# Reducing the Number of Homogeneous Linear Equations in Finding Annihilators

• Deepak Kumar Dalai
• Subhamoy Maitra
Conference paper

DOI: 10.1007/11863854_33

Part of the Lecture Notes in Computer Science book series (LNCS, volume 4086)
Cite this paper as:
Dalai D.K., Maitra S. (2006) Reducing the Number of Homogeneous Linear Equations in Finding Annihilators. In: Gong G., Helleseth T., Song HY., Yang K. (eds) Sequences and Their Applications – SETA 2006. SETA 2006. Lecture Notes in Computer Science, vol 4086. Springer, Berlin, Heidelberg

## Abstract

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 $$\nu_f \times (\sum_{i=0}^{d} \binom{n}{i} -- \mu_f)$$, where, νf = |{x | wt(x) > d, f(x) = 1}| and μf = |{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 μh = |{x | h(x) = 1, wt(x) ≤d}| is maximized (and in turn νh 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.

### Keywords

Algebraic Attacks Algebraic Normal Form Annihilators Boolean Functions Homogeneous Linear Equations