Algorithmic Number Theory

This volume presents the refereed proceedings of the First Algorithmic Number Theory Symposium, ANTS-I, held at Cornell University, Ithaca, NY in May 1994.
The 35 papers accepted for inclusion in this book address many current issues of algorithmic, computational and complexity-theoretic aspects of number theory and thus report the state-of-the-art in this exciting area of research; the book also contributes essentially to foundational research in cryptology and coding.
Of particular value is a collection entitled "Open Problems in Number Theoretic Complexity, II" contributed by Len Adleman and Kevin McCurley. This survey presents on 32 pages 36 central open problems and relates them to the literature by means of some 160 references.

Inhalt
On the difficulty of finding reliable witnesses.- Density computations for real quadratic 2-class groups.- Lattice sieving and trial division.- A subexponential algorithm for discrete logarithms over the rational subgroup of the Jacobians of large genus hyperelliptic curves over finite fields.- Computing rates of growth of division fields on CM Abelian varieties.- Algorithms for CM-Fields.- Schoof's algorithm and isogeny cycles.- Integer points on rational elliptic curves.- Counting the number of points on elliptic curves over finite fields of characteristic greater than three.- Straight-line complexity and integer factorization.- Decomposition of algebraic functions.- A new modular interpolation algorithm for factoring multivariate polynomials.- The function field sieve.- Heegner point computations.- Computing the degree of a modular parametrization.- Galois representations from the cohomology of SL(3,?).- An analysis of the Gaussian algorithm for lattice reduction.- A fast variant of the Gaussian reduction algorithm.- Reducing lattice bases by means of approximations.- Analysis of a left-shift binary GCD algorithm.- The complexity of greatest common divisor computations.- Explicit formulas for units in certain quadratic number fields.- Factorization of polynomials over finite fields in subexponential time under GRH.- On orders of optimal normal basis generators.- Computing in the jacobian of a plane algebraic curve.- Under the assumption of the Generalized Riemann Hypothesis verifying the class number belongs to NP ? co-NP.- Calculating the class number of certain Hilbert class fields.- Efficient checking of computations in number theory.- Constructing elliptic curves with given group order over large finite fields.- Computing ?(x), M(x) and ?(x).- On someapplications of finitely generated semi-groups.- Improved incremental prime number sieves.- Polynomial time algorithms for discrete logarithms and factoring on a quantum computer.- On dispersion and Markov constants.- Open problems in number theoretic complexity, II.