From Gauss to G|del, mathematicians have sought an efficientalgorithm to distinguish prime numbers from compositenumbers. This book presents a random polynomial timealgorithm for the problem. The methods used are fromarithmetic algebraic geometry, algebraic number theory andanalyticnumber theory. In particular, the theory of twodimensional Abelian varieties over finite fields isdeveloped.The book will be of interest to both researchers andgraduate students in number theory and theoretical computerscience.