A new construction of odd-variable rotation symmetric Boolean functions with optimal algebraic immunity and higher nonlinearity
Introduction
Boolean functions used in cryptographic systems should satisfy a variety of cryptographic properties to resist many attacks. These are balancedness, optimal algebraic immunity, high nonlinearity, high algebraic degree, and good immunity to fast algebraic attacks. Algebraic attack was proposed by Courtois and Meier in 2003 [6]. Since then, several classes of Boolean functions with optimal algebraic immunity have been investigated and constructed (see [2], [3], [7] and the references therein). Boolean functions in cryptography should be kept away from affine functions as far as possible, that is, they should have high nonlinearity. The nonlinearity is strongly related to the resistance of affine approximation attack [8] and fast correlation attack [1].
A Boolean function is said to be rotation symmetric if it is invariant under the action of cyclic group. As we know, a variety of rotation symmetric Boolean functions with optimal algebraic immunity have been proposed [5], [9], [10], [12], [13], [14], [15], [16]. Very recently, in 2019, there were three new works about constructing n-variable rotation symmetric Boolean functions with optimal algebraic immunity, where n is an odd integer. Firstly, Zhao et al. constructed a class of n-variable rotation symmetric Boolean functions with optimal algebraic immunity and nonlinearity [18]. Secondly, Zhang et al. constructed a class of n-variable rotation symmetric Boolean functions with optimal algebraic immunity and nonlinearity [17]. Thirdly, the authors in paper [4] constructed a class of rotation symmetric Boolean functions on odd variables with optimal algebraic immunity and higher nonlinearity, whose nonlinearity can not be simply expressed.
In this paper, we present a new construction of odd-variable rotation symmetric Boolean functions with optimal algebraic immunity by modifying the support of the majority function. The nonlinearity of the newly constructed functions is higher than the nonlinearities of all the known odd-variable rotation symmetric Boolean functions with optimal algebraic immunity. In addition, the algebraic degree and the fast algebraic immunity of these functions on n variables are also analyzed for small values of n.
The rest of this paper is organized as follows. Some basic definitions and necessary preliminaries are reviewed in Section 2. In Section 3, the construction of odd-variable rotation symmetric Boolean functions with optimal algebraic immunity is presented. At the same time, the nonlinearity, the algebraic degree, and the fast algebraic immunity of the newly constructed rotation symmetric Boolean functions are also discussed. Section 4 concludes this paper.
Section snippets
Preliminaries
Let be the n-dimensional vector space over the finite field . Given a vector , define its support as the set , and its Hamming weight as the cardinality of its support, i.e., . Given two vectors and , if for all , then α is said to be covered by β, denoted by .
An n-variable Boolean function f is a mapping from into . We denote by the set of all n-variable Boolean
Construction of RSBFs with optimal algebraic immunity
In this section, we provide a new construction of n-variable RSBFs with optimal algebraic immunity, where with . The nonlinearity, the algebraic degree, and the fast algebraic immunity of the newly constructed RSBFs are also determined.
Conclusion
In this paper, we present a new construction of odd-variable RSBFs with optimal algebraic immunity, whose cryptographic properties such as algebraic immunity, nonlinearity, algebraic degree, and fast algebraic immunity are studied. The nonlinearity of the newly constructed RSBFs is higher than the nonlinearities of all the known odd-variable RSBFs with optimal algebraic immunity, and the algebraic degree is also high enough. How to construct RSBFs with better cryptographic properties is our
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors would like to thank the editors and the anonymous reviewers for their constructive comments and suggestions which improved the quality of the paper. This work is supported by the Key Scientific Research Project of Colleges and Universities in Henan Province (Grant No. 21A413003) and the National Natural Science Foundation of China (Grant No. 61502147).
References (18)
- et al.
Constructing odd-variable RSBFs with optimal algebraic immunity, good nonlinearity and good behavior against fast algebraic attacks
Discrete Appl. Math.
(2019) - et al.
Rotation symmetric Boolean functions-count and cryptographic properties
Discrete Appl. Math.
(2008) - et al.
Balanced 2p-variable rotation symmetric Boolean functions with optimal algebraic immunity
Discrete Appl. Math.
(2016) - et al.
A new construction of rotation symmetric Boolean function with optimal algebraic immunity and higher nonlinearity
Discrete Appl. Math.
(2019) - et al.
Improved fast correlation attacks using parity-check equations of weight 4 and 5
- et al.
Algebraic immunity for cryptographically significant Boolean functions: analysis and construction
IEEE Trans. Inf. Theory
(2006) - et al.
Further properties of several classes of boolean functions with optimum algebraic immunity
Des. Codes Cryptogr.
(2009) - et al.
Construction of even-variable rotation symmetric Boolean functions with optimal algebraic immunity
J. Cryptol.
(2014) - et al.
Algebraic attacks on stream ciphers with linear feedback
Cited by (1)
Balanced odd-variable rotation symmetric Boolean functions with optimal algebraic immunity and higher nonlinearity
2023, Discrete Applied MathematicsCitation Excerpt :Several studies have focused on the construction of RS bent functions [11,25] and RS resilient functions [8,9,20]. In particular, some constructions of RS Boolean functions with optimal AI have been presented in [3,10,19,22–24,28]. The reminder of this paper is organized as follows.