Modular dynamic evaluation

Abstract

In this paper we propose an efficient modular method for Dynamic Evaluation (DE) [2][3][5]. For efficient computation over algebraic number fields, we have to design algorithms very carefully. For example, over algebraic number fields, the polynomial factorization is harder and the coefficient swell is more serious than those over the rationals. In DE, the defining polynomial of an algebraic number to be added to a ring L need not to be irreducible, which avoids expensive irreducible factorizations. Instead, the obtained extension may have zero-divisors in general. Such a zero-divisor can be detected dynamically during a computation of the inverse or a zero-recognition of an element and the ring is then decomposed into several components. The known method simply applies the Euclid algorithm to detect zero-divisors, but it often causes coefficient swell. In this paper we give a new formulation of DE by using the notion of ideal quotient, which enables us to apply efficient modular methods to decompose the ring.