Ki-Seng Tan

Lifting Elliptic Curves and Solving the Elliptic Curve Discrete Logarithm Problem

Essentially all subexponential time algorithms for the discrete logarithm problem over finite fields are based on the index calculus idea. In proposing cryptosystems based on the elliptic curve discrete logarithm problem (ECDLP) Miller [6] also gave heuristic reasoning as to why the index calculus idea may not extend to solve the analogous problem on elliptic curves. A careful analysis by Silverman and Suzuki provides strong theoretical and numerical evidence in support of Miller’s arguments. An alternative approach recently proposed by Silverman, dubbed ‘xedni calculus’, for attacking the ECDLP was also shown unlikely to work asymptotically by Silverman himself and others in a subsequent analysis. The results in this paper strengthen the observations of Miller, Silverman and others by deriving necessary but difficult-to-satisfy conditions for index-calculus type of methods to solve the ECDLP in subexponential time. Our analysis highlights the fundamental obstruction as being the necessity to lift an asymptotically increasing number of random points on an elliptic curve over a finite field to rational points of reasonably bounded height on an elliptic curve over ℚ. This difficulty is underscored by the fact that a method that meets the requirement implies, by virtue of a theorem we prove, a method for constructing elliptic curves over ℚ of arbitrarily large rank.

A generalized Mazur's theorem and its applications

We generalize a theorem of Mazur concerning the universal norms of an abelian variety over a ℤ d p -extension of a complete local field. Then we apply it to the proof of a control theorem for abelian varieties over global function fields.

Deciding finiteness for matrix groups over function fields

LetF be a field andt an indeterminate. In this paper we consider aspects of the problem of deciding if a finitely generated subgroup of GL(n,F(t)) is finite. WhenF is a number field, the analysis may be easily reduced to deciding finiteness for subgroups of GL(n,F), for which the results of [1] can be applied. WhenF is a finite field, the situation is more subtle. In this case our main results are a structure theorem generalizing a theorem of Weil and upper bounds on the size of a finite subgroup generated by a fixed number of generators with examples of constructions almost achieving the bounds. We use these results to then give exponential deterministic algorithms for deciding finiteness as well as some preliminary results towards more efficient randomized algorithms.

A note on the order of finite subgroups of GL(n, ℤ)

A simple upper bound on the size of nite subgroups of GL(n; Z) is given. Only elementary number-theoretic arguments are used, thereby avoiding previous methods depending on estimates of the Jordan number and the classiication of nite simple groups.

The Iwasawa Main Conjecture for constant ordinary abelian varieties over function fields

We study a geometric analogue of the Iwasawa Main Conjecture for constant ordinary abelian varieties over

Generalized Stark formulae over function fields

\ZZ_p^d

Nonlinear approximation theory on compact groups

-extensions of function fields ramifying at a finite set of places.

Twists of matrix algebras and Brauer groups

We establish formulae of Stark type for the Stickelberger elements in the function field setting. Our result generalizes a work of Hayes and a conjecture of Gross. It is used to deduce a p-adic version of Rubin-Stark Conjecture and Burns Conjecture.

Integral points on elliptic curves over function fields

In this article we extend to the setting of band-limited functions on compact groups previous results bounding from below the percentage of energy, contained in the low frequency portion of the spectrum of a positive function defined on a cyclic group. Connections to signal recovery for positive functions, as well as partial spectral analysis, are also discussed.

On Some Brauer Subgroups Arising From Twists of Matrix Algebras

Abstract We study the twists of matrix algebras by some continuous characters which are regarded as 1-cocycles. Using results of Tate and Brauer-Hasse-Noether, we show that the whole Brauer group of a number field can be obtained by such twists of matrix algebras.