This article is devoted to Boolean and vectorial bent functions and their duals. Our ultimate objective is to increase such functions' corpus by designing new ones covering many previous bent ...functions' constructions. To this end, we provide several new infinite families of bent functions, including idempotent bent functions of any algebraic degree, bent functions in univariate trace form, and self-dual bent functions. Those bent functions are of great theoretical and practical interest because of their special structures and relationship with self-dual codes. In particular, many well-known bent functions are special cases of our bent functions. Moreover, we extend our results to vectorial bent functions and obtain three new infinite classes of vectorial bent functions of any possible degree by determining the explicit duals of three classes of well-known bent functions.
In this study, we propose a flexible construction of complementary sequences (CSs) that can contain zero-valued elements. To derive the construction, we use Boolean functions to represent a ...polynomial generated with a recursion. By applying this representation to recursive CS constructions, we show the impact of construction parameters such as sign, amplitude, phase rotation used in the recursion on the elements of the synthesized CS. As a result, we extend Davis and Jedwab's CS construction by obtaining independent functions for the amplitude and phase of each element of the CS, and the seed sequence positions in the CS. The proposed construction shows that a set of distinct CSs compatible with non-contiguous resource allocations for orthogonal frequency-division multiplexing (OFDM) and various constellations can be synthesized systematically. It also leads to a low peak-to-mean-envelope-power ratio (PMEPR) multiple accessing scheme in the uplink and a low-complexity recursive decoder. We demonstrate the performance of the proposed encoder and decoder through comprehensive simulations.
Due to their beautiful correlation properties and low peak-to-average power ratio (PAPR), Golay-zero correlation zone (ZCZ) sequences are of great value in channel estimation, synchronization, and ...PAPR reduction for orthogonal frequency division multiplexing (OFDM). Inspired by recent constructions of Golay-ZCZ sequences by Chen and Wu, we focus on a new direct construction of Golay-ZCZ sequences with lower PAPR based on generalized Boolean functions (GBFs). The proposed construction leads to a family of cyclically shift distinct ZCZ sequences for which each resultant sequence is also a Golay sequence thus yields PAPR of at most 2.
The field of cryptography has given a lot of attention to rotation symmetric Boolean functions (RSBFs) because they possess special structures and include many functions with good cryptographic ...properties. It is difficult to construct even-variable balanced RSBFs with optimal algebraic immunity in the study on RSBFs. Recently, Mesnager et al. proposed the first and only construction of balanced RSBFs with optimal algebraic immunity for arbitrary even variables (Des. Codes Cryptogr. 89 (1) (2021) 1-17). The nonlinearity of their functions is not high. In this paper, we develop further research based on their construction and present a fresh design of n-variable balanced RSBFs with optimal algebraic immunity for arbitrary even n. Our functions have higher nonlinearity compared to their functions. Furthermore, the algebraic degree and fast algebraic immunity of our functions are not less than n−2 for n not bigger than 16.
A major drawback of orthogonal frequency division multiplexing (OFDM) systems is their high peak-to-mean envelope power ratio (PMEPR). The PMEPR problem can be solved by adopting large codebooks ...consisting of complementary sequences with low PMEPR. In this paper, we present a new construction of polyphase complementary sets (CSs) using generalized Boolean functions (GBFs), which generalizes Schmidt's construction in 2007, Paterson's construction in 2000 and Golay complementary pairs (GCPs) given by Davis and Jedwab in 1999. Compared with Schmidt's approach, our proposed CSs lead to lower PMEPR with higher code-rate for sequences constructed from higher-order (≥ 3) GBFs. We obtain polyphase complementary sequences with maximum PMEPR of <inline-formula> <tex-math notation="LaTeX">2^{k+1} </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">2^{k+2}-2M </tex-math></inline-formula> where <inline-formula> <tex-math notation="LaTeX">k,M </tex-math></inline-formula> are non-negative integers that can be easily derived from the GBF associated with the CS.
We consider the problem of checking the generalized affine equivalence of two given Boolean functions. This problem arises in various computer-aided design (CAD) and cryptographic applications, such ...as circuit synthesis and Reed-Muller codes. Based on the theory of affine group acting on the Boolean functions, we define the coefficient spaces and transition relations, and transform the checking problem into reachability analysis of finite state machines. Two methods are proposed to check the affine equivalence of Boolean functions using Binary Decision Diagrams (BDDs) and Property Directed Reachability (PDR), respectively. Both methods can check affine equivalence of bent and semi-bent functions, which state-of-the-art methods can hardly handle. Furthermore, existing methods only consider the case of affine equivalence, while our methods can handle the generalized affine equivalence of subspaces of Boolean functions. In the application of circuit synthesis, our methods can significantly reduce the size of the library compared to Boolean matching. To classify Boolean functions up to the generalized affine equivalence, we propose a method to obtain a complete classification based on BDDs. In the experiments, we have successfully applied our methods to some examples that can hardly be solved by using the previous methods, thus validating the effectiveness of our methods.
Nowadays, the resistance against algebraic attacks and fast algebraic attacks are considered as an important cryptographic property for Boolean functions used in stream ciphers. Both attacks are very ...powerful analysis concepts and can be applied to symmetric cryptographic algorithms used in stream ciphers. The notion of algebraic immunity has received wide attention since it is a powerful tool to measure the resistance of a Boolean function to standard algebraic attacks. Nevertheless, an algebraic tool to handle the resistance to fast algebraic attacks is not clearly identified in the literature. In the current paper, we propose a new parameter to measure a Boolean function's resistance to fast algebraic attack. We also introduce the notion of fast immunity profile and show that it informs both on the resistance to standard and fast algebraic attacks. Further, we evaluate our parameter for two secondary constructions of Boolean functions. Moreover, A coding-theory approach to the characterization of perfect algebraic immune functions is presented. Via this characterization, infinite families of binary linear complementary dual codes (or LCD codes for short) are obtained from perfect algebraic immune functions. Some of the binary LCD codes presented in this paper are optimal. These binary LCD codes have applications in armoring implementations against so-called side-channel attacks (SCA) and fault non-invasive attacks, in addition to their applications in communication and data storage systems.
Boolean functions and vectorial Boolean functions are the most important nonlinear components of stream ciphers. They should satisfy several criteria such as high nonlinearity, proper resiliency and ...so on to guarantee the security of the whole system. However, there are some constraints among the criteria, and how to achieve a trade-off between them is an important issue. In this paper, some nonlinear Boolean functions possessing simple algebraic normal form with special Walsh spectrum are proposed. By using these functions, we provide two construction methods on balanced and resilient Boolean functions with high nonlinearity. In addition, based on the disjoint linear codes and vector matrices with special properties, some resilient vectorial Boolean functions with currently best-known nonlinearity have also been given.
In cryptography, rotation symmetric Boolean functions (RSBFs) have very significant research value. In this paper, based on the knowledge of integer splitting, a new class of even-variable RSBFs with ...optimal algebraic immunity (AI) was constructed. The new functions have a nonlinearity of
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, which is the highest among all existing RSBFs with optimal AI as well as existing nonlinearity. The algebraic degree of our new construction is as well the highest. Additionally, the results indicate that within the computing capacity of computer, the new class of even-variable RSBFs have almost optimal fast algebraic immunity (FAI).
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