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 = g k . 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(2 n ). It appears that in order to be safe from attacks using these algorithms, the value of n for which GF(2 n ) 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(2 n ) are much easier to compute than in fields GF(p) with p prime. Hence the fields GF(2 n ) ought to be avoided in all cryptographic applications. On the other hand, ...

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The Cray-2 is capable of performing matrix multiplication at very high rates. Using library routines provided byCray Research, Inc., performance rates of 300 to 425 MFLOPS can be obtained on a single processor, depending on system load. Considerably higher rates can be achieved with all four processors running simultaneously. This article describes how matrix multiplication can be performed even faster, up to twice the above rates. This can be achieved by (1) employing Strassen's matrix multiplication algorithm to reduce the number of oating-point operations performed and (2) utilizing local memory on the Cray-2 to avoid performance losses due to memory bank contention. The numerical stability and potential for parallel application of this procedure are also discussed.

A new parallel algorithm is given to evaluate a straight line program. The algorithm evaluates a program over a commutative semi-ring R of degree d and size n in time O(log n(log nd)) using M(n) processors, where M(n) is the number of processors required for multiplying n \Theta n matrices over the semi-ring R in O(log n) time. Appears in SIAM J. Comput., 17/4, pp. 687--695 (1988). Preliminary version of this paper appeared in [6]. y Research supported in part by National Science Foundation Grant MCS-800756 A01. z Research supported by NSF under ECS-8404866, the Semiconductor Research Corporation under RSCH 84-06-049-6, and by an IBM Faculty Development Award. x Research Supported in part by NSF Grant DCR-8504391 and by an IBM Faculty Development Award. 1 INTRODUCTION 1 1 Introduction In this paper we consider the problem of dynamic evaluation of a straight line program in parallel. This is a generalization of the result of Valiant et al [10]. They consider the problem of ta...

An efficient context-free parsing algorithln is preseuted that can parse sentences with unknown parts of unknown length. It produc in finite form all possible parses (often infinite in number) that could account for the missing parts. The algorithm is a variation on the construction due to Earley. ltowever, its presentation is such that it can readily be adapted to any chart parsing schema (top- down, bottom-up, etc...).

foremost recognition of technical contributions to the computing community. The citation of Cook's achievements noted that "Dr. Cook has advanced our understanding of the complexity of computation in a significant and profound way. His seminal paper, The Complexity of Theorem Proving Procedures, presented at the 1971 ACM SIGACT Symposium on the Theory of Computing, laid the foundations for the theory of NP-completeness. The ensuing exploration of the boundaries and nature of the NP-complete class of problems has been one of the most active and important research activities in computer science for the last decade. Cook is well known for his influential results in fundamental areas of computer science. He has made significant contributions to complexity theory, to time-space tradeoffs in computation, and to logics for programming languages. His work is characterized by elegance and insights and has illuminated the very nature of computation." During 1970-1979, Cook did extensive work under grants from the

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The communication cost of algorithms (also known as I/Ocomplexity) is shown to be closely related to the expansion properties of the corresponding computation graphs. We demonstrate this on Strassen’s and other fast matrix multiplication algorithms, and obtain the first lower bounds on their communication costs. For sequential algorithms these bounds are attainable and so optimal. 1.

Abstract. The Laplacian matrices of graphs are fundamental. In addition to facilitating the application of linear algebra to graph theory, they arise in many practical problems. In this talk we survey recent progress on the design of provably fast algorithms for solving linear equations in the Laplacian matrices of graphs. These algorithms motivate and rely upon fascinating primitives in graph theory, including low-stretch spanning trees, graph sparsifiers, ultra-sparsifiers, and local graph clustering. These are all connected by a definition of what it means for one graph to approximate another. While this definition is dictated by Numerical Linear Algebra, it proves useful and natural from a graph theoretic perspective.