Invited Papers -- Running Time Predictions for Factoring Algorithms -- A New Look at an Old Equation -- Elliptic Curves Cryptology and Generalizations -- Abelian Varieties with Prescribed Embedding Degree -- Almost Prime Orders of CM Elliptic Curves Modulo p -- Efficiently Computable Distortion Maps for Supersingular Curves -- On Prime-Order Elliptic Curves with Embedding Degrees k?=?3, 4, and 6 -- Arithmetic of Elliptic Curves -- Computing in Component Groups of Elliptic Curves -- Some Improvements to 4-Descent on an Elliptic Curve -- Computing a Lower Bound for the Canonical Height on Elliptic Curves over Totally Real Number Fields -- Faster Multiplication in GF(2)[x] -- Integer Factorization -- Predicting the Sieving Effort for the Number Field Sieve -- Improved Stage 2 to P ± 1 Factoring Algorithms -- K3 Surfaces -- Shimura Curve Computations Via K3 Surfaces of Néron-Severi Rank at Least 19 -- K3 Surfaces of Picard Rank One and Degree Two -- Number Fields -- Number Fields Ramified at One Prime -- An Explicit Construction of Initial Perfect Quadratic Forms over Some Families of Totally Real Number Fields -- Functorial Properties of Stark Units in Multiquadratic Extensions -- Enumeration of Totally Real Number Fields of Bounded Root Discriminant -- Point Counting -- Computing Hilbert Class Polynomials -- Computing Zeta Functions in Families of C a,b Curves Using Deformation -- Computing L-Series of Hyperelliptic Curves -- Point Counting on Singular Hypersurfaces -- Arithmetic of Function Fields -- Efficient Hyperelliptic Arithmetic Using Balanced Representation for Divisors -- Tabulation of Cubic Function Fields with Imaginary and Unusual Hessian -- Modular Forms -- Computing Hilbert Modular Forms over Fields with Nontrivial Class Group -- Hecke Operators and Hilbert Modular Forms.-Cryptography -- A Birthday Paradox for Markov Chains, with an Optimal Bound for Collision in the Pollard Rho Algorithm for Discrete Logarithm -- An Improved Multi-set Algorithm for the Dense Subset Sum Problem -- Number Theory -- On the Diophantine Equation x 2?+?2 ? 5 ? 13 ? ?=?y n -- Non-vanishing of Dirichlet L-functions at the Central Point.

This book constitutes the refereed proceedings of the 8th International Algorithmic Number Theory Symposium, ANTS 2008, held in Banff, Canada, in May 2008. The 28 revised full papers presented together with 2 invited papers were carefully reviewed and selected for inclusion in the book. The papers are organized in topical sections on elliptic curves cryptology and generalizations, arithmetic of elliptic curves, integer factorization, K3 surfaces, number fields, point counting, arithmetic of function fields, modular forms, cryptography, and number theory.