Feedback functions for generating cycles over a finite alphabet

AbstractIn this note, first there are established simple formulas enabling the calculation of feedback functions that generate a cycle of given length over a given finite field. A theorem communicated in the appendix says that feedback functions producing cycles over a finite field can also be utilized for constructing general feedback functions yielding cycles (in particular, de Bruijn cycles) over an arbitrarily given finite alphabet

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

Avances en el estudio de la complejidad lineal del filtrado no lineal

La mayoría de autores que abordan el tema de la complejidad lineal de secuencias binarias generadas para su aplicación criptográfica se limitan a definir familias muy específicas de filtrados no lineales para los que se puede acotar la complejidad lineal de sus correspondientes secuencias generadas. Aquí se introduce una nueva línea original para el problema de la complejidad lineal para un rango amplio y general de filtrados no lineales. Nuevos conceptos permiten determinar cotas inferiores y superiores al valor de la complejidad linea