Algorithmic Number Theory

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9510244 Bach The goal of this project is to study the computational complexity of number-theoretic problems. Algorithms for solving these problems are useful in the design of systems for random number generation, cryptography, and computer algebra. In addition, many of the classic questions in complexity theory (such as whether randomization helps in efficient computation) are motivated by computational number theory. Our proposed investigations fall into three categories: a) Computations and explicit bounds in analytic number theory. b) The design and analysis of number-theoretic algorithms. c) Constructive techniques in algebraic geometry and commutative algebra.