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 $\phi(x)=\la w,x\ra$ 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.

翻译：整数线性系统 $Ax=b$（其中矩阵 $A$、$b$ 以及解 $x$ 均要求为整数）可通过 $A$ 的不变因子（即计算 $A$ 的史密斯标准型）求解。本文探讨一个在应用中产生的新问题：给定 $\zz_n$ 中的 $A$ 和 $b$，寻求求解模线性系统 $Ax=b\rem n$（其中 $x\in\zz_n$）的条件，并要求存在某个解 $x$ 使得线性函数 $\phi(x)=\la w,x\ra$ 的值与 $n$ 互素。本文发展了将系统分解为 $n$ 的互素模除数 $p^{r(p)}$ 的方法，并展示了这种分解如何简化史密斯形式的计算。这将著名的计算离散对数的指标演算方法（其中为在素域上求解约化系统，假设约化线性系统的模数为素数）推广至模数的因子为素数幂 $p^{r(p)}$ 的情形。研究表明，通过利用增广矩阵 $[A,-p^{r(p)}I]$ 的不变因子和史密斯形式，并结合 $w$ 满足的模 $p$ 条件（其中 $p^{r(p)}$ 取遍 $n$ 的所有素因子幂），可高效处理该问题。