Spherically Punctured Reed-Muller Codes

Dumer, Ilya; Kapralova, Olga

doi:10.1109/tit.2017.2673827

Cited by 2 publications

(2 citation statements)

References 10 publications

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“…As we already explained, the algebraic immunity when G is a power function being not very good, we tried functions G of the form G(y) = y d + y −1 . We obtained the following results: n AI(f F ) 8 4 10 5 12 5 14 6 16 6…”

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confidence: 81%

Parameterization of Boolean functions by vectorial functions and associated constructions

Carlet¹

2024

AMC

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Too few general constructions of Boolean functions satisfying all cryptographic criteria are known. We investigate the construction in which the support of <inline-formula><tex-math id="M1">\begin{document}$ f $\end{document}</tex-math></inline-formula> equals the image set of an injective vectorial function <inline-formula><tex-math id="M2">\begin{document}$ F $\end{document}</tex-math></inline-formula>, that we call a parameterization of <inline-formula><tex-math id="M3">\begin{document}$ f $\end{document}</tex-math></inline-formula>. Every balanced Boolean function can be obtained this way. We study five illustrations of this general construction. The three first correspond to known classes (Maiorana-McFarland, majority functions and balanced functions in odd numbers of variables with optimal algebraic immunity). The two last correspond to new classes:- the sums of indicators of disjoint graphs of <inline-formula><tex-math id="M4">\begin{document}$ (k,n-k $\end{document}</tex-math></inline-formula>)-functions,- functions parameterized through <inline-formula><tex-math id="M5">\begin{document}$ (n-1,n) $\end{document}</tex-math></inline-formula>-functions due to Beelen and Leander.We study the cryptographic parameters of balanced Boolean functions, according to those of their parameterizations: the algebraic degree of <inline-formula><tex-math id="M6">\begin{document}$ f $\end{document}</tex-math></inline-formula>, that we relate to the algebraic degrees of <inline-formula><tex-math id="M7">\begin{document}$ F $\end{document}</tex-math></inline-formula> and of its graph indicator, the nonlinearity of <inline-formula><tex-math id="M8">\begin{document}$ f $\end{document}</tex-math></inline-formula>, that we relate by a bound to the nonlinearity of <inline-formula><tex-math id="M9">\begin{document}$ F $\end{document}</tex-math></inline-formula>, and the algebraic immunity (AI), whose optimality is related to a natural question in linear algebra, and which may be approached (in two ways) by using the graph indicator of <inline-formula><tex-math id="M10">\begin{document}$ F $\end{document}</tex-math></inline-formula>. We revisit each of the five classes for each criterion. The fourth class is very promising, thanks to a lower bound on the nonlinearity by means of the nonlinearity of the chosen <inline-formula><tex-math id="M11">\begin{document}$ (k,n-k $\end{document}</tex-math></inline-formula>)-functions. The sub-class of the sums of indicators of affine functions, for which we prove an upper bound and a lower bound on the nonlinearity, seems also interesting. The fifth class includes functions with an optimal algebraic degree, good nonlinearity and good AI.

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confidence: 81%

Parameterization of Boolean functions by vectorial functions and associated constructions

Carlet¹

2024

AMC

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“…Reed Muller codes RM (r, n) are binary codes of length 2 n whose codewords are the evaluations of all Boolean functions of algebraic degree at most r in n variables on their 2 n entries. Fixing the Hamming weight to the entries to k gives the spherically punctured Reed-Muller codes studied by Kapralova and Dumer [DK13,DK17]. The properties of these codes are connected to Boolean functions with fixed weight entries.…”

Section: Spherically Punctured Reed-muller Codesmentioning

confidence: 99%

On the weightwise nonlinearity of weightwise perfectly balanced functions

Gini

Méaux

2022

Discrete Applied Mathematics

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Spherically Punctured Reed-Muller Codes

Cited by 2 publications

References 10 publications

Parameterization of Boolean functions by vectorial functions and associated constructions

Parameterization of Boolean functions by vectorial functions and associated constructions

On the weightwise nonlinearity of weightwise perfectly balanced functions

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