Cryptology ePrint Archive: Report 2014/004

MaxMinMax problem and sparse equations over finite fields

Igor Semaev

Abstract: Asymptotical complexity of sparse equation systems over finite field $F_q$ is studied. Let the variable sets belong to a fixed family $\mathcal{X}=\{X_1,\ldots,X_m\}$ while the polynomials $f_i(X_i)$ are taken independently and uniformly at random from the set of all polynomials of degree $\leq q-1$ in each of the variables in $X_i$. In particular, for $|X_i|\le3$, $m=n$, we prove the average complexity of finding all solutions to $f_i(X_i)=0, i=1,\ldots,m$ by Gluing algorithm ( Semaev, Des. Codes Cryptogr., vol. 49 (2008), pp.47--60) is at most $ q^{\frac{n}{5.7883}+O(\log n)}$ for arbitrary $\mathcal{X}$ and $q$. The proof results from a detailed analysis of 3-MaxMinMax problem, a novel problem for hyper-graphs.

Category / Keywords: complexity theory, equation solving, cryptanalysis

Date: received 2 Jan 2014

Contact author: igor at ii uib no

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Version: 20140103:085826 (All versions of this report)

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