Methods for Solving Problems of Mathematical Safes on Matrices with Different Types of Locks
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METHODS FOR SOLVING PROBLEMS OF MATHEMATICAL SAFES ON MATRICES WITH DIFFERENT TYPES OF LOCKS
UDC 519.1
A. L. Gurin
Abstract. This paper deals with the problem of mathematical safes on matrices with locks of different types. The method of subsystems separation is used, which was developed and substantiated in previous author’s works for simpler safe types. The problem for safes with locks of two types is investigated. Examples of solving the problem are given. Keywords: state matrix, solution matrix, subsystem of the first kind, subsystem of the second kind, inverse matrix, safe of the same type, simple module, compound module. The concept of a mathematical safe was introduced in [1]. We will use some definitions from this work. A mathematical safe on a matrix is understood to be a system of locks Z = ( z ij ) mn such that if the key in a lock z kl is rotated, then the same rotation occurs in all the locks in the k th row and the lth column. For each mathematical safe, there is the initial state that is specified by a matrix B = ( bij ) mn , where 0 £ bij £ k ij - 1 , and, in this case, k ij corresponds to the number of states of the lock z ij . The problem is as follows: proceeding from the initial state of a safe, find a matrix X = ( x ij ) mn such that, after executing x ij rotations in the corresponding locks, the safe will be at the state B fin = ( bij = 0) mn . If all k ij are equal to one another, i.e., k ij = K , then these safes are called safes with one-type locks. Such safes were considered in [2] for K = 2 and in [3] for arbitrary prime K . We introduce the designations b = ( b11 , b12 , ... , b1n ,... , b21 , b22 , ... , b2 n ,... , b p1 , b p 2 ,... , b pn ,... , bm1 , bm 2 ,... , bmn ) and x = ( x11 , x12 , ..., x1n , x 21 , x 22 , K , x 2 n , K , x m1 , x m 2 , ..., x mn ) . The solution of the safe problem with one-type locks is reduced to the solution of the following system of linear equations:
Ax + b = 0 (mod K ).
(1)
A matrix A of size mn ´ mn is of special form and consists of m 2 cells,
æ Á n E n E n ... ... E n ö ç ÷ ç E n Á n E n ... ... E n ÷ (2) A = ç E n E n Á n ... ... E n ÷ , ç ... ... ... ... ... ... ÷ ç ÷ è E n E n E n ... ... Á n ø where Á n is a matrix of size n ´ n that consists of units and E n is a unit matrix of size n ´ n . The problem of solving the stated problem consists of finding the matrix inverse to A . In [4], due to the specifics of the matrix A , it has been possible to directly find its inverse expression A -1 = Tm ,n ( a 1 , a 2 , a 3 , a 4 ) in the form of the following so-called T -matrix consisting of m 2 submatrices:
æ H n ( a, b ) H n ( g , d) ç H ( g , d) H n ( a, b ) Tm , n ( a, b, g , d) = ç n ... ... ç ç H ( g , d) H ( g , d) n è n
... H n ( g , d) ö ÷ ... H n ( g , d) ÷ , ... ... ÷ ... H n ( a, b ) ÷ø
(3)
National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute,” Kyiv, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 4, July–August, 2019, pp. 166–175. Original article submitted October 17
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