Extension of simultaneous Diophantine approximation algorithm for partial approximate common divisor variants

Abstract

A simultaneous Diophantine approximation (SDA) algorithm takes instances of the partial approximate common divisor (PACD) problem as input and outputs a solution. While several encryption schemes have been published and their securities depend on the presumed hardness of variant of the PACD problem, fewer studies have attempted to extend the SDA algorithm to be applicable to these variants. In this study, the SDA algorithm is extended to solve the general PACD problem. In order to proceed, first the variants of the PACD problem are classified and how to extend the SDA algorithm for each is suggested. Technically, the authors show that a short vector of some lattice used in the SDA algorithm gives an algebraic relation between secret parameters. Then, all the secret parameters can be recovered by finding this short vector. It is also confirmed experimentally that this algorithm works well.