Algorithmic Number Theory, Vol. 1: Efficient Algorithms by Eric Bach and Jeffrey Shallit.
From the preface:
This is the first volume of a projected two-volume set on algorithmic number theory, the design and analysis of algorithms for problems from the theory of numbers. This volume focuses primarily on those problems from number theory that admit relatively efficient solutions. The second volume will largely focus on problems for which efficient algorithms are not known, and applications thereof.
We hope that the material in this book will be useful for readers at many levels, from the beginning graduate student to experts in the area. The early chapters assume that the reader is familiar with the topics in an undergraduate algebra course: groups, rings, and fields. Later chapters assume some familiarity with Galois theory.
As stated above, this book discusses the current state of the art in algorithmic number theory. This book is not an elementary number theory textbook, and so we frequently do not give detailed proofs of results whose central focus is not computational. Choosing otherwise would have made this book twice as long.
The webpage offers the BibTeX files for the bibliography, which includes more than 1800 papers and books.
BTW, Amazon notes that Volume 2 was never published.
Now that high performance computing resources are easily available, perhaps you can start working on your own volume 2. Yes?
I first saw this in a tweet by Alvaro Videla.