Final Report Summary - ANTICS (Algorithmic Number Theory in Computer Science) Data security and privacy protection are major challenges in the digital world. Cryptology contributes to solutions, and one of the goals of ANTICS was to develop the next generation public key cryptosystems, based on algebraic curves and abelian varieties. Challenges to be tackled were the complexity of computations, certification of the computed results and parallelisation, addressed by introducing more informatics into algorithmic number theory.The project has made important contributions to the complexity of problems in algorithmic number theory. We have worked on complex multiplication constructions, which yield elliptic and more complicated algebraic curves for use in cryptosystems. With a variety of tools, we have obtained algorithms with a quasi-linear complexity in their output size, which means that the time they take for their full computations is essentially proportional to the time needed for writing down the result. To this purpose, we use algebraic, exact approaches, but also floating point computations, which due to potential error accumulation could lead to wrong results, but which are easy to check for correctness probabilistically. We have furthermore worked on the arithmetic of elliptic curves and higher-dimensional abelian varieties, and in particular on pairings that serve to implement novel cryptographic primitives. In another direction, the project has developed leading algorithms for the computation of invariants of number fields (class and unit groups, euclidianity) and quaternion algebras.Some of the algorithms have been conceived or modified to reach certified correct results. This concerns floating point computations of number theoretic functions using interval or ball arithmetic, but also the construction of number theoretic objects such as cryptographically suitable abelian varieties or Galois representations which internally depend on floating point operations, but where the final object can be proved to have the desired properties.Our algorithms have been validated by the publication of free software, which we use to carry out record computations. The implementations are partly distributed through stand-alone software, but mainly through the PARI/GP computer algebra system, the world leader for number theoretic computations, which is composed of the C library PARI and the interactive command interpreter GP.
