Representing the Integer Factorization Problem Using Ordered Binary Decision Diagrams

A method is given to reduce the problem of finding a nontrivial factorization of a positive integer alpha, greater than one, to the problem of finding a solution to a system of Boolean equations, that is, a system of equations such that each equation is of the form f=g where f and g are Boolean functions, meaning {0,1}-valued functions in zero or more Boolean ({0,1}-valued) variables. Our system is obtained by applying a sequence of reductions to an initial system of equations of the form {fi((x) over right arrow,(y) over right arrow),=alpha i|i is an element of{0,,2n}} where for each i fi((x) over right arrow,(y) over right arrow)=fi(x0...,,xn,y0...,,yn) gives the coefficient of 2i in the binary expansion of (x0+2x1+...+2(n)xn)(y0+2y1+...+2nyn), alpha i gives the coefficient of 2i in the binary expansion of alpha, and xi and yi are {0,1}-valued variables. That is, the initial system represents a binary multiplier whose output bits have been set equal to the bits of alpha. It is shown that each Boolean function in our reduced system, that is, each Boolean function (g-h) mod2={0 double left right arrow g=h/1 double left right arrow g not equal h such that g=h is an equation in the reduced system, can be represented by a type of graph called an ordered binary decision diagram (OBDD) with an upper bound on its number of vertices of O(log2(alpha/log2(alpha))3. Previous work has shown that the initial system has at least one Boolean function with an OBDD representation with number of vertices exponential in log2 (alpha).