Let n ≥ 3 be an odd integer,d1=3n-1/2-1, d2=3n-2 andube an element of the finite field F3n. This paper shows thatfu(x)=uxd1+xd2is an almost perfect nonlinear (APN) function on F3n if and only if χ(u+1)= χ(u-1)=χ(u), where χ(∙) denotes the quadratic character of F3n. This settles the open problem raised by Ness and Helleseth in IEEE Trans. Inf. Theory 53(7): 2581-2586, 2007, where only the sufficiency part of the result was proved. Furthermore, we investigate the differential spectra offu(x)for elements u satisfying χ(u+1)=χ(u-1) and express them in terms of several quadratic character sums of cubic polynomials.

Let$n$be a positive integer and$p$a prime. The power mapping$x^{p^{n}-3}$over${\mathbb {F}}_{p^{n}}$has desirable differential properties, and its differential spectra for$p=2,\,3$have been determined. In this paper, for any odd prime$p$, by investigating certain quadratic character sums and some equations over${\mathbb {F}}_{p^{n}}$, we determine the differential spectrum of$x^{p^{n}-3}$with a unified approach. The obtained result shows that for any given odd prime$p$, the differential spectrum can be expressed explicitly in terms of$n$. Compared with previous results, a special elliptic curve over${\mathbb {F}}_{p}$plays an important role in our computation for the general case$p \ge 5$.

Let p be a prime, n = 2m and d = 3pm- 2 with m ≥ 2, and gcd(d, pn- 1) = 1. In this paper, the correlation distribution between a p-ary m-sequence of period pn- 1 and its d-decimation sequence is investigated in a unified approach. Some results for the binary case are extended to the general case. It is shown that the problem of determining the correlation distribution for d can be reduced to that of solving two combinatorial problems related to the unit circle of the finite field Fpn. For an arbitrary odd prime p, it seems difficult to solve these two problems. However, for p = 3, by studying the weight distribution of the ternary Zetterberg code and counting the numbers of solutions of some equations over F3n, the two problems are solved, and thus, the corresponding correlation distribution for d is completely determined. It is noteworthy that this is the first time that the correlation distribution for a non-binary Niho decimation has been determined since 1976.

In this paper, let n = 2k and d = 3 · 2k- 2 with k ≥ 3 and gcd(d, 2n- 1) = 1. Based on some analysis of certain equations over finite fields and the number of codewords with Hamming weight five in Zetterberg code, the correlation distribution between a binary m-sequence of period 2n- 1 and its d-decimation sequence is completely determined. This solves a ten-year-old open problem proposed by Dobbertin et al.

For an odd prime p and two positive integers m, k such that m/gcd (k,m) ≥ 3 is odd, let d be a positive integer satisfying d(pk+1)=2(mod pm-1). In this paper, the cross-correlation between a p-ary m-sequence and its d-decimated sequences is investigated, and the cross correlation distribution is completely determined. The relationship between the decimations d considered in this paper and some known ones is also studied. This paper generalizes some previous results, and also gives new decimations, which lead to low cross correlation.

In this paper, for an odd prime$p$and an integer$n \geq 3$, a new family of$p$-ary sequences of period${{p^{n}-1} \over {2}}$with low correlation is constructed. The new family is constructed by shifts and additions of two decimated sequences of a$p$-ary$m$-sequence, and its family size is$2(p^{n}-1)$. The complete correlation distribution of this new family is derived. It is also shown that the family is optimal with respect to the parameter$\theta_{{\rm rms}}$, which denotes the root mean square of all nontrivial correlations. Compared with the known sequence families, our sequence family is new and has a larger family size, which is four times of its period.