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

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

A description of 148 algorithms fundamental to number-theoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring. The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods, including a detailed description of the number field sieve algorithm. The whole is rounded off with a description of available computer packages and some useful tables, backed by numerous exercises. Written by an authority in the field, and one with great practical and teaching experience, this is certain to become the standard and indispensable reference on the subject.

A Course in Computational Algebraic Number Theory

A major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists, since expander graphs, the protagonists of our story, come up in numerous and often surprising contexts in both fields But, perhaps, we should start with a few words about graphs in general They are, of course, one of the prime objects of study in Discrete Mathematics However, graphs are among the most ubiquitous models of both natural and human-made structures In the natural and social sciences they model relations among species, societies, companies, etc In computer science, they represent networks of communication, data organization, computational devices as well as the flow of computation, and more In mathematics, Cayley graphs are useful in Group Theory Graphs carry a natural metric and are therefore useful in Geometry, and though they are “just” one-dimensional complexes, they are useful in certain parts of Topology, eg Knot Theory In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems The study of these models calls, then, for the comprehension of the significant structural properties of the relevant graphs But are there nontrivial structural properties which are universally important? Expansion of a graph requires that it is simultaneously sparse and highly connected Expander graphs were first defined by Bassalygo and Pinsker, and their existence first proved by Pinsker in the early ’70s The property of being an expander seems significant in many of these mathematical, computational and physical contexts It is not surprising that expanders are useful in the design and analysis of communication networks What is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness In mathematics, we will encounter eg their role in the study of metric embeddings, and in particular in work around the Baum-Connes Conjecture Expansion is closely related to the convergence rates of Markov Chains, and so they play a key role in the study of Monte-Carlo algorithms in statistical mechanics and in a host of practical computational applications The list of such interesting and fruitful connections goes on and on with so many applications we will not even

/pdf/expander-graphs-and-their-applications-wbsb8gxfw2.pdf

Expander graphs and their applications

Sphere Packings, Lattices and Groups

There are infinitely many Carmichael numbers

Throughout this paper d will denote a fundamental discriminant, and χd the associated primitive real character to the modulus |d|. We investigate here the distribution of values of L(1, χd) as d varies over all fundamental discriminants with |d| ≤ x. Our main concern is to compare the distribution of values of L(1, χd) with the distribution of “random Euler products” L(1,X) = ∏ p(1 − X(p)/p)−1 where the X(p)’s are independent random variables taking values 0 or ±1 with suitable probabilities, described below. For example, we shall give asymptotics for the probability that L(1, χd) exceeds eγτ , and the probability that L(1, χd) ≤ π2 6 1 eγτ uniformly in a wide range of τ . These results are sufficiently uniform to prove slightly more than a recent conjecture of H.L. Montgomery and R.C. Vaughan [MV]. One important motivation for our work is to make progress towards resolving the discrepancy between extreme values that may be exhibited (the omega results of S.D. Chowla described below) and the conditional bounds on these extreme values (the O-results of J.E. Littlewood, see below). The uniformity of our results provides evidence that the omega results of Chowla may represent the true nature of extreme values of L(1, χd). These questions have been studied by many authors, most notably by P.D.T.A. Elliott [E1,2,3] and Montgomery and Vaughan [MV]. We begin by reviewing some of the history of the subject which will help place our results in context. Throughout this paper logj will denote the j-fold iterated logarithm; that is, log2 = log log, log3 = log log log, and so on. In [Li] Littlewood showed that on the assumption of the Generalized Riemann Hypothesis (GRH)

/pdf/the-distribution-of-values-of-l-1-kh-d-2gpm5nsuk1.pdf

The distribution of values of L(1, χ d )

We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a p-block with defect 0, completing an investigation of many authors. The only finite simple groups whose defect zero p-blocks remained unclassified were the alternating groups An. Here we show that these all have a p-block with defect 0 for every prime p ≥ 5. This follows from proving the same result for every symmetric group Sn, which in turn follows as a consequence of the t-core partition conjecture, that every non-negative integer possesses at least one t-core partition, for any t ≥ 4. For t ≥ 17, we reduce this problem to Lagrange's Theorem that every non-negative integer can be written as the sum of four squares. The only case with t < 17, that was not covered in previous work, was the case t = 13. This we prove with a very different argument, by interpreting the generating function for t-core partitions in terms of modular forms, and then controlling the size of the coefficients using Deligne's Theorem (nee the Weil Conjectures). We also consider congruences for the number of p-blocks of Sn, proving a conjecture of Garvan, that establishes certain multiplicative congruences when 5 < p < 23. By using a result of Serre concerning the divisibility of coefficients of modular forms, we show that for any given prime p and positive integer m, the number of p-blocks with defect 0 in Sn is a multiple of m for almost all n. We also establish that any given prime p divides the number of p-modularly irreducible representations of Sn, for almost all n. © 1996 American Mathematical Society.

/pdf/defect-zero-p-blocks-for-finite-simple-groups-1ddg9rt4dz.pdf

Defect zero p-blocks for finite simple groups

The analysis of many number theoretic algorithms turns on the role played by integers which have only small prime factors; such integers are known as “smooth numbers”. To be able to determine which algorithm is faster than which, it has turned out to be important to have accurate estimates for the number of smooth numbers in various sequences. In this chapter, we will first survey the important estimates for application to computational number theory questions, results as well as conjectures, before moving on to give sketches of the proofs of many of the most important results. After this, we will describe applications of smooth numbers to various problems in different areas of number theory. More complete surveys, with many more references, though with a different focus, were given by Norton [1971] and Hildebrand and Tenenbaum [1993a].

/pdf/smooth-numbers-computational-number-theory-and-beyond-1vvg0yjdkk.pdf

Smooth numbers: computational number theory and beyond

In 1918 Polya and Vinogradov gave an upper bound for the maximal size of character sums, which still remains the best known general estimate. One of the main results of this paper provides a substantial improvement of the Polya-Vinogradov bound for characters of odd, bounded order. In 1977 Montgomery and Vaughan showed how the Polya-Vinogradov inequality may be sharpened assuming the Generalized Riemann Hypothesis. We give a simple proof of their estimate and provide an improvement for characters of odd, bounded order. The paper also gives characterizations of the characters for which the maximal character sum is large, and it finds a hidden structure among these characters

/pdf/large-character-sums-pretentious-characters-and-the-polya-2g16jkxeto.pdf

Andrew Granville

Papers

There are infinitely many Carmichael numbers

The distribution of values of L(1, χ d )

Defect zero p-blocks for finite simple groups

Smooth numbers: computational number theory and beyond

Large character sums: Pretentious characters and the Pólya-Vinogradov theorem