The Adjacency Graphs of Linear Feedback Shift Registers with Primitive-like Characteristic Polynomials

We consider the adjacency graphs of the linear feedback shift registers (LFSRs) with characteristic polynomials of the form l(x)p(x), where l(x) is a polynomial of small degree and p(x) is a primitive polynomial. It is shown that, their adjacency graphs are closely related to the association graph of l(x) and the cyclotomic numbers over finite fields. By using this connection, we give a unified method to determine their adjacency graphs. As an application of this method, we explicitly calculate the adjacency graphs of LFSRs with characteristic polynomials of the form (1+x+x^3+x^4)p(x), and construct a large class of De Bruijn sequences from them

De Bruijn Sequences from Joining Cycles of Nonlinear Feedback Shift Registers

De Bruijn sequences are a class of nonlinear recurring sequences that have wide applications in cryptography and modern communication systems. One main method for constructing them is to join the cycles of a feedback shift register (FSR) into a full cycle, which is called the cycle joining method. Jansen et al. (IEEE Trans on Information Theory 1991) proposed an algorithm for joining cycles of an arbitrary FSR. This classical algorithm is further studied in this paper. Motivated by their work, we propose a new algorithm for joining cycles, which doubles the efficiency of the classical cycle joining algorithm. Since both algorithms need FSRs that only generate short cycles, we also propose efficient ways to construct short-cycle FSRs. These FSRs are nonlinear and are easy to obtain. As a result, a large number of de Bruijn sequences are constructed from them. We explicitly determine the size of these de Bruijn sequences. Besides, we show a property of the pure circulating register, which is important for searching for short-cycle FSRs

Codes and Sequences for Information Retrieval and Stream Ciphers

Given a self-similar structure in codes and de Bruijn sequences, recursive techniques may be used to analyze and construct them. Batch codes partition the indices of code words into m buckets, where recovery of t symbols is accomplished by accessing at most tau in each bucket. This finds use in the retrieval of information spread over several devices. We introduce the concept of optimal batch codes, showing that binary Hamming codes and first order Reed-Muller codes are optimal. Then we study batch properties of binary Reed-Muller codes which have order less than half their length. Cartesian codes are defined by the evaluation of polynomials at a subset of points in F_q. We partition F_q into buckets defined by the quotient with a subspace V. Several properties equivalent to (V intersect ) = {0} for all i,j between 1 and mu are explored. With this framework, a code in F_q^(mu-1) capable of reconstructing mu indices is expanded to one in F_q^(mu) capable of reconstructing mu+1 indices. Using a base case in F_q^3, we are able to prove batch properties for codes in F_q. We generalize this to Cartesian Codes with a limit on the degree mu of the polynomials. De Bruijn sequences are cyclic sequences of length q^n that contain every q-ary word of length n exactly once. The pseudorandom properties of such sequences make them useful for stream ciphers. Under a particular homomorphism, the preimages of a binary de Bruijn sequence form two cycles. We examine a method for identifying points where these sequences may be joined to make a de Bruijn sequence of order n. Using the recursive structure of this construction, we are able to calculate sums of subsequences in O(n^4 log(n)) time, and the location of a word in O(n^5 log(n)) time. Together, these functions allow us to check the validity of any potential toggle point, which provides a method for efficiently generating a recursive specification. Each successful step takes O(k^5 log(k)), for k from 3 to n

Proceedings of AUTOMATA 2010: 16th International workshop on cellular automata and discrete complex systems

International audienceThese local proceedings hold the papers of two catgeories: (a) Short, non-reviewed papers (b) Full paper