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.

A unified construction for yielding optimal and balanced quaternary sequences from ideal/optimal balanced binary sequences was proposed by Zeng et al. In this paper, the linear complexity over finite field 𝔽2, 𝔽4 and Galois ring ℤ4 of the quaternary sequences are discussed, respectively. The exact values of linear complexity of sequences obtained by Legendre sequence pair, twin-prime sequence pair and Hall's sextic sequence pair are derived.

Periodic sequences, used as keys in cryptosystems, plays an important role in cryptography. Such periodic sequences should possess high linear complexity to resist B-M algorithm. Sequences constructed by cyclotomic cosets have been widely studied in the past few years. In this paper, the linear complexity of n-periodic cyclotomic sequences of order 2 and 4 over 𝔽p has been calculated, where n and p are two distinct odd primes. The conclusions reveal that the presented sequences have high linear complexity in many cases, which indicates that the sequences can resist the linear attack.

Linear complexity and the k-error linear complexity of periodic sequences are the important security indices of stream cipher systems. This paper focuses on the distribution of p-error linear complexity of p-ary sequences with period pn. For p-ary sequences of period pn with linear complexity pn-p+1, n≥1, we present all possible values of the p-error linear complexity, and derive the exact formulas to count the number of the sequences with any given p-error linear complexity.

In this paper, new classes of binary generalized cyclotomic sequences of period 2pm+1qn+1 are constructed. These sequences are balanced. We calculate the linear complexity of the constructed sequences with a simple method. The results show that the linear complexity of such sequences attains the maximum.

Pseudorandom number generators are required to generate pseudorandom numbers which have good statistical properties as well as unpredictability in cryptography. An m-sequence is a linear feedback shift register sequence with maximal period over a finite field. M-sequences have good statistical properties, however we must nonlinearize m-sequences for cryptographic purposes. A geometric sequence is a sequence given by applying a nonlinear feedforward function to an m-sequence. Nogami, Tada and Uehara proposed a geometric sequence whose nonlinear feedforward function is given by the Legendre symbol, and showed the period, periodic autocorrelation and linear complexity of the sequence. Furthermore, Nogami et al. proposed a generalization of the sequence, and showed the period and periodic autocorrelation. In this paper, we first investigate linear complexity of the geometric sequences. In the case that the Chan-Games formula which describes linear complexity of geometric sequences does not hold, we show the new formula by considering the sequence of complement numbers, Hasse derivative and cyclotomic classes. Under some conditions, we can ensure that the geometric sequences have a large linear complexity from the results on linear complexity of Sidel'nikov sequences. The geometric sequences have a long period and large linear complexity under some conditions, however they do not have the balance property. In order to construct sequences that have the balance property, we propose interleaved sequences of the geometric sequence and its complement. Furthermore, we show the periodic autocorrelation and linear complexity of the proposed sequences. The proposed sequences have the balance property, and have a large linear complexity if the geometric sequences have a large one.

The k-error linear complexity of a sequence is a fundamental concept for assessing the stability of the linear complexity. After computing the k-error linear complexity of a sequence, those bits that cause the linear complexity reduced also need to be determined. For binary sequences with period 2pn, where p is an odd prime and 2 is a primitive root modulo p2, we present an algorithm which computes the minimum number k such that the k-error linear complexity is not greater than a given constant c. The corresponding error sequence is also obtained.

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.

Pseudorandom number generators have been widely used in Monte Carlo methods, communication systems, cryptography and so on. For cryptographic applications, pseudorandom number generators are required to generate sequences which have good statistical properties, long period and unpredictability. A Dickson generator is a nonlinear congruential generator whose recurrence function is the Dickson polynomial. Aly and Winterhof obtained a lower bound on the linear complexity profile of a Dickson generator. Moreover Vasiga and Shallit studied the state diagram given by the Dickson polynomial of degree two. However, they do not specify sets of initial values which generate a long period sequence. In this paper, we show conditions for parameters and initial values to generate long period sequences, and asymptotic properties for periods by numerical experiments. We specify sets of initial values which generate a long period sequence. For suitable parameters, every element of this set occurs exactly once as a component of generating sequence in one period. In order to obtain sets of initial values, we consider a logistic generator proposed by Miyazaki, Araki, Uehara and Nogami, which is obtained from a Dickson generator of degree two with a linear transformation. Moreover, we remark on the linear complexity profile of the logistic generator. The sets of initial values are described by values of the Legendre symbol. The main idea is to introduce a structure of a hyperbola to the sets of initial values. Our results ensure that generating sequences of Dickson generator of degree two have long period. As a consequence, the Dickson generator of degree two has some good properties for cryptographic applications.

Based on the generalized cyclotomy of order two with respect to n=p1e1+1p2e2+1…ptet+1, where p1, p2, …,pt are pairwise distinct odd primes and e1, e2,…, et are non-negative integers satisfying gcd (piei (pi-1), pjej (pj-1)) = 2 for all i ≠ j, this paper constructs a new family of generalized cyclotomic sequences of order two with length n and investigates their linear complexity. In the view of cascade theory, this paper obtains the linear complexity of a representative sequence.

Based on a unified representation of generalized cyclotomic classes, every generalized cyclotomic sequence of order d over $Z_{p_{1}^{e_{1}}p_{2}^{e_{2}}cdots p_{r}^{e_{r}}}$ is shown to be a sum of d-residue sequences over $Z_{p_{s}^{e_{s}}}$ for $sin {1,2,cdots,r }$. For d=2, by the multi-rate approach, several generalized cyclotomic sequences are explicitly expressed by Legendre sequences, and their linear complexity properties are analyzed.

For an odd prime p and a positive integer r, new classes of binary sequences with period pr+1 are proposed from Euler quotients in this letter, which include several known classes of binary sequences derived from Fermat quotients and Euler quotients as special cases. The advantage of the new constructions is that they allow one to choose their support sets freely. Furthermore, with some constrains on the support set, the new sequences are proved to possess large linear complexities under the assumption of 2p-1 ≢ 1 mod p2.

Let r be an odd prime, such that r≥5 and r≠p, m be the order of r modulo p. Then, there exists a 2pth root of unity in the extension field Frm. Let G(x) be the generating polynomial of the considered quaternary sequences over Fq[x] with q=rm. By explicitly computing the number of zeros of the generating polynomial G(x) over Frm, we can determine the degree of the minimal polynomial, of the quaternary sequences which in turn represents the linear complexity. In this paper, we show that the minimal value of the linear complexity is equal to $ rac{1}{2}(3p-1) $ which is more than p, the half of the period 2p. According to Berlekamp-Massey algorithm, these sequences viewed as enough good for the use in cryptography.

The linear complexity of binary sequences plays a fundamental part in cryptography. In the paper, we construct more general forms of generalized cyclotomic binary sequences with period 2pm+1qn+1. Furthermore, we establish the formula of the linear complexity of proposed sequences. The results reveal that such sequences with period 2pm+1qn+1 have a good balance property and high linear complexity.

The stability theory of stream ciphers plays an important role in designing good stream cipher systems. Two algorithms are presented, to determine the optimal shift and the minimum linear complexity of the sequence, that differs from a given sequence over Fq with period qn-1 by one digit. We also describe how the linear complexity changes with respect to one digit differing from a given sequence.

Jacobi sequences have good cryptography properties. Li et al. [X. Li et al., Linear Complexity of a New Generalized Cyclotomic Sequence of Order Two of Length pq*, IEICE Trans. Fundamentals, vol.E96-A, no.5, pp.1001-1005, 2013] defined a new modified Jacobi sequence of order two and got its linear complexity. In this corresponding, we determine the linear complexity and minimal polynomials of the new modified Jacobi sequence of order d. Our results show that the sequence is good from the viewpoint of linear complexity.

The k-error linear complexity of periodic sequences is an important security index of stream cipher systems. By using an interesting decomposing approach, we investigate the intrinsic structure for the set of 2n-periodic binary sequences with fixed complexity measures. For k ≤ 4, we construct the complete set of error vectors that give the k-error linear complexity. As auxiliary results we obtain the counting functions of the k-error linear complexity of 2n-periodic binary sequences for k ≤ 4, as well as the expectations of the k-error linear complexity of a random sequence for k ≤ 3. Moreover, we study the 2t-error linear complexity of the set of 2n-periodic binary sequences with some fixed linear complexity L, where t < n-1 and the Hamming weight of the binary representation of 2n-L is t. Also, we extend some results to pn-periodic sequences over Fp. Finally, we discuss some potential applications.

In this paper, the linear complexity and minimal polynomials of Legendre sequences over Fq have been calculated, where q = pm and p is a prime number. Our results show that Legendre sequences have high linear complexity over Fq for a large part of prime power number q so that they can resist the linear attack method.

Pseudo-random sequences with high linear complexity play important roles in many domains. We give linear complexity of generalized cyclotomic quaternary sequences with period pq over Z4 via the weights of its Fourier spectral sequence. The results show that such sequences have high linear complexity.