"[Algorithmic Number Theory] is an enormous achievement and an extremely valuable reference." - Donald E. Knuth, Emeritus, Stanford University Algorithmic Number Theory provides a thorough introduction to the design and analysis of algorithms for problems from the theory of numbers. Although not an elementary textbook, it includes over 300 exercises with suggested solutions. Every theorem not proved in the text or left as an exercise has a reference in the notes section that appears at the end of each chapter. The bibliography contains over 1,750 citations to the literature. Finally, it successfully blends computational theory with practice by covering some of the practical aspects of algorithm implementations. The subject of algorithmic number theory represents the marriage of number theory with the theory of computational complexity. It may be briefly defined as finding integer solutions to equations, or proving their non-existence, making efficient use of resources such as time and space. Implicit in this definition is the question of how to efficiently represent the objects in question on a computer. The problems of algorithmic number theory are important both for their intrinsic mathematical interest and their application to random number generation, codes for reliable and secure information transmission, computer algebra, and other areas. The first volume focuses on problems for which relatively efficient solutions can be found. The second (forthcoming) volume will take up problems and applications for which efficient algorithms are currently not known. Together, the two volumes cover the current state of the art in algorithmic number theory and will be particularly useful to researchers and students with a special interest in theory of computation, number theory, algebra, and cryptography.