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A b s t r a c t We show how to find sufficiently small integer solutions to a polynomial in a single variable modulo N, and to a polynomial in two variables over the integers. The methods sometimes extend to more variables. As applications: RSA encryption with exponent 3 is vulnerable if the opponent knows 2/3 of the message, or if two messages agree over 8/9 of their length; and we can find the factors of N = P Q if we are given the high order (log 2 N)/4 bits of P. The algorithm entails a novel application of lattice basis reduction methods (Lenstra Lenstra Lovasz). A b s t r a c t Hybrid symbolic-numeric algorithms have been applied with some success in numerical definite integration. Symbolic analysis techniques have been used for the detection of singularities and for singularlty-handling, allowing the purely numerical phase to proceed efficiently even at high precision requirements. A progress report will be presented on the use of sinc methods for the approximate representation of indefinite integrals and for numerical multiple integration. For the solution of differential equations, the application of symbolic analysis techniques for the generation of good numerical update formulas is being based on a divided-difference calculus proposed by W. Kahan. Further investigations will study the role of sinc methods in the numerical solution of differential equations. A b s t r a c t Finding the solution of a system of algebraic equations over a given field is a fundamental problem in symbolic and scientific computing. The importance of good solvers for algebraic systems has risen considerably in different application domains. Polynomial equations are ubiquitously used for representing models and constraints in computer graphics, vision, robotics, economics, computational biology and manufacturing. This include spline models and their boolean combinations, geometric constraints, kinematics formulation, object localiza-tion and path planning etc. Many fundamental problems in these areas related to model building and display, object recognition, inverse kinematics, ray-tracing, confor-mation search reduce to symbolic, numeric and geometric manipulation of algebraic systems. In this talk we survey some recent work on solving algebraic systems using multi-polynomial resultants and matrix computations. The resultant of a polynomial system is expressed in terms of matrices and determinants and we reduce the problem to computing eigenvalues of matrix polynomials. Based on this formulation, we present algorithms for numerically solving zero and one dimensional algebraic systems. These algorithms …

1996 East Coast Computer Algebra Day Abstracts, Yorktown Heights, New York, April 13, 1996