Reducing the Number of Homogeneous Linear Equations in Finding Annihilators

摘要

Given a Boolean function f on n-variables, we find a reduced set of homogeneous linear equations by solving which one can decide whether there exist annihilators at degree d or not. Using our method the size of the associated matrix becomes v_f x (∑_(i=0)~d (_i~n) - μ_f), where, v_f = |{x|wt(x) > d, f(x) = 1}| and μ_f = |{x|wt(x) ≤ d, f(x) = 1}| and the time required to construct the matrix is same as the size of the matrix. This is a preprocessing step before the exact solution strategy (to decide on the existence of the annihilators) that requires to solve the set of homogeneous linear equations (basically to calculate the rank) and this can be improved when the number of variables and the number of equations are minimized. As the linear transformation on the input variables of the Boolean function keeps the degree of the annihilators invariant, our preprocessing step can be more efficiently applied if one can find an affine transformation over f(x) to get h(x) = f(Bx + b) such that μ_h = |{x|h(x) = 1, wt(x) ≤ d}| is maximized (and in turn v_h is minimized too). We present an efficient heuristic towards this. Our study also shows for what kind of Boolean functions the asymptotic reduction in the size of the matrix is possible and when the reduction is not asymptotic but constant.