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.

This paper proposes a new multi-value sequence generated by utilizing primitive element, trace, and power residue symbol over odd characteristic finite field. In detail, let p and k be an odd prime number as the characteristic and a prime factor of p-1, respectively. Our proposal generates k-value sequence T={ti | ti=fk(Tr(ωi)+A)}, where ω is a primitive element in the extension field $F{p}{m}$, Tr(⋅) is the trace function that maps $F{p}{m} ightarrow {p}$, A is a non-zero scalar in the prime field ${p}$, and fk(⋅) is a certain mapping function based on k-th power residue symbol. Thus, the proposed sequence has four parameters as p, m, k, and A. Then, this paper theoretically shows its period, autocorrelation, and cross-correlation. In addition, this paper discusses its linear complexity based on experimental results. Then, these features of the proposed sequence are observed with some examples.

A pseudorandom number generator is widely used in cryptography. A cryptographic pseudorandom number generator is required to generate pseudorandom numbers which have good statistical properties as well as unpredictability. 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 binary 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. They showed the geometric sequences have good properties for the period, periodic autocorrelation and linear complexity. However, the geometric sequences do not have the balance property. In this paper, we introduce geometric sequences of two types and show some properties of interleaved sequences of the geometric sequences of two types. These interleaved sequences have the balance property and double the period of the geometric sequences by the interleaved structure. Moreover, we show correlation properties and linear complexity of the interleaved sequences. A key of our observation is that the second type geometric sequence is the complement of the left shift of the first type geometric sequence by half-period positions.

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.