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: 2.62). 09/2007; DOI: 10.1109/TIT.2007.901189 Source: IEEE Xplore

- IACR Cryptology ePrint Archive. 01/2012; 2012:13.
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**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 $(2\wt(n)+1)2^{\lfloor \log_2 n \rfloor}$IEEE Transactions on Information Theory - TIT. 11/2011; 58(8). -
##### Article: Constructions of 1-resilient Boolean functions on odd number of variables with a high nonlinearity

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**ABSTRACT:**In this paper, we concentrate on the design of 1-resilient Boolean functions with desirable cryptographic properties. Firstly, we put forward a novel secondary construction to obtain 1-resilient functions. Next, we present the relationships between the properties of these constructed 1-resilient functions and that of the initial functions. Based on the construction and a class of bent functions on n variables, we can obtain a class of (n + 3)-variable 1-resilient non-separable cryptographic functions with a high algebraic immunity, whose nonlinearity is equal to the bent concatenation bound 2n + 2 − 2(n + 2)/2. Furthermore, we propose a set of 1-resilient non-separable functions on odd number of variables with an optimal algebraic degree, a high algebraic immunity, and a high nonlinearity. Copyright © 2011 John Wiley & Sons, Ltd.Security and Communication Networks 06/2012; 5(6). · 0.43 Impact Factor

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