In this survey we summarize properties of pseudorandomness and non-randomness of some number-theoretic sequences and present results on their behaviour under the following measures of pseudorandomness: balance, linear complexity, correlation measure of order k, expansion complexity and 2-adic complexity. The number-theoretic sequences are the Legendre sequence and the two-prime generator, the Thue-Morse sequence and its sub-sequence along squares, and the prime omega sequences for integers and polynomials.

Pseudorandom sequences with large linear complexity can resist the linear attack. The trace representation plays an important role in analysis and design of pseudorandom sequences. In this letter, we present the construction of a family of new binary sequences derived from Euler quotients modulo pq, where pq is a product of two primes and p divides q-1. Firstly, the linear complexity of the sequences are investigated. It is proved that the sequences have larger linear complexity and can resist the attack of Berlekamp-Massey algorithm. Then, we give the trace representation of the proposed sequences by determining the corresponding defining pair. Moreover, we generalize the result to the Euler quotients modulo pmqn with m≤n. Results indicate that the generalized sequences still have high linear complexity. We also give the trace representation of the generalized sequences by determining the corresponding defining pair. The result will be helpful for the implementation and the pseudorandom properties analysis of the sequences.

Compressed sensing is an effective compression algorithm. It is widely used to measure signals in distributed sensor networks (DSNs). Considering the limited resources of DSNs, the measurement matrices used in DSNs must be simple. In this paper, we construct a deterministic measurement matrix based on Gordon-Mills-Welch (GMW) sequence. The column vectors of the proposed measurement matrix are generated by cyclically shifting a GMW sequence. Compared with some state-of-the-art measurement matrices, the proposed measurement matrix has relative lower computational complexity and needs less storage space. It is suitable for resource-constrained DSNs. Moreover, because the proposed measurement matrix can be realized by using simple shift register, it is more practical. The simulation result shows that, in terms of recovery quality, the proposed measurement matrix performs better than some state-of-the-art measurement matrices.

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.

In this paper we study relationships between the linear complexities of a sequence when treated as a sequence over two distinct fields. We obtain bounds for one linear complexity in the form of a constant multiple of the other, where the constant depends only on the fields, not on the particular sequence.

Recently, a family of 4-phase sequences (alphabet {1,j,-1,-j}) was discovered having the same size 2r+1 and period 2r-1 as the family of binary (i.e., {+1, -1}) Gold sequences, but whose maximum nontrivial correlation is smaller by a factor of 2. In addition, the worst-case correlation magnitude remains the same for r odd or even, unlike in the case of Gold sequences. The family is asymptotically optimal with respect to the Welch lower bound on Cmax for complex-valued sequences and the sequences within the family are easily generated using shift registers. This paper aims to provide a more accessible description of these sequences.