Samuel S. Wagstaff

The irregular primes to 125000

We have determined the irregular primes below 125000 and tabulated their distribution. Two primes of index five of irregularity were found, namely 78233 and 94693. Fermats Last Theorem has been verified for all exponents up to 125000. We computed the cyclotomic invariants,M.p, Ap , and found that -O 0 for all p < 125000. The complete factorizations of the numerators of the Bernoulli numbers B2k for 2k < 60 and of the Euler numbers E2k for 2k < 42 are given.

An Efficient Time-Bound Hierarchical Key Management Scheme for Secure Broadcasting

In electronic subscription and pay TV systems, data can be organized and encrypted using symmetric key algorithms according to predefined time periods and user privileges and then broadcast to users. This requires an efficient way of managing the encryption keys. In this scenario, time-bound key management schemes for a hierarchy were proposed by Tzeng and Chien in 2002 and 2005, respectively. Both schemes are insecure against collusion attacks. In this paper, we propose a new key assignment scheme for access control, which is both efficient and secure. Elliptic-curve cryptography is deployed in this scheme. We also provide the analysis of the scheme with respect to security and efficiency issues.

Divisors of Mersenne numbers

We add to the heuristic and empirical evidence for a conjecture of Gillies about the distribution of the prime divisors of Mersenne numbers. We list some large prime divisors of Mersenne numbers Mp in the range 17000

Greatest of the least primes in arithmetic progressions having a given modulus

We give a heuristic argument, supported by numerical evidence, which suggests that the maximum, taken over the reduced residue classes modulo k, of the least prime in the class, is usually about 0(k) log k log 0(k), where 0 is Eulers phi-function.

A practical analysis of the elliptic curve factoring algorithm

Much asymptotic analysis has been devoted to factoring algorithms. We present a practical analysis of the complexity of the elliptic curve algorithm, suggesting optimal parameter selection and run-time guidelines. The parameter selection is aided by a novel use of Bayesian statistical decision techniques as applied to random algorithms. We discuss how frequently the elliptic curve algorithm succeeds in practice and compare it with the quadratic sieve algorithm.

Sums of Squares of Integers

Introduction Prerequisites Outline of Chapters 2 - 8 Elementary Methods Introduction Some Lemmas Two Fundamental Identities Eulers Recurrence for Sigma(n) More Identities Sums of Two Squares Sums of Four Squares Still More Identities Sums of Three Squares An Alternate Method Sums of Polygonal Numbers Exercises Bernoulli Numbers Overview Definition of the Bernoulli Numbers The Euler-MacLaurin Sum Formula The Riemann Zeta Function Signs of Bernoulli Numbers Alternate The von Staudt-Clausen Theorem Congruences of Voronoi and Kummer Irregular Primes Fractional Parts of Bernoulli Numbers Exercises Examples of Modular Forms Introduction An Example of Jacobi and Smith An Example of Ramanujan and Mordell An Example of Wilton: t (n) Modulo 23 An Example of Hamburger Exercises Heckes Theory of Modular Forms Introduction Modular Group ? and its Subgroup ? 0 (N) Fundamental Domains For ? and ? 0 (N) Integral Modular Forms Modular Forms of Type Mk(? 0(N) chi) and Euler-Poincare series Hecke Operators Dirichlet Series and Their Functional Equation The Petersson Inner Product The Method of Poincare Series Fourier Coefficients of Poincare Series A Classical Bound for the Ramanujan t function The Eichler-Selberg Trace Formula l-adic Representations and the Ramanujan Conjecture Exercises Representation of Numbers as Sums of Squares Introduction The Circle Method and Poincare Series Explicit Formulas for the Singular Series The Singular Series Exact Formulas for the Number of Representations Examples: Quadratic Forms and Sums of Squares Liouvilles Methods and Elliptic Modular Forms Exercises Arithmetic Progressions Introduction Van der Waerdens Theorem Roths Theorem t 3 = 0 Szemeredis Proof of Roths Theorem Bipartite Graphs Configurations More Definitions The Choice of tm Well-Saturated K-tuples Szemeredis Theorem Arithmetic Progressions of Squares Exercises Applications Factoring Integers Computing Sums of Two Squares Computing Sums of Three Squares Computing Sums of Four Squares Computing rs(n) Resonant Cavities Diamond Cutting Cryptanalysis of a Signature Scheme Exercises References Index

Watermarking with quadratic residues

We describe novel methods of watermarking data using quadratic residues and random numbers. Our methods are fast, generic and improve the security of the watermark in most known watermarking techniques.

MPQS with Three Large Primes

We report the factorization of a 135-digit integer by the triple-large-prime variation of the multiple polynomial quadratic sieve. Previous workers [6][10] had suggested that using more than two large primes would be counterproductive, because of the greatly increased number of false reports from the sievers. We provide evidence that, for this number and our implementation, using three large primes is approximately 1.7 times as fast as using only two. The gain in efficiency comes from a sudden growth in the number of cycles arising from relations which contain three large primes. This effect, which more than compensates for the false reports, was not anticipated by the authors of [6][10] but has become quite familiar from factorizations obtained using the number field sieve. We characterize the various types of cycles present, and give a semi-quantitative description of their rather mysterious behaviour.

Aurifeuillian factorizations and the period of the Bell numbers modulo a prime

We show that the minimum period modulo p of the Bell exponential integers is (p p − 1)/(p − 1) for all primes p < 102 and several larger p. Our proof of this result requires the prime factorization of these periods. For some primes p the factoring is aided by an algebraic formula called an Aurifeuillian factorization. We explain how the coefficients of the factors in these formulas may be computed.

Some numerical results on Fekete polynomials

It is known that if X is a real residue character modulo k with X(P) -1 k n for the first five primes p, then the corresponding Fekete polynomial Yn=l X(n)x changes sign on (0, 1). In this paper it is shown that the condition that X(P) be 1 for the first four primes p is not sufficient to guarantee such a sign change. More specifically, if X is the real nonprincipal character modulo either 1277 or 1973, it is shown that the corresponding Fekete polynomial is positive throughout (0, 1) even though X(2) X(3) = X(5) = x(7) = 1.