In this paper, we modify the Legendre-Sidelnikov sequence which was defined by M. Su and A. Winterhof and consider its exact autocorrelation values. This new sequence is balanced for any p,q and proved to possess low autocorrelation values in most cases.

This correspondence contributes to some d-form functions and d-form sequences. A property of d-form functions is obtained firstly. Then we present a way to construct d-form sequences and extended d-form sequences with ideal autocorrelation. Based on our result, many sequences with ideal autocorrelation can be constructed by the corresponding difference-balanced d-form functions.

Binary sequences with good autocorrelation and large linear complexity have found many applications in communication systems. A construction of almost difference sets was given by Cai and Ding in 2009. Many classes of binary sequences with three-level autocorrelation could be obtained by this construction and the linear complexity of two classes of binary sequences from the construction have been determined by Wang in 2010. Inspired by the analysis of Wang, we deternime the linear complexity and the minimal polynomials of another class of binary sequences, i.e., the class based on the WG difference set, from the construction by Cai and Ding. Furthermore, a generalized version of the construction by Cai and Ding is also presented.

The concept of witness hiding was proposed by Feige and Shamir as a natural relaxation of zero-knowledge. Prior constructions of witness hiding protocol for general hard distribution on NP language consist of at least three rounds. In this paper we construct a two-round witness hiding protocol for all hard distributions on NP language. Our construction is based on two primitives: point obfuscation and adaptive witness encryption scheme.

Binary sequences with low autocorrelation have important applications in communication systems and cryptography. In this paper, the autocorrelation values of binary Whiteman generalized cyclotomic sequences of order six and period pq are discussed. Our result shows that the autocorrelation of these sequences is four-valued and that the corresponding values are in {-1,3,-5,pq} if the parameters are chosen carefully.

In the past two decades, many generalized cyclotomic sequences have been constructed and they have been used in cryptography and communication systems for their high linear complexity and low autocorrelation. But there are a few of papers focusing on the 2-adic complexities of such sequences. In this paper, we first give a property of a class of Gaussian periods based on Whiteman's generalized cyclotomic classes of order 4. Then, as an application of this property, we study the 2-adic complexity of a class of Whiteman's generalized cyclotomic sequences constructed from two distinct primes p and q. We prove that the 2-adic complexity of this class of sequences of period pq is lower bounded by pq-p-q-1. This lower bound is at least greater than one half of its period and thus it shows that this class of sequences can resist against the rational approximation algorithm (RAA) attack.

Binary sequences with high linear complexity and high 2-adic complexity have important applications in communication and cryptography. In this paper, the 2-adic complexity of a class of balanced Whiteman generalized cyclotomic sequences which have high linear complexity is considered. Through calculating the determinant of the circulant matrix constructed by one of these sequences, the result shows that the 2-adic complexity of this class of sequences is large enough to resist the attack of the rational approximation algorithm (RAA) for feedback with carry shift registers (FCSRs).