On the constructions of resilient Boolean functions with five-valued Walsh spectra and resilient semi-bent functions
Introduction
In the design of cryptographic Boolean functions, it is necessary to consider their various cryptographic characteristics, such as balancedness, resilience, high nonlinearity, high algebraic degree. The importance of each characteristic depends on the choice of the cryptosystem. Boolean functions with low Walsh transform values have large Hamming distances to the set of affine Boolean functions, so they are highly nonlinear. In this paper, when we say Walsh transform values, we mean the values of Walsh spectral transform coefficients. Bent functions were first introduced by Rothaus in 1976 as Boolean functions having constant magnitude of Walsh transform values [17]. Such functions have been extensively studied because of their important applications in coding theory [1], [11], cryptography [2], sequence designs [16], and graph theory [6], [21]. Notice that bent functions exist only for an even number of variables and cannot be balanced. In fact, nonlinearity and resiliency belong to the number of the most important cryptographic characteristics of Boolean functions required for the resistance of cryptosystems with Boolean functions as building blocks against cryptographic attacks. Therefore, it is very desirable that functions used in ciphers have high nonlinearity and resiliency simultaneously.
The term of semi-bent function has been introduced by Chee, Lee, and Kim at Asiacrypt’94 [5]. Semi-bent functions form an important subclass of Boolean functions with three-valued Walsh spectra. Semi-bent functions are studied in cryptography because they can have desirable properties, such as resilience and high algebraic degree, in addition to the Walsh spectral transform coefficients of low value that provide protection against fast correlation attacks [14] and linear cryptanalysis [13]. Besides their applications in cryptography, it has been stated that they have a wide use in code-division multiple access (CDMA) communication systems [3]. A lot of research has been devoted to finding new families of semi-bent functions (see [3], [7], [9], [15], [22] and the references therein). For example, it is well known that the Gold function Tr is semi-bent [8]. The authors in [10] determined the case where an -linear combination of Gold functions was semi-bent over for odd . The result was generalized to even by Charpin et al. [4]. However, there is only a few known constructions of semi-bent functions, and most of semi-bent functions obtained by primary constructions have low algebraic degrees. In general, it is difficult to characterize all functions with few Walsh transform values.
It is actual the problem of the constructing of Boolean functions that have high nonlinearity and resiliency simultaneously. In this paper, we first present the construction of resilient five-valued functions by modifying the support of Rothaus’s bent function. Secondly, the method of constructing resilient semi-bent functions is presented, and the number of the resilient semi-bent functions is also determined. Lastly, we give the construction of resilient semi-bent functions with maximal algebraic degree.
The rest of this paper is organized as follows. In Section 2, some basic notations and definitions of Boolean functions, five-valued functions, and semi-bent functions are reviewed. In Section 3, the construction of resilient five-valued functions is presented. In Section 4, the method of constructing resilient semi-bent functions with maximal algebraic degree is given. Section 5 concludes this paper.
Section snippets
Preliminaries
Let be the finite field with two elements, be a positive integer, and be the -dimensional vector space over . Given a vector , we define its support as the set and its Hamming weight as the cardinality of its support, i.e., . In this paper, for simplicity, we do not distinguish the vector and the integer if the context is clear, since they are one-to-one corresponding. Given another
Construction of resilient five-valued Boolean functions
In this section, we present the methods of constructing five-valued Boolean functions and resilient five-valued Boolean functions on even number of variables by modifying the support of Rothaus’s bent function.
Construction of resilient semi-bent functions
In this section, we present the method of constructing resilient semi-bent functions on even number of variables. At the same time, the number of the newly constructed resilient semi-bent functions is determined. At last, how to construct resilient semi-bent functions with maximal algebraic degree is given.
Conclusion
In this paper, we first present a construction of resilient five-valued functions by modifying the support of Rothaus’s bent function. Secondly, the method of constructing th order resilient semi-bent functions is presented. At the same time, the number of the newly constructed th order resilient semi-bent functions is determined. Lastly, we give the method of constructing th order resilient semi-bent functions with maximal algebraic degree. How to construct higher order resilient semi-bent
Acknowledgments
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, China (Grant No. 21A413003) and the National Natural Science Foundation of China (Grant No. 61502147).
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