# Classifying Problems on Linear Congruences and Abelian Permutation Groups Using Logspace Counting Classes

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## Abstract.

In this paper we classify the complexity of several problems based on Abelian permutation groups and linear congruences using logspace counting classes. The problems we consider were defined by McKenzie & Cook (1987).

Central to our study is the problem LCON: given as input (*A*, **b**, *q*), where \(A \in {\mathbb{Z}}^{m \times n}\) and **b** \(\in {\mathbb{Z}}^m\), the problem is to determine if *Ax* = **b** is a feasible system of linear equations over \({\mathbb{Z}}_q\). We assume that *q* is given by its prime factorization \(q = p^{e_1}_{1} p^{e_2}_{2} \cdot \cdot \cdot p^{e_k}_{k}\), such that each \(p^{e_i}_i\) is tiny (i.e. given in unary). We give a randomized NC^{2} algorithm for LCON. More precisely, LCON is in the nonuniform class L^{GapL}/poly. As LCON is hard for L^{GapL} we get a fairly tight characterization of LCON in terms of logspace counting classes. We derive the same upper bound for computing a basis for the nullspace of a linear map from \({\mathbb{Z}}^n_q\) to \({\mathbb{Z}}^m_q\). A number of Abelian permutation group problems studied in McKenzie & Cook (1987) turn out to be logspace Turing equivalent to these linear-algebraic problems. Consequently, the upper and lower bounds also carry over to these problems.

## Keywords.

Logspace counting classes abelian permutation groups linear congruences randomized computation## Subject classification.

68Q15 15A03 15A06 20B05## Preview

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