The injection of ideas from number theory and algebraic geometry has brought about remarkable progress in recent years in the study of primality testing and integer factoring, and in computational number theory in general. It is plan to continue the exploration of arithmetric and geometric methods as a fundamental tool in the study of computational problems, particularly those that are related to number theory. The goal is to develop efficient algorithms for them, to study complexity theoretic relation among them, and to explore application to cryptography, as well as other problems in computational complexity. The primary focus will be: (1) Further investigation into primality testing and integer factoring, and other fundamental problems in computational number theory. (2) Development of efficient algorithms for computing with curves and higher dimensional algebraic varieties, particularly as they apply to the computational problems mentioned above. (3) Application of arithmetic and geometric techniques in the analysis of security of public key cryptosystems. Development of new cryptosystems based on geometric ideas.
