Solving Modular Linear Systems with a Constraint by parallel decomposition of the Smith form and extended Euclidean division modulo powers of primes divisors

Abstract

Integral linear systems $Ax=b$ with matrices $A$, $b$ and solutions $x$ are also required to be in integers, can be solved using invariant factors of $A$ (by computing the Smith Canonical Form of $A$). This paper explores a new problem which arises in applications, that of obtaining conditions for solving the Modular Linear System $Ax=b\rem n$ given $A,b$ in $\zz_n$ for $x$ in $\zz_n$ along with the constraint that the value of the linear function $ϕ(x)=\la w,x\ra$ is coprime to $n$ for some solution $x$. In this paper we develop decomposition of the system to coprime moduli $p^{r(p)}$ which are divisors of $n$ and show how such a decomposition simplifies the computation of Smith form. This extends the well known index calculus method of computing the discrete logarithm where the moduli over which the linear system is reduced were assumed to be prime (to solve the reduced systems over prime fields) to the case when the factors of the modulus are prime powers $p^{r(p)}$. It is shown how this problem can be addressed effciently using the invariant factors and Smith form of the augmented matrix $[A,-p^{r(p)}I]$ and conditions modulo $p$ satisfied by $w$, where $p^{r(p)}$ vary over all divisors of $n$ with $p$ prime.