### On a conjecture about binary strings distribution

Jean-Pierre Flori, Hugues Randriambololona, Gérard Cohen, and Sihem Mesnager

##### Abstract

It is a difficult challenge to find Boolean functions used in stream ciphers achieving all of the necessary criteria and the research of such functions has taken a significant delay with respect to cryptanalyses. A lot of attacks has led to design criteria for these functions; mainly: balancedness, a high algebraic degree, a high nonlinearity, a good behavior against Fast Algebraic Attacks and also a high algebraic immunity (which is now an absolutely necessary criterion (but not sufficient) for cryptographic Boolean functions). Very recently, an infinite class of Boolean functions has been proposed by Tu and Deng having many very nice cryptographic properties under the assumption that the following combinatorial conjecture about binary strings is true: \begin{cjt} \label{cjt:original} Let $\Stk$ be the following set: $\Stk=\set{(a,b) \in \left(\Zk\right)^2 | a + b = t \text{ and } w(a) + w(b) < k} .$ Then: $\abs{\Stk} \leq 2^{k-1} .$ \end{cjt} The main contribution of the present paper is the reformulation of the problem in terms of {\em carries} which gives more insight on it than simple counting arguments. Successful applications of our tools include explicit formulas of $\abs{\Stk}$ for numbers whose binary expansion is made of one block (see theorem \ref{thm:one}), a proof that the conjecture is {\em asymptotically} true (see theorem \ref{thm:asymptotic}) and a proof that a family of numbers (whose binary expansion has a high number of \textttup{1}s and isolated \textttup{0}s) reaches the bound of the conjecture (see theorem \ref{thm:extremal}). We also conjecture that the numbers in that family are the only ones reaching the bound (see conjecture \ref{cjt:extremal}).

Note: Corrected authors' affiliations.

Available format(s)
Publication info
Published elsewhere. Unknown where it was published
Keywords
boolean functions combinatorics
Contact author(s)
flori @ enst fr
History
2010-04-01: last of 3 revisions
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Short URL
https://ia.cr/2010/170 CC BY

BibTeX

@misc{cryptoeprint:2010/170,
author = {Jean-Pierre Flori and Hugues Randriambololona and Gérard Cohen and Sihem Mesnager},
title = {On a conjecture about binary strings distribution},
howpublished = {Cryptology ePrint Archive, Paper 2010/170},
year = {2010},
note = {\url{https://eprint.iacr.org/2010/170}},
url = {https://eprint.iacr.org/2010/170}
} Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.