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.

BEAN is a newly proposed lightweight stream cipher adopting Fibonacci FCSRs. It is designed for very constrained environments and aims at providing a balance between security, efficiency and cost. A weakness in BEAN was first found by Å gren and Hell in 2011, resulting in a key recovery attack slightly better than brute force. In this paper, we present new correlations between state and keystream with large statistical advantage, leading to a much more efficient key recovery attack. The time and data complexities of this attack are 257.53 and 259.94, respectively. Moreover, two new output functions are provided as alternatives, which are more efficent than the function used in BEAN and are immune to all attacks proposed on the cipher. Also, suggestions for improving the FCSRs are given.

In the construction of a no-linear key-stream generator, self-shrinking is an established way of getting the binary pseudo-random periodic sequences in cryptography design. In this paper, using the theoretical analysis, we mainly study the self-shrinking sequence based on the l-sequence, and the theoretical results reflect its good cryptography properties accurately, such that it has the last period T = pe(p-1)/2 when T is an odd number, and the expected value of its autocorrelation belongs to {0,1/T and the variance is O(T/ln4T). Furthermore, we find that the 2-adic complexity of the self-shrinking sequence based on the l-sequence is large enough to resist the Rational Approximation attack.

The 2-adic complexity of binary periodic sequences plays an important role in cryptology. In this paper, by means of the usual Fourier transform, we give a simpler form of the upper bound for 2-adic complexity than related result before. For pn-periodic sequences, we discuss the relation between sequences and their Fourier coefficients. Furthermore, based on the relation, we get the lower bound for the number of pn-periodic sequences with given 2-adic complexity.