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21 References# A Note on Symmetric Boolean Functions With Maximum Algebraic Immunity in Odd Number of Variables

**Abstract**

In this note, it is proved that for each odd positive integer n there are exactly two n-variable symmetric Boolean functions with maximum algebraic immunity.

- CitationsCitations45
- ReferencesReferences21

- "They can be represented in a very compact way, both in their algebraic normal forms and in their truth table form [3], and they can be implemented efficiently [17]. Previous work has demonstrated that rotation symmetric Boolean functions is a class of functions, which contains Boolean functions with excellent cryptographic properties [18, 22, 25, 26, 27, 28]. However, as we will show, they are vulnerable to higher order algebraic attacks. "

- "In [13] it was proved that the majority function G n and its complement G n + 1 are the only two trivially balanced symmetric Boolean functions with maximum AI. Finally, it was proved that the number of symmetric Boolean functions with maximum AI is exactly two [16]. For the case where n is even, the situation becomes very complex. "

[Show abstract] [Hide abstract]**ABSTRACT:**Algebraic immunity of Boolean function $f$ is defined as the minimal degree of a nonzero $g$ such that $fg=0$ or $(f+1)g=0$. Given a positive even integer $n$, it is found that the weight distribution of any $n$-variable symmetric Boolean function with maximum algebraic immunity $\frac{n}{2}$ is determined by the binary expansion of $n$. Based on the foregoing, all $n$-variable symmetric Boolean functions with maximum algebraic immunity are constructed. The amount is $(2\wt(n)+1)2^{\lfloor \log_2 n \rfloor}$- "Among all Boolean functions, symmetric Boolean function is an interesting class and their properties are well studied [9], [12], [13]. In [12], [13] , the authors proved that there are only two symmetric Boolean functions on odd number of variables with maximum AI. In Braeken's thesis [15], some symmetric Boolean functions on even variables with maximum AI are constructed. "

[Show abstract] [Hide abstract]**ABSTRACT:**In this paper, we explicitly construct a large class of symmetric Boolean functions on $2k$ variables with algebraic immunity not less than $d$, where integer $k$ is given arbitrarily and $d$ is a given suffix of $k$ in binary representation. If let $d = k$, our constructed functions achieve the maximum algebraic immunity. Remarkably, $2^{\lfloor \log_2{k} \rfloor + 2}$ symmetric Boolean functions on $2k$ variables with maximum algebraic immunity are constructed, which is much more than the previous constructions. Based on our construction, a lower bound of symmetric Boolean functions with algebraic immunity not less than $d$ is derived, which is $2^{\lfloor \log_2{d} \rfloor + 2(k-d+1)}$. As far as we know, this is the first lower bound of this kind.

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