 2011_04_21

## April 14, Zong, 13:00 @HU1012

 Ying Zong [10] (University of Toronto) GANITA Seminar Thursday April 14 13:00 HU1012 Title: Elliptic Cartan subalgebra Abstract: Given a semi-simple Lie algebra s and an irreducible representation E of s, over a field k, does s have a Cartan sub-algebra acting irreducibly on E? With Kumar Murty, we give a solution to this question when k is a p-adic field, and apply it to a problem of splitting abelian varieties. [ arXiv] 2011_04_14_Zong_Notes 2011_04_14

## March 31, Chow, 13:00 @HU1012

 Aaron Chow [11] (University of Toronto) GANITA Seminar Thursday March 31 13:00 HU1012 Title: Side-Channel Attacks on SHA-3 Candidates (Part 2) Abstract: This talk is a follow up on a side-channel attack using CPA against HMACs based on the current SHA-3 candidates. [ arXiv] 2011_03_31_Chow_Notes 2011_03_31

## March 24, Lim, 13:00 @HU1012

 Meng Fai Lim [12] (University of Toronto) GANITA Seminar Thursday March 24 13:00 HU1012 Title: The growth of the Tate-Shafarevich group in p-Hilbert class field towers Abstract: The existence of an infinite p-Hilbert class field tower was first established by Golod and Shaferevich in 1964. Stark then asked whether the p-class rank of the layers in an infinite p-class field tower tends to infinity. It turns out that this should be the case by a conjecture of Fontaine and Mazur. In a paper of Murty and Ouyang, they consider an analog of Stark's question and show that for a certain class of elliptic curves with complex multiplication, its p-rank of the Selmer group in the p-class field tower tends to infinity. In this talk, we will consider another analog of the above problems, namely, we will show that, under certain assumptions on p, the p-rank of the Tate-Shafarevich group of an abelian variety with complex multiplication in the infinite p-class field tower tends to infinity. This is a joint work with Prof. Kumar Murty. [ arXiv] 2011_03_24_Lim_Notes 2011_03_24

## March 17, Hanson, 13:00 @HU1012

 Brandon Hanson [13] (University of Toronto) GANITA Seminar Thursday March 17 13:00 HU1012 Title: Probabilistic reasoning and the number of prime divisors Abstract: The function $\omega(n)$ which counts the number of distinct prime divisors has been attacked in the past with some nice probabilistic arguments. The theorem of Hardy-Ramanujan and Turan gives an analog of Law of Large Numbers, that $\omega(n)$ has normal order $\log\log n$. Even more interesting, the theorem of Erdos-Kac tells us that $\omega(n)$ is $\log\log n$ + a normal random variable. We'll discuss these results. [ arXiv] 2011_03_17_Hanson_Notes 2011_03_17

## March 10, George, 13:00 @HU1012

 William George [14] (University of Toronto) GANITA Seminar Thursday March 10 13:00 HU1012 Title: The Bivariate Polynomial Injection Problem Abstract: One can find examples of polynomials in two variables with integer coefficients that yield an injection from $\mathbb{Z}^2$ to $\mathbb{Z}$; however, it is surprisingly difficult to determine whether there is such a polynomial injection from $\mathbb{Q}^2$ to $\mathbb{Q}$. Indeed, while it is widely believed that such a polynomial should exist, the best results towards this question reduce the problem to an analogue of the abc conjecture or, via a recent result of Bjorn Poonen, to the Bombieri-Lang conjecture. We will look at these results and talk about how this problem relates to issues of Diophantine definability. [ arXiv] 2011_03_10_George_Notes 2011_03_10

## March 03, Burko, 14:00 @HU1012

 Robby Burko [15] (University of Toronto) GANITA Seminar Thursday March 03 14:00 HU1012 Title: On various Gauss-Manin connections Abstract: [ arXiv] 2011_03_03_Burko_Notes 2011_03_03

## March 03, Eskandari, 13:00 @HU1012

 Payman Eskandari [16] (University of Toronto) GANITA Seminar Thursday March 03 13:00 HU1012 Title: Small gaps between primes: A result of Goldstein, Pintz and Yildirim Abstract: The difference between two consecutive primes has been interesting mathematicians since the ancient times. The most famous unsolved problem in this direction is the twin prime conjecture, which can be stated as $\liminf_{r} p_{r+1}-p_r =2$, where $p_r$ denotes the rth prime. Let $N$ be a large number. By the prime number theorem, there are about $N/\log N$ primes between $N$ and $2N$, so the average gap between two consecutive primes in this range is $\log N$, which is almost $log p$ for any prime in the range. In this sense, the gap $p_{r+1}-p_r$ is in average about $\log p_r$. This motivates the study of $\liminf_{r}\frac{p_{r+1}-p_r}{\log p_r}$. It was first Erdos that proved (unconditionally) that this $\liminf$ is less than $1$, and several mathematicians including Ricci, Bombieri and Davenport improved the upper bound for this $\liminf$. In 2006, finally it was proved, by Goldston, Pintz and Yildirim that this $\liminf$ is indeed zero. My goal is to give a sketch of the proof of GPY result and discuss some of its ideas. Time permitting, I might also briefly discuss some parts of Soundararajan's review paper and Tao's blog article on the result. [ arXiv] 2011_03_03_Eskandari_Notes 2011_03_03

## February 10, Laptyeva, 13:00 @HU1012

 Nataliya Laptyeva [17] (University of Toronto) GANITA Seminar Thursday February 10 13:00 HU1012 Title: On the Brun-Titchmarch Theorem Abstract: [ arXiv] 2011_02_10_Laptyeva_Notes 2011_02_10

## February 03, Mourtada, 13:00 @HU1012

 Mariam Mourtada [18] (University of Toronto) GANITA Seminar Thursday February 03 13:00 HU1012 Title: Nonvanishing of quadraric Dirichlet L-functions at $s=\frac{1}{2}$ Abstract: Soundararajan proved that for at least 87.5% of odd-squarefree integers $d$, we have $L(\frac{1}{2},\chi_{8d}) \neq 0$ . This striking proportion of nonvanishing has been proved through the introduction of a "Mollifier". Historically, this was used by Selberg when he proved that a positive proportion of zeros of the Riemann Zeta function lie on the critical line. In my talk, I will state and prove some of the lemmas involved, in order to conclude the theorem. One thing to know, is that the central point is not special here, a similar result can be proved for any point $\sigma+it$ in the critical strip, and for fundamental discriminants in any arithmetic progression $a \pmod{b}$. [ arXiv] 2011_02_03_Mourtada_Notes 2011_02_03

## January 27, Chow, 13:00 @HU1012

 Aaron Chow [19] (University of Toronto) GANITA Seminar Thursday January 27 13:00 HU1012 Title: Side-Channel Attacks on SHA-3 Candidates Abstract: A message authentication code (MAC) is used to authenticate a message with a secret key. It protects both a message's integrity and authenticity. A hash-based message authentication code (HMAC) is one based on a hash function. In this talk, a side-channel attack using CPA on HMACs based on the current SHA-3 candidates will be discussed. This has an impact especially on hardware-based implementations. [ arXiv] 2011_01_27_Chow_Notes 2011_01_27

## January 20, Anghel, 14:00 @HU1012

 Catalina Anghel [20] (University of Toronto) GANITA Seminar Thursday January 20 14:00 HU1012 Title: Proof of one of Bourgain's theorems on exponential sums Abstract: [ arXiv] 2011_01_20_Anghel_Notes 2011_01_20

## January 20, Usefi, 13:00 @HU1012

 Hamid Usefi [21] (University of Toronto) GANITA Seminar Thursday January 20 13:00 HU1012 Title: Algebraic Eraser and Non-Abelian Group-Based Cryptography Abstract: [ arXiv] 2011_01_20_Usefi_Notes 2011_01_20