No abstract

Algorithms are proposed for computing the basis of the solution set of a system of linear Diophantine homogeneous or inhomogeneous equations in the residue field modulo a prime number. Keywords: residue field, linear Diophantine equation, basis of a solution set. Linear Diophantine equations and their systems often occur in various application areas of computer science. Problems of pattern recognition and mathematical games [1], cryptography [2], unification [3], cycle parallelization [4], etc.are reduced to the solution of such equations and their systems. In this case, the sets to which belong the coefficients of such equations are the set of integers or the residue ring or residue field modulo some number, and the sets in which solutions are searched for are the set of natural numbers or the same rings and fields. Algorithms of searching for solutions of systems of linear Diophantine equations in the set of natural numbers are described in many publications [5][6][7][8][9][10][11][12]. In the present article, algorithms are considered that solve systems of linear Diophantine equations in the residue field modulo a prime number. The proposed algorithms are based on the TSS method used for constructing the minimum generating solution set of a system of linear homogeneous Diophantine equations in the set of natural numbers [12].

Algorithms are described that solve homogeneous systems of linear Diophantine equations over natural numbers and over the set {0, 1}. Properties of the algorithms and their time estimates are given.