Attack On the Markov Problem

In 2000 Ko gave potential hard problem is proposed called the Markov problem. We give an algorithm, for certain parameters, for solution of the Markov problem. The Markov problem is related to the knot recognition problem. Hence we also a new algorithm the knot recognition problem . This knot recognition algorithm may be used for previously proposed cryptosystem that uses knots. 1 Attack On the Markov Problem 1.1 The Markov Problem The goal of this paper, is to show that the suggested hard problem [2] is solvable, for certain parameters, and we describe probabilistic practical implementation of this algorithm that may give a practical solution to it. There is no known, polynomial complexity, algorithm to solve,for all parameters, the Markov problem. As an aside to solving this problem we get a new knot recognition, algorithm. Every knot can be represented as the closure of a braid. Two di¤erent braids can represent the same knot. We know a knot is ambient isotopic to a closed braid by Alexanders Theorem. A knot is transformed into an equivalent knot by a chain of Reidemeister moves. A knot can be constructed from a braid. Markov in the 1930’s discovered the closure of any two braids is equivalent if they undergo a chain of his Markov moves. A full proof of his idea came later. In 1961 [6] the …rst algorithm given to compare knots was given by Haken. A cryptosystem based on knots was proposed in [5]. Our knot algorithm, given in the section below, may be used with protocol based on knots, for example, possibly with scheme in [5],or a variant of [5]. The Markov problem as stated in [2] is the following (and using the notation of [2]): Instance: y 2 Bn such that y is conjugate to a braid of the form w 1 n 1 for w 2 Bn 1 Objective: Find (z; w) 2 (Bn; Bn 1) such that we have the equation one. zyz 1 = w n 1 where e = 1 or e = 1 (1)