From the Publisher:
A valuable reference for the novice as well as for the expert who needs a wider scope of coverage within the area of cryptography, this book provides easy and rapid access of information and includes more than 200 algorithms and protocols; more than 200 tables and figures; more than 1,000 numbered definitions, facts, examples, notes, and remarks; and over 1,250 significant references, including brief comments on each paper.

Handbook of Applied Cryptography

Finite fields

Introduction Arithmetic functions Elementary theory of prime numbers Characters Summation formulas Classical analytic theory of $L$-functions Elementary sieve methods Bilinear forms and the large sieve Exponential sums The Dirichlet polynomials Zero-density estimates Sums over finite fields Character sums Sums over primes Holomorphic modular forms Spectral theory of automorphic forms Sums of Kloosterman sums Primes in arithmetic progressions The least prime in an arithmetic progression The Goldbach problem The circle method Equidistribution Imaginary quadratic fields Effective bounds for the class number The critical zeros of the Riemann zeta function The spacing of zeros of the Riemann zeta-function Central values of $L$-functions Bibliography Index.

/pdf/analytic-number-theory-3xy3ackp22.pdf

Analytic Number Theory

good, does not support this; the label ‘Example’ is given in a rather small font followed by a ‘PROOF,’ and the body of an example is nonitalic, utterly unlike other statements accompanied by demonstrations. The proofs of some theorems are continued after the statements and proofs of needed lemmas with no helpful demarcations (“Continuation of proof of Theorem . . . ”). Finally, at least one lemma (3.34) has a corollary with no theorem or proposition in sight; the status of Lemma 3.36 is also unclear. Lemma 1.16 on page 12 giving a closed formula for Stirling numbers of the second kind is important enough to figure in the proof of capstone Theorem 3.71 (Schlömilch’s formula) on page 114. This nonstandard approach to chapter organization may impede some readers’ comprehending the structure of the author’s highly purposeful discussion. Review of 3 Enumerative Combinatorics Author of Book: Charalambos A. Charalambides Publisher: Chapman & Hall/CRC ISBN L-58488-290-5, Hard Cover, 609 pages

Enumerative Combinatorics

The first part of this book presents an introduction to the theory of finite fields, with emphasis on those aspects that are relevant for applications. The second part is devoted to a discussion of the most important applications of finite fields especially information theory, algebraic coding theory and cryptology (including some very recent material that has never before appeared in book form). There is also a chapter on applications within mathematics, such as finite geometries. combinatorics. and pseudorandom sequences. Worked-out examples and list of exercises found throughout the book make it useful as a textbook.

Introduction to finite fields and their applications

Bridging the gap between elementary number theory and the systematic study of advanced topics, A Classical Introduction to Modern Number Theory is a well-developed and accessible text that requires only a familiarity with basic abstract algebra Historical development is stressed throughout, along with wide-ranging coverage of significant results with comparatively elementary proofs, some of them new An extensive bibliography and many challenging exercises are also included This second edition has been corrected and contains two new chapters which provide a complete proof of the Mordell-Weil theorem for elliptic curves over the rational numbers, and an overview of recent progress on the arithmetic of elliptic curves

/pdf/a-classical-introduction-to-modern-number-theory-1tszqvyyai.pdf

A classical introduction to modern number theory

Polynomials over Finite Fields.- Primes, Arithmetic Functions, and the Zeta Function.- The Reciprocity Law.- Dirichlet L-series and Primes in an Arithmetic Progression.- Algebraic Function Fields and Global Function Fields.- Weil Differentials and the Canonical Class.- Extensions of Function Fields, Riemann-Hurwitz, and the ABC Theorem.- Constant Field Extensions.- Galois Extensions - Artin and Hecke L- functions.- Artin's Primitive Root Conjecture.- The Behavior of the Class Group in Constant Field Extensions.- Cyclotomic Function Fields.- Drinfeld Modules, An Introduction.- S-Units, S-Class Group, and the Corresponding L-functions.- The Brumer-Stark Conjecture.- Class Number Formulas in Quadratic and Cyclotomic Function Fields.- Average Value Theorems in Function Fields.

Number Theory in Function Fields

https://link.springer.com/content/pdf/10.1007/BF01442878.pdf

Idempotent relations and factors of Jacobians

Mertens' proved an interesting and useful theorem about the partial product of the Riemann zeta function at s = 1. Namely.

A Generalization of Mertens' Theorem

Nagao has recently given a conjectural limit formula for the rank of an elliptic surface E in terms of a weighted average of fibral Frobenius trace values. We show that Tate's conjecture on the order of vanishing of L2(E,s) essentially implies Nagao's formula; in particular, we prove Nagao's formula for rational elliptic surfaces. In the case that E is a twist, we reduce Nagao's and Tate's conjectures to the case of products of curves, and we verify the conjectures for many new classes of elliptic surfaces of Kodaira dimension 0 and 1.

Michael Rosen

Papers

A classical introduction to modern number theory

Number Theory in Function Fields

Idempotent relations and factors of Jacobians

A Generalization of Mertens' Theorem

On the rank of an elliptic surface