A Course in Number Theory and Cryptography By Neal Koblitz
This is a fundamentally revised and updated introduction to the arithmetic, ancient and modern topics that are central to applications of number theory, especially in cryptography. Therefore, no background in algebra or number theory is assumed, and the book begins with a discussion of the underlying number theory required. The approach taken is algorithmic, emphasizing estimates of the efficiency of techniques derived from theory, and a special feature is the inclusion of recent applications of elliptic curve theory. Intensive exercises and careful answers are an integral part of all chapters.
Table of contents:
I. Some Topics in Elementary Number Theory.- 1. Time estimates for doing arithmetic.- 2. Divisibility and the Euclidean algorithm.- 3. Congruences.- 4. Some applications to factoring.
II. Finite Fields and Quadratic Residues.- 1. Finite fields.- 2. Quadratic residues and reciprocity.
III. Cryptography.- 1. Some simple cryptosystems.- 2. Enciphering matrices.
IV. Public Key.- 1. The idea of public key cryptography.- 2. RSA.- 3. Discrete log.- 4. Knapsack.- 5 Zero-knowledge protocols and oblivious transfer.
V. Primality and Factoring.- 1. Pseudoprimes.- 2. The rho method.- 3. Fermat factorization and factor bases.- 4. The continued fraction method.- 5. The quadratic sieve method.
VI. Elliptic Curves.- 1. Basic facts.- 2. Elliptic curve cryptosystems.- 3. Elliptic curve primality test.- 4. Elliptic curve factorization.- Answers to Exercises.
A course in number theory and cryptography
Author(s): Koblitz, Neal
Series: Graduate texts in mathematics 114
Publisher: Springer, Year: 2012
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