A class of Boolean functions with four-valued Walsh spectra

Authors:
Yonghong Xie;Lei Hu;Wenfeng Jiang;Xiangyong Zeng
Affiliations:
State Key Laboratory of Information Security, Graduate University of Chinese Academy of Sciences, Beijing, China;State Key Laboratory of Information Security, Graduate University of Chinese Academy of Sciences, Beijing, China;State Key Laboratory of Information Security, Graduate University of Chinese Academy of Sciences, Beijing, China;Faculty of Mathematics and Computer Science, Hubei University, Wuhan, China
Venue:
APCC'09 Proceedings of the 15th Asia-Pacific conference on Communications
Year:
2009

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Abstract

In this paper, we propose an infinite class of Boolean functions with four-valued Walsh spectra. These functions have a simple trace expression of the form f(x) = tr1n(xd(2n+1))+ tr12n(bx) for b ∈ F22n and d satisfying d(2l+1) = 2i(mod 2n-1) with integers l and i, where x ∈ F22n. Their cryptographic properties, including balancedness, spectrum distribution, nonlinearity, algebraic degree and algebraic immunity, are investigated. We prove that the proposed functions have high nonlinearity, and algebraic degrees n - gcd(n, l) + 2. Our computer simulation shows these functions have optimal or suboptimal algebraic immunity.