On an Efficient Algorithm for Big Rational Number Computations by Parallel p-adics

This paper presents an algorithm for evaluating an arithmetic expression over "big" rational numbers. The method exploits p-adic arithmetic and parallelism to achieve efficiency. Roughly, the algorithm begins by mapping the input rational numbers to the related p-adic codes for several prime bases, then the corresponding expression is evaluated over these codes and finally the rational output is recovered from the resulting codes. All these three steps are parallelized. An efficient algorithm to compute the recovery step is proposed. Let p be the maximum of the prime bases for the codes, let r be their truncation order and let k be the number of processors. The asymptotic bit-wise complexity of the proposed recovery step is O(r (k log p)log23. The previous known bound was O(r2(k log p)log23). Let n be the number of operations in the given expression: the asymptotic bitwise complexity of the proposed algorithm for evaluating arithmetic expressions is O(n r2log p).