Functions with good differential-uniformity properties have important applications in coding theory and sequence design in addition to the applications in cryptography. The differential spectrum of a cryptographic function is useful for estimating its resistance to some variants of differential cryptanalysis. The objective of this paper is to determine the differential spectrum of the power function $ x^{p^{2k}-p^k+1} $ over $ \mathbb F_{p^n} $, where $ p $ is an odd prime, $ n, k, e $ are integers with $ \gcd(n,k) = e $ and $ \frac{n}{e} $ being odd. In particular, when $ n $ is odd and $ e = 1 $, our result includes a recent one (IEEE Trans. Inform. Theory 65(10): 6819-6826) as a special case.

Let $p$ be a prime. We obtain good bounds for the $p$ -adic sizes of the coefficients of the divided universal Bernoulli number $\tfrac {\hat {B}_n}{n}$ when $n$ is divisible by $p-1$ . As an application, we give a simple proof of Clarke’s 1989 universal von Staudt theorem. We also establish the universal Kummer congruences modulo $p$ for the divided universal Bernoulli numbers for the case $(p-1)|n$ , which is a new result.

In this paper, we propose a new class of optimal one-coincidence FHS (OC-FHS) sets with respect to the Peng-Fan bounds, including prime sequence sets and HMC sequence sets as special cases. Thereafter, through investigating their properties, we determine all of the FHS distances in the OC-FHS set. Finally, for a given positive integer, we also propose a new class of wide-gap one-coincidence FHS (WG-OC-FHS) sets where the FHS gap is larger than the given positive integer. Moreover, such a WG-OC-FHS set is optimal with respect to the WG-Lempel-Greenberger bound and the WG-Peng-Fan bounds simultaneously.

For an odd prime p which is congruent to 3 module 4 and an odd integer k, we investigate the upper bound on the magnitude of cross correlation values of a p-ary m-sequence s(t) and its decimated sequences s(dt + l) for a decimation value d. Using the above upper bound of the magnitude of cross correlation values of pary m-sequence and its decimated sequences, we construct a new class of p-ary sequence families with low correlation property via m-sequence.

Z-complementary pairs (ZCPs) have been widely used in different communication systems. In this paper, we first investigate the odd-periodic correlation property of ZCPs, and propose a new class of ZCPs, called ZOC-ZCPs with zero correlation zone (ZCZ) width $ Z $ and zero odd-period correlation zone (ZOCZ) width $ Z_{odd} = Z $ by horizontal concatenation of a certain combination of some known ZCPs. Particularly, based on any known Golay pair, we can generate a class of GCPs of more flexible length whose ZOCZ width is larger than a quarter of the sequence length.