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.