The Complexity of Solving Linear Equations over a Finite Ring [chapter]

V. Arvind, T. C. Vijayaraghavan
2005 Lecture Notes in Computer Science  
In this paper we first examine the computational complexity of the problem LCON defined as follows: given a matrix A and a column vector b over Z, determine if Ax = b is a feasible system of linear equations over Zq. Here q is also given as part of the input by its prime factorization q = p e 1 1 p e 2 2 . . . p e k k , such that each p e i i is tiny (i.e. given in unary). In [MC87] an NC 3 algorithm is given for this problem. We show that in fact the problem can be solved in randomized NC 2 .
more » ... ore precisely, we show that LCON is in the nonuniform class L GapL /poly. Combined with the hardness of LCON for L GapL , we have a fairly tight characterization of the complexity of LCON in terms of logspace counting classes. We prove the same upper bound results for the problem of testing feasibility of Ax = b over finite rings R with unity, where R is given as part of the input as a table.
doi:10.1007/978-3-540-31856-9_39 fatcat:iqesp3pmsbhltk22sdouahgqyq