IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
Online ISSN : 1745-1337
Print ISSN : 0916-8508
Regular Section
Constructing Rotation Symmetric Boolean Functions with Maximum Algebraic Immunity on an Odd Number of Variables
Jie PENGHaibin KAN
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2012 Volume E95.A Issue 6 Pages 1056-1064

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Abstract

It is well known that Boolean functions used in stream and block ciphers should have high algebraic immunity to resist algebraic attacks. Up to now, there have been many constructions of Boolean functions achieving the maximum algebraic immunity. In this paper, we present several constructions of rotation symmetric Boolean functions with maximum algebraic immunity on an odd number of variables which are not symmetric, via a study of invertible cyclic matrices over the binary field. In particular, we generalize the existing results and introduce a new method to construct all the rotation symmetric Boolean functions that differ from the majority function on two orbits. Moreover, we prove that their nonlinearities are upper bounded by $2^{n-1}-\binom{n-1}{\lfloor\frac{n}{2}\rfloor}+2(n-6)$.

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© 2012 The Institute of Electronics, Information and Communication Engineers
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