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Construction and Count of Boolean Functions of an Odd Number of Variables with Maximum Algebraic Immunity

Na Li·Wen-Feng Qi
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Abstract

Algebraic immunity has been proposed as an important property of Boolean functions. To resist algebraic attack, a Boolean function should possess high algebraic immunity. It is well known now that the algebraic immunity of an $n$-variable Boolean function is upper bounded by $\left\lceil {\frac{n}{2}} \right\rceil $. In this paper, for an odd integer $n$, we present a construction method which can efficiently generate a Boolean function of $n$ variables with maximum algebraic immunity, and we also show that any such function can be generated by this method. Moreover, the number of such Boolean functions is greater than $2^{2^{n-1}}$.

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Na Li, Wen-Feng Qi. 2006-05-30. Construction and Count of Boolean Functions of an Odd Number of Variables with Maximum Algebraic Immunity. https://arxiv.org/abs/cs/0605139

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