Recently the word-based stream ciphers have been the subject of a considerable amount of research. The theory of such stream ciphers requires the study of the complexity of a multisequence. Let S1, S2, . . . , Sm be m N-ary sequences of period T, i.e., a multisequence. The relationship between the joint N-adic complexity and the number of the nonzero columns of the generalized Fourier transform for the N-ary multisequence is determined which generalizes the well-known result about the joint linear complexity and the generalized Fourier transform for a multisequence to the case of the joint N-adic complexity.

Feedback with carry shift registers (FCSRs) implemented using Galois representation have been found to have a weakness called LFSRization. It leads to powerful attacks against the stream ciphers based on them. A new representation called ring representation has been proposed to avoid the attacks. It was considered to circumvent the weaknesses of Galois FCSRs. This correspondence presents a class of ring FCSRs, which meet the implementation criteria, but are still possible to maintain linear behavior for several clock cycles. Their LFSRization probability and how to improve their security are also mentioned.

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.

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.

We regard a De Bruijn sequence of order n as a bijection on $mathbb{F}_2^n$ and consider the transition mappings between them. It is shown that there are only two conjugate transformations that always transfer De Bruijn sequences to De Bruijn sequences.

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.

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.