Advanced Topics in Computational Number Theory (Graduate Texts in Mathematics, 193)

Rating:
Summary: A first class job, just as in the first volume
Review: The author continues his excellent overview of computational number theory in this book. And, as in the first volume, the writing is first-rate and gives the reader a comprehensive overview of the more advanced algorithms in the subject. Contrary to the first volume, I have not used many of the algorithms in this book and cannot attest to their quality, but the author gives the detailed background on each of them, enhancing their credibility. I read this book mostly to gain more insight into algebraic geometry and its connection with coding theory and cryptography. The following algorithms were ones that I found helpful and interesting but only a few of which I coded myself: (1) The algorithm for generating a random element from an ideal. (2) The compositum of two number fields (3) Valuation of a prime ideal. (4)Ideal factorization. (5)Smith normal form for finite groups. (6) Quotient of groups. (7) Group extensions. (8) Right four-term exact sequences. (9) Image and inverse image of a subgroup. (10)Subgroups with prime index. (11) Solving linear systems in integers. (12) Algorithms involving p-adic logarithms. (13) Computation of the Dedekind eta-function. (14) Unramified Abelian extensions using complex multiplication. (15) Computation of quasi-periods for elliptic functions. (16) Computation of the sigma-function for an elliptic curve.