In this paper, we consider the spectra of Boolean functions with respect to the nega-Hadamard transform. Based on the properties of the nega-Hadamard transform and the solutions of the Diophantine equations, we investigate all possibilities of the nega-Hadamard transform of Boolean functions with exactly two distinct nega-Hadamard coefficients.

In this paper, we first recall the concept of 2-tuples distribution matrix, and further study its properties. Based on these properties, we find four special classes of 2-tuples distribution matrices. Then, we provide a new sufficient and necessary condition for n-variable rotation symmetric Boolean functions to be 2-correlation immune. Finally, we give a new method for constructing such functions when n=4t - 1 is prime, and we show an illustrative example.

Boolean functions play an important role in symmetric ciphers. One of important open problems on Boolean functions is determining the maximum possible resiliency order of n-variable Boolean functions with optimal algebraic immunity. In this letter, we search Boolean functions in the rotation symmetric class, and determine the maximum possible resiliency order of 9-variable Boolean functions with optimal algebraic immunity. Moreover, the maximum possible nonlinearity of 9-variable rotation symmetric Boolean functions with optimal algebraic immunity-resiliency trade-off is determined to be 224.

In this letter, we propose a novel direct construction of three-phase Z-complementary triads with flexible lengths and various widths of the zero-correlation zone based on extended Boolean functions. The maximum width ratio of the zero-correlation zone of the construction can reach 3/4. And the proposed sequences can exist for all lengths other than powers of three. We also investigate the peak-to-average power ratio properties of the proposed ZCTs.

Boolean functions with a few Walsh spectral values have important applications in sequence ciphers and coding theory. In this paper, we first construct a class of Boolean functions with at most five-valued Walsh spectra by using the secondary construction of Boolean functions, in particular, plateaued functions are included. Then, we construct three classes of Boolean functions with five-valued Walsh spectra using Kasami functions and investigate the Walsh spectrum distributions of the new functions. Finally, three classes of minimal linear codes with five-weights are obtained, which can be used to design secret sharing scheme with good access structures.

Permutation polynomials have important applications in cryptography, coding theory and combinatorial designs. In this letter, we construct four classes of permutation polynomials over 𝔽2n × 𝔽2n, where 𝔽2n is the finite field with 2n elements.

Low confusion coefficient values can make side-channel attacks harder for vector Boolean functions in Block cipher. In this paper, we give new results of confusion coefficient for f ⊞ g, f ⊡ g, f ⊕ g and fg for different Boolean functions f and g, respectively. And we deduce a relationship on the sum-of-squares of the confusion coefficient between one n-variable function and two (n - 1)-variable decomposition functions. Finally, we find that the confusion coefficient of vector Boolean functions is affine invariant.

In this paper, we study the properties of the sum-of-squares indicator of vectorial Boolean functions. Firstly, we give the upper bound of $sum_{uin mathbb{F}_2^n,vin mathbb{F}_2^m}mathcal{W}_F^3(u,v)$. Secondly, based on the Walsh-Hadamard transform, we give a secondary construction of vectorial bent functions. Further, three kinds of sum-of-squares indicators of vectorial Boolean functions are defined by autocorrelation function and the lower and upper bounds of the sum-of-squares indicators are derived. Finally, we study the sum-of-squares indicators with respect to several equivalence relations, and get the sum-of-squares indicator which have the best cryptographic properties.

We show that every polynomial threshold function that sign-represents the ODD-MAXBITn function has total absolute weight 2Ω(n1/3). The bound is tight up to a logarithmic factor in the exponent.

Let f be a Boolean function in n variables. The Möbius transform and its converse of f can describe the transformation behaviors between the truth table of f and the coefficients of the monomials in the algebraic normal form representation of f. In this letter, we develop the Möbius transform and its converse into a more generalized form, which also includes the known result given by Reed in 1954. We hope that our new result can be used in the design of decoding schemes for linear codes and the cryptanalysis for symmetric cryptography. We also apply our new result to verify the basic idea of the cube attack in a very simple way, in which the cube attack is a powerful technique on the cryptanalysis for symmetric cryptography.

In this paper, some properties of Boolean functions via the unitary transform and c-correlation functions are presented. Based on the unitary transform, we present two classes of secondary constructions for c-bent4 functions. Also, by using the c-correlation functions, a direct link between c-autocorrelation function and the unitary transform of Boolean functions is provided, and the relationship among c-crosscorrelation functions of arbitrary four Boolean functions can be obtained.

Vectorial Boolean functions play an important role in cryptography, sequences and coding theory. Both affine equivalence and EA-equivalence are well known equivalence relations between vectorial Boolean functions. In this paper, we give an exact formula for the number of affine equivalence classes, and an asymptotic formula for the number of EA-equivalence classes of vectorial Boolean functions.

Construction of resilient Boolean functions in odd variables having strictly almost optimal (SAO) nonlinearity appears to be a rather difficult task in stream cipher and coding theory. In this paper, based on the modified High-Meets-Low technique, a general construction to obtain odd-variable SAO resilient Boolean functions without directly using PW functions or KY functions is presented. It is shown that the new class of functions possess higher resiliency order than the known functions while keeping higher SAO nonlinearity, and in addition the resiliency order increases rapidly with the variable number n.

This letter provides a direct construction of binary even-length Z-complementary pairs. To date, the maximum zero correlation zone ratio of Type-I Z-complementary pairs has reached 6/7, but no direct construction of Z-complementary pairs can achieve the zero correlation zone ratio of 6/7. In this letter, based on Boolean function, we give a direct construction of binary even-length Z-complementary pairs with zero correlation zone ratio 6/7. The length of constructed Z-complementary pairs is 2m+3 + 2m + 2+2m+1 and the width of zero correlation zone is 2m+3 + 2m+2.

The r-th nonlinearity of Boolean functions is an important cryptographic criterion associated with higher order linearity attacks on stream and block ciphers. In this paper, we tighten the lower bound of the second-order nonlinearity of a class of Boolean function over finite field F2n, fλ(x)=Tr(λxd), where λ∈F*2r, d=22r+2r+1 and n=7r. This bound is much better than the lower bound of Iwata-Kurosawa.

Linear codes with a few-weight have important applications in combinatorial design, strongly regular graphs and cryptography. In this paper, we first construct a class of Boolean functions with at most five-valued Walsh spectra, and determine their spectrum distribution. Then, we derive two classes of linear codes with at most six-weight from the new functions. Meanwhile, the length, dimension and weight distributions of the codes are obtained. Results show that both of the new codes are minimal and among them, one is wide minimal code and the other is a narrow minimal code and thus can be used to design secret sharing scheme with good access structures. Finally, some Magma programs are used to verify the correctness of our results.

Construction of multiple output functions is one of the most important problems in the design and analysis of stream ciphers. Generally, such a function has to be satisfied with several criteria, such as high nonlinearity, resiliency and high algebraic degree. But there are mutual restraints among the cryptographic parameters. Finding a way to achieve the optimization is always regarded as a hard task. In this paper, by using the disjoint linear codes and disjoint spectral functions, two classes of resilient multiple output functions are obtained. It has been proved that the obtained functions have high nonlinearity and high algebraic degree.

The notion of the signal-to-noise ratio (SNR), proposed by Guilley, et al. in 2004, is a property that attempts to characterize the resilience of (n, m)-functions F=(f1,...,fm) (cryptographic S-boxes) against differential power analysis. But how to study the signal-to-noise ratio for a Boolean function still appears to be an important direction. In this paper, we give a tight upper and tight lower bounds on SNR for any (balanced) Boolean function. We also deduce some tight upper bounds on SNR for balanced Boolean function satisfying propagation criterion. Moreover, we obtain a SNR relationship between an n-variable Boolean function and two (n-1)-variable decomposition functions. Meanwhile, we give SNR(f⊞g) and SNR(f⊡g) for any balanced Boolean functions f, g. Finally, we give a lower bound on SNR(F), which determined by SNR(fi) (1≤i≤m), for (n, m)-function F=(f1,f2,…,fm).

Semi-bent functions have almost maximal nonlinearity. In this paper, two classes of semi-bent functions are constructed by modifying the supports of two quadratic Boolean functions $f_1(x_1,x_2,cdots,x_n)=igopluslimits^{k}_{i=1}x_{2i-1}x_{2i}$ with $n=2k+1geq3$ and $f_2(x_1,x_2,cdots,x_n)=igopluslimits^{k}_{i=1}x_{2i-1}x_{2i}$ with $n=2k+2geq4$. Meanwhile, the algebraic normal forms of the newly constructed semi-bent functions are determined.

In an OFDM communication system using quadrature amplitude modulation (QAM) signals, peak envelope powers (PEPs) of the transmitted signals can be well controlled by using QAM Golay complementary sequence pairs (CSPs). In this letter, by making use of a new construction, a family of new 16-QAM Golay CSPs of length N=2m (integer m≥2) with binary inputs is presented, and all the resultant pairs have the PEP upper bound 2N. However, in the existing such pairs from other references their PEP upper bounds can arrive at 3.6N when the worst case happens. In this sense, novel pairs are good candidates for OFDM applications.