### 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.

Available format(s)
Publication info
Preprint. MINOR revision.
Keywords
complexity theoryequation solvingcryptanalysis
Contact author(s)
igor @ ii uib no
History
Short URL
https://ia.cr/2014/004

CC BY

BibTeX

@misc{cryptoeprint:2014/004,
author = {Igor Semaev},
title = {MaxMinMax problem  and  sparse equations over finite fields},
howpublished = {Cryptology ePrint Archive, Paper 2014/004},
year = {2014},
note = {\url{https://eprint.iacr.org/2014/004}},
url = {https://eprint.iacr.org/2014/004}
}

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