Andrew M. Odlyzko

Singularity analysis of generating functions

This work presents a class of methods by which one can translate, on a term-by-term basis, an asymptotic expansion of a function around a dominant singularity into a corresponding asymptotic expansion for the Taylor coefficients of the function. This approach is based on contour integration using Cauchy’s formula and Hankel-like contours. It constitutes an alternative to either Darboux’s method or Tauberian theorems that appears to be well suited to combinatorial enumerations, and a few applications in this area are outlined.

Paris metro pricing for the internet

A simple approach, called PMP (Paris Metro Pricing), is suggested for providing differentiated services in packet networks such as the Internet. It is to partition a network into several logically separate channels, each of which would treat all packets equally on a best effort basis. There would be no formal guarantees of quality of service. These channels would differ only in the prices paid for using them. Channels with higher prices would attract less traffic, and thereby provide better service. Price would be the primary tool of traffic management. PMP is the simplest differentiated services solution. It is designed to accommodate user preferences at the cost of sacrificing some of the utilization efficiency of the network.

Algebraic Properties of Cellular Automata

Cellular automata are discrete dynamical systems, of simple construction but complex and varied behaviour. Algebraic techniques are used to give an extensive analysis of the global properties of a class of finite cellular automata. The complete structure of state transition diagrams is derived in terms of algebraic and number theoretical quantities. The systems are usually irreversible, and are found to evolve through transients to attractors consisting of cycles sometimes containing a large number of configurations.

String overlaps, pattern matching, and nontransitive games

Abstract This paper studies several topics concerning the way strings can overlap. The key notion of the correlation of two strings is introduced, which is a representation of how the second string can overlap into the first. This notion is then used to state and prove a formula for the generating function that enumerates the q -ary strings of length n which contain none of a given finite set of patterns. Various generalizations of this basic result are also discussed. This formula is next used to study a wide variety of seemingly unrelated problems. The first application is to the nontransitive dominance relations arising out of a probabilistic coin-tossing game. Another application shows that no algorithm can check for the presence of a given pattern in a text without examining essentially all characters of the text in the worst case. Finally, a class of polynomials arising in connection with the main result are shown to be irreducible.

Discrete logarithms in finite fields and their cryptographic significance

Given a primitive element g of a finite field GF(q), the discrete logarithm of a nonzero element u ? GF(q) is that integer k, 1 ? k ? q-1, for which u = gk. The well-known problem of computing discrete logarithms in finite fields has acquired additional importance in recent years due to its applicability in cryptography. Several cryptographic systems would become insecure if an efficient discrete logarithm algorithm were discovered. This paper surveys and analyzes known algorithms in this area, with special attention devoted to algorithms for the fields GF(2n). It appears that in order to be safe from attacks using these algorithms, the value of n for which GF(2n) is used in a cryptosystem has to be very large and carefully chosen. Due in large part to recent discoveries, discrete logarithms in fields GF(2n) are much easier to compute than in fields GF(p) with p prime. Hence the fields GF(2n) ought to be avoided in all cryptographic applications. On the other hand, the fields GF(p) with p prime appear to offer relatively high levels of security.

Discrete logarithms in GF ( p )

Several related algorithms are presented for computing logarithms in fieldsGF(p),p a prime. Heuristic arguments predict a running time of exp((1+o(1))

On the distribution of spacings between zeros of the zeta function

\sqrt {\log p \log \log p}

Improved low-density subset sum algorithms

) for the initial precomputation phase that is needed for eachp, and much shorter running times for computing individual logarithms once the precomputation is done. The running time of the precomputation is roughly the same as that of the fastest known algorithms for factoring integers of size aboutp. The algorithms use the well known basic scheme of obtaining linear equations for logarithms of small primes and then solving them to obtain a database to be used for the computation of individual logarithms. The novel ingredients are new ways of obtaining linear equations and new methods of solving these linear equations by adaptations of sparse matrix methods from numerical analysis to the case of finite rings. While some of the new logarithm algorithms are adaptations of known integer factorization algorithms, others are new and can be adapted to yield integer factorization algorithms.