**Efficient probabilistic algorithm for estimating the algebraic properties of Boolean functions for large $n$**

*Yongzhuang Wei and Enes Pasalic and Fengrong Zhang and Samir Hod\v zić*

**Abstract: **Although several methods for estimating the resistance of a random Boolean function against (fast) algebraic attacks
were proposed, these methods are usually infeasible in practice for relative large input variables $n$ (for instance $n\geq 30)$ due to increased computational complexity.
An efficient estimation the resistance of Boolean function (with relative large input variables $n$) against (fast) algebraic attacks appears to be a rather difficult task. In this paper, the concept of partial linear relations decomposition is introduced, which decomposes any given nonlinear Boolean function into
many linear (affine) subfunctions by using the disjoint sets of input variables. Based on this result, a general probabilistic decomposition algorithm for nonlinear Boolean functions is presented which gives a new framework for estimating the resistance of Boolean function against (fast) algebraic attacks. It is shown that our new probabilistic method gives very tight estimates (lower and upper bound) and it only requires about $O(n^22^n)$ operations for a random Boolean function with $n$ variables, thus having much less time complexity than previously known algorithms.

**Category / Keywords: **Stream ciphers, fast algebraic attacks, time complexity, algebraic immunity.

**Date: **received 2 Jul 2016, last revised 4 Jul 2016

**Contact author: **walker_wei at msn com;zhfl203@cumt edu cn

**Available format(s): **PDF | BibTeX Citation

**Version: **20160706:054628 (All versions of this report)

**Short URL: **ia.cr/2016/671

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