Article
A Note on Symmetric Boolean Functions With Maximum Algebraic Immunity in Odd Number of Variables
Nat. Univ. of Defense Technol., ChangSha
IEEE Transactions on Information Theory (Impact Factor: 2.33). 09/2007; 53(8):2908  2910. DOI: 10.1109/TIT.2007.901189 Source: IEEE Xplore
ABSTRACT
In this note, it is proved that for each odd positive integer n there are exactly two nvariable symmetric Boolean functions with maximum algebraic immunity.

 "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. "

 "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. "
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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 $2^{\lfloor \log_2{d} \rfloor + 2(kd+1)}$. As far as we know, this is the first lower bound of this kind. 
 "The majority function achieves MAI [8] [3]. For odd n, the majority function is the only symmetric MAI functions, up to addition of a constant [12] [18]. However, the majority function was found vulnerable to fast algebraic attacks in [2] at EUROCRYPT'06. "
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ABSTRACT: In this correspondence, first we give a decomposition of symmetric Boolean functions, then we show that almost all symmetric Boolean functions, including these functions with good algebraic immunity, behave badly against fast algebraic attacks. Besides, we improve the relations between algebraic degree and algebraic immunity of symmetric Boolean functions.
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