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.

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this letter, a class of four-weight binary cyclic codes are presented. Their weight distributions of these cyclic codes are also settled.

S-box is one of the core components of symmetric cryptographic algorithms, but differential distribution table (DDT) is an important tool to research some properties of S-boxes to resist differential attacks. In this paper, we give a relationship between the sum-of-squares of DDT and the sum-of-squares indicator of (n, m)-functions based on the autocorrelation coefficients. We also get some upper and lower bounds on the sum-of-squares of DDT of balanced (n, m)-functions, and prove that the sum-of-squares of DDT of (n, m)-functions is affine invariant under affine affine equivalent. Furthermore, we obtain a relationship between the sum-of-squares of DDT and the signal-to-noise ratio of (n, m)-functions. In addition, we calculate the distributions of the sum-of-squares of DDT for all 3-bit S-boxes, the 4-bit optimal S-boxes and all 302 balanced S-boxes (up to affine equivalence), data experiments verify our results.

To resist algebraic and fast algebraic attacks, Boolean functions used in stream ciphers should have optimal algebraic immunity and good fast algebraic immunity. One challenge of cryptographic Boolean functions is to determine their ability to resist fast algebraic attacks, which can be measured by their fast algebraic immunities. In this letter, we determine the exact values of fast algebraic immunity of the majority function of 2m and 2m+1 variables. This is the first time that the exact values of the fast algebraic immunity of an infinite class of symmetric Boolean functions with optimal algebraic immunity are determined.

Pseudo-random sequences with good statistical property, such as low autocorrelation, high linear complexity and 2-adic complexity, have been widely applied to designing reliable stream ciphers. In this paper, we explicitly determine the 2-adic complexities of two classes of generalized cyclotomic binary sequences with order 4. Our results show that the 2-adic complexities of both of the sequences attain the maximum. Thus, they are large enough to resist the attack of the rational approximation algorithm for feedback with carry shift registers. We also present some examples to illustrate the validity of the results by Magma programs.

In this letter, we examine the linear complexity of some 3-ary sequences, proposed by No, of period 3n-1(n=3ek, e, k integer) with the ideal autocorrelation property. The exact value of linear complexity k(6e)w is determined when the parameter r =. Furthermore, the upper bound of the linear complexity is given when the other forms of the value r is taken. Finally, a Maple program is designed to illustrate the validity of the results.

We determine the linear complexity of binary sequences derived from the polynomial quotient modulo p defined by $F(u)equiv rac{f(u)-f_p(u)}{p} ~(mod~ p), qquad 0 le F(u) le p-1,~uge 0,$ where fp(u)≡f(u) (mod p), for general polynomials $f(x)in mathbb{Z}[x]$. The linear complexity equals to one of the following values {p2-p,p2-p+1,p2-1,p2} if 2 is a primitive root modulo p2, depending on p≡1 or 3 modulo 4 and the number of solutions of f'(u)≡0 (mod) p, where f'(x) is the derivative of f(x). Furthermore, we extend the constructions to d-ary sequences for prime d|(p-1) and d being a primitive root modulo p2.

Let p be an odd prime number. We define a family of quaternary sequences of period 2p using generalized cyclotomic classes over the residue class ring modulo 2p. We compute exact values of the linear complexity, which are larger than half of the period. Such sequences are 'good' enough from the viewpoint of linear complexity.

Based on the cyclotomy classes of extension fields, a family of binary cyclotomic sequences are constructed and their pseudorandom measures (i.e., the well-distribution measure and the correlation measure of order k) are estimated using certain exponential sums. A lower bound on the linear complexity profile is also presented in terms of the correlation measure.

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.

Low-hit-zone frequency-hopping sequences (LHZ-FHSs) are frequency-hopping sequences with low Hamming correlation in a low-hit-zone (LHZ), which have important applications in quasi-synchronous communication systems. However, the strict quasi-synchronization may be hard to maintain at all times in practical FHMA networks, it is also necessary to minimize the Hamming correlation for time-shifts outside of the LHZ. The main objective of this letter is to propose a lower bound on the maximum correlation magnitude outside the low-hit-zone for LHZ-FHS sets. It turns out that the proposed bound is tight or almost tight in the sense that it can be achieved by some LHZ-FHS sets.

Some new generalized cyclotomic sequences defined by C. Ding and T. Helleseth are proven to exhibit a number of good randomness properties. In this paper, we determine the defining pairs of these sequences of length pm (p prime, m ≥ 2) with order two, then from which we obtain their trace representation. Thus their linear complexity can be derived using Key's method.

MICKEY-family ciphers are lightweight cryptographic primitives and include a register R determined by two related maximal-period linear transformations. Provided that primitivity is efficiently decided in finite fields, it is shown by quantitative analysis that potential parameters for R can be found in probabilistic polynomial time.

Linear codes with a few weights have many applications in secret sharing schemes, authentication codes, association schemes and strongly regular graphs, and they are also of importance in consumer electronics, communications and data storage systems. In this paper, based on the theory of defining sets, we present a class of five-weight linear codes over $mathbb{F}_p$(p is an odd prime), which include an almost optimal code with respect to the Griesmer bound. Then, we use exponential sums to determine the weight distribution.

The global avalanche characteristics measure the overall avalanche properties of Boolean functions, an n-variable balanced Boolean function of the sum-of-square indicator reaching σƒ=22n+2n+3 is an open problem. In this paper, we prove that there does not exist a balanced Boolean function with σƒ=22n+2n+3 for n≥4, if the hamming weight of one decomposition function belongs to the interval Q*. Some upper bounds on the order of propagation criterion of balanced Boolean functions with n (3≤n≤100) variables are given, if the number of vectors of propagation criterion is equal and less than 7·2n-3-1. Two lower bounds on the sum-of-square indicator for balanced Boolean functions with optimal autocorrelation distribution are obtained. Furthermore, the relationship between the sum-of-squares indicator and nonlinearity of balanced Boolean functions is deduced, the new nonlinearity improves the previously known nonlinearity.

Codebooks with good parameters are preferred in many practical applications, such as direct spread CDMA communications and compressed sensing. In this letter, an upper bound on the set size of a codebook is introduced by modifying the Levenstein bound on the maximum amplitudes of such a codebook. Based on an estimate of a class of character sums over a finite field by Katz, a family of codebooks nearly meeting the modified bound is proposed.

We introduce the r-ary sequence with period 2p2 derived from Euler quotients modulo 2p (p is an odd prime) where r is an odd prime divisor of (p-1). Then based on the cyclotomic theory and the theory of trace function in finite fields, we give the trace representation of the proposed sequence by determining the corresponding defining polynomial. Our results will be help for the implementation and the pseudo-random properties analysis of the sequences.

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.

A family of quaternary sequences over Z4 is defined based on the Ding-Helleseth generalized cyclotomic classes modulo pq for two distinct odd primes p and q. The linear complexity is determined by computing the defining polynomial of the sequences, which is in fact connected with the discrete Fourier transform of the sequences. The results show that the sequences possess large linear complexity and are “good” sequences from the viewpoint of cryptography.

We define a family of 2e+1-periodic binary threshold sequences and a family of p2-periodic binary threshold sequences by using Carmichael quotients modulo 2e (e > 2) and 2p (p is an odd prime), respectively. These are extensions of the construction derived from Fermat quotients modulo an odd prime in our earlier work. We determine exact values of the linear complexity, which are larger than half of the period. For cryptographic purpose, the linear complexities of the sequences in this letter are of desired values.