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 Prikl. Diskr. Mat. Suppl., 2016, Issue 9, Pages 36–38 (Mi pdma262)  Discrete Functions

On quadratic form rank distribution and asymptotic bounds of affinity level

A. V. Cheremushkinab

a Academy of Cryptography of Russian Federation, Moscow
b Research Institute "Kvant", Moscow

Abstract: An affinity level $\operatorname{la}(f)$ of a Boolean function $f$ is defined as minimal number of variable fixations, that produce an affine function. A general affinity level $\mathcal La(f)$ of Boolean function $f$ is defined as minimal number of fixations of variable linear combination values, that produce an affine function. The general affinity level of quadratic form equals $r$ iff the rank of quadratic form equals $2r$. The paper contains some asymptotic properties of the rank of quadratic forms. As a corollary some asymptotic bounds of the general affinity level of quadratic forms are formulated. Let $0\le c\le1$. If $n=2k+\varepsilon$, $0\le\varepsilon\le1$, then for almost all quadratic forms of $n$ variables,
$$\operatorname{la}(f)\geq\mathcal La(f)\geq k-\lceil\sqrt{\frac12\log_2 k}+\frac{c+(-1)^\varepsilon}2\rceil+1$$
as $n\to\infty$.

Keywords: Boolean functions, affinity level, quadratic form.

DOI: https://doi.org/10.17223/2226308X/9/15  Full text: PDF file (494 kB) References: PDF file   HTML file

UDC: 519.719.325

Citation: A. V. Cheremushkin, “On quadratic form rank distribution and asymptotic bounds of affinity level”, Prikl. Diskr. Mat. Suppl., 2016, no. 9, 36–38 Citation in format AMSBIB
\Bibitem{Che16} \by A.~V.~Cheremushkin \paper On quadratic form rank distribution and asymptotic bounds of affinity level \jour Prikl. Diskr. Mat. Suppl. \yr 2016 \issue 9 \pages 36--38 \mathnet{http://mi.mathnet.ru/pdma262} \crossref{https://doi.org/10.17223/2226308X/9/15} 

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