V. Kumar Murty

Non-vanishing of L-functions and applications

Award-winning monograph of the Ferran Sunyer i Balagure Prize 1996. This book systematically develops some methods for proving the non-vanishing of certain L-functions at points in the critical strip. Researchers in number theory, graduate students who wish to enter into the area and non-specialists who wish to acquire an introduction to the subject will benefit by a study of this book. One of the most attractive features of the monograph is that it begins at a very basic level and quickly develops enough aspects of the theory to bring the reader to a point the latest discoveries as are presented in the final chapters can be fully appreciated.

Refinements of Miller's algorithm for computing the Weil/Tate pairing

The efficient computation of the Weil and Tate pairings is of significant interest in the implementation of certain recently developed cryptographic protocols. The standard method of such computations has been the Miller algorithm. Three refinements to Millers algorithm are given in this work. The first refinement is an overall improvement. If the binary expansion of the involved integer has relatively high Hamming weight, the second improvement suggested shows significant gains. The third improvement is especially efficient when the underlying elliptic curve is over a finite field of characteristic three, which is a case of particular cryptographic interest. Comment on the performance analysis and characteristics of the refinements are given.

An Application of Sieve Methods to Elliptic Curves

Let E be an elliptic curve defined over the rationals. Koblitz conjectured that the number of primes p ? x such that the number of points |E(Fp)| on the curve over the finite field of p elements has prime order is asymptotic to CEx/(log x)2 for some constant CE. We consider curves without complex multiplication. Assuming the GRH (that is, the Riemann Hypothesis for Dedekind zeta functions) we prove that for »x/(log x)2 primes p ? x, the group order |E(Fp)| has at most 16 prime divisors. We also show (again, assuming the GRH) that for a random prime p, the group order |E(Fp)| has log log p prime divisors.

Nonadjacent Radix-

In his seminal papers, Koblitz proposed curves for cryptographic use. For fast operations on thesecurves,thesepapersalsoinitiatedastudyoftheradix-� expansionofintegersinthenumberfields Q( √ −3) and Q( √ −7). The (window) nonadjacent form of �-expansion of integers in Q( √ −7) was first investigated by Solinas. For integers in Q( √ −3), the nonadjacent form and the window nonadjacent form of the �-expansion were studied. These are used for efficient point multiplications on Koblitz curves. In this paper, we complete the picture by producing the (window) nonadjacent radix-� expansions for integers in all Euclidean imaginary quadratic number fields.

\tau

Let L/K be a finite Galois extension of number fields, with group G. Let H be a subgroup of G. Under the assumption of the Generalized Riemann Hypothesis, we obtain bounds for the least prime ofK which does not split completely in L. This bound is sharper than the earlier result of Lagarias and Odlyzko. Undersome conditions, we also obtain a bound for the least prime p such that the Frobenius automorphism σρ is disjoint from H. We apply this to study effectively torsion on elliptic curves. 1991 Mathematics Subject Classification: 11R44, 11R42, 11G05, 11M41. §

Expansions of Integers in Euclidean Imaginary Quadratic Number Fields

Let K be an algebraic number field and f a complex-valued function on the ideal class group of K. Then, f extends in a natural way to the set of all non-zero ideals of the ring of integers of K and we can consider the Dirichlet series

Transcendental values of class group L -functions

{L(s,f) =\sum_{{\mathfrak a}} f({\mathfrak a}){\bf N}({\mathfrak a})^{-s}}

On the Enumeration of Finite Groups

which converges for