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.