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Paper: Reducing the Number of Homogeneous Linear Equations in Finding Annihilators

Authors:
Deepak Kumar Dalai
Subhamoy Maitra
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URL: http://eprint.iacr.org/2006/032
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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, $\nu_f = |\{x | wt(x) > d, f(x) = 1\}|$ and $\mu_f = |\{x | wt(x) \leq d, f(x) = 1\}|$ 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 $\mu_h = |\{x | h(x) = 1, wt(x) \leq d\}|$ is maximized (and in turn $\nu_h$ 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.
BibTeX
@misc{eprint-2006-21525,
  title={Reducing the Number of Homogeneous Linear Equations in Finding Annihilators},
  booktitle={IACR Eprint archive},
  keywords={secret-key cryptography /},
  url={http://eprint.iacr.org/2006/032},
  note={The paper has been accepted in SETA 2006 subho@isical.ac.in 13332 received 27 Jan 2006, last revised 3 Jul 2006},
  author={Deepak Kumar Dalai and Subhamoy Maitra},
  year=2006
}