Abstract not found

Feedback shift registers with carry operation (FCSR’s) are described, implemented, and analyzed with respect to memory requirements, initial loading, period, and distributional properties of their output sequences. Many parallels with the theory of linear feedback shift registers (LFSR’s) are presented, including a synthesis algorithm (analogous to the Berlekamp-Massey algorithm for LFSR’s) which, for any pseudorandom sequence, constructs the smallest FCSR which will generate the sequence. These techniques are used to attack the summation cipher. This analysis gives a unified approach to the study of pseudorandom sequences, arithmetic codes, combiners with memory, and the Marsaglia-Zaman random number generator. Possible variations on the FCSR architecture are indicated at the end. Index Terms – Binary sequence, shift register, stream cipher, combiner with memory, cryptanalysis, 2-adic numbers, arithmetic code, 1/q sequence, linear span. 1

The Berlekamp-Massey algorithm takes a sequence of elements from a field and finds the shortest linear recurrence (or linear feedback shift register) that can generate the sequence. In this paper we extend the algorithm to the case when the elements of the sequence are integers modulo m, where m is an arbitrary integer with known prime decomposition.

Hankel matrices consisting of Catalan numbers have been analyzed by various authors. DesainteCatherine and Viennot found their determinant to be # 1#i#j#k i+j+2n i+j and related them to the Bender - Knuth conjecture. The similar determinant formula # 1#i#j#k i+j-1+2n i+j-1 can be shown to hold for Hankel matrices whose entries are successive middle binomial coe#cients # 2m+1 m # . Generalizing the Catalan numbers in a di#erent direction, it can be shown that determinants of Hankel matrices consisting of numbers 1 3m+1 # 3m+1 m # yield an alternate expression of two Mills -- Robbins -- Rumsey determinants important in the enumeration of plane partitions and alternating sign matrices. Hankel matrices with determinant 1 were studied by Aigner in the definition of Catalan -- like numbers. The well - known relation of Hankel matrices to orthogonal polynomials further yields a combinatorial application of the famous Berlekamp -- Massey algorithm in Coding Theory, which can be applied in order to calculate the coe#cients in the three -- term recurrence of the family of orthogonal polynomials related to the sequence of Hankel matrices.

This is an expository account of a constructive theorem on shortest linear recurrences over an arbitrary integral domain R. A generalisation of rational approximation, which we call 'realization', plays a key role throughout the paper. We also give the associated 'minimal realization' algorithm, which has a simple control structure and is division-free. It is easy to show that the number of R-multiplications required is O(n 2 ), where n is the length of the input sequence. Our approach is algebraic and independent of any particular application. We view a linear recurring sequence as a torsion element in a natural R[X]-module. The standard R[X]-module of Laurent polynomials over R underlies our approach to finite sequences. The prerequisites are nominal and we use short Fibonacci sequences as running examples.

Abstract—We show that the random number generator of Marsaglia and Zaman produces the successive digits of a rational-adic number. (The-adic number system generalizes-adic numbers to an arbitrary integer base.) Using continued fractions, we derive an efficient prediction algorithm for this generator. Index Terms — Continued fractions, inductive inference,-adic numbers, pseudorandom sequences.

We develop a theory of minimal realizations of a finite sequence over an integral domain R, from first principles. Our notion of a minimal realization is closely related to that of a linear recurring sequence and of a partial realization (as in Mathematical Systems Theory). From this theory, we derive Algorithm MR which computes a minimal realization of a sequence of L elements using at most L(5L + 1)=2 R{multiplications. We also characterize all minimal realizations of a given sequence in terms of the computed minimal realization. This algorithm computes the linear complexity of an R sequence, solves non-singular linear systems over R (extending Wiedemann's method), computes the minimal polynomial of an R-matrix, transfer/growth functions and symbolic Padé approximations. There are also a number of applications to Coding Theory. We thus provide a common framework for solving some well-known problems in Systems Theory, Symbolic/Algebraic Computation and Coding Theory.

We propose a slight modification of the Berlekamp-Massey Algorithm for obtaining the minimal polynomial of a given linearly recurrent sequence. Such a modification enables to explain it in a simpler way and to adapt it to lazy evaluation.

Let R be a commutative ring and let n 1: We study (s), the generating function and Ann(s), the ideal of characteristic polynomials of s, an n-dimensional sequence over R.

A Hankel matrix (or persymmetric matrix) is a matrix (a ij ) in which for every r the entries on the diagonal i + j = r are the same, i.e., a i,r-i = c r for some c r . For a sequence c 0 , c 1 , c 2 , . . . of real numbers we consider the collection of Hankel matrices n , k = 0, 1, . . ., n = 1, 2, . . ., where # # # # # . . . . . . . . . . . . # # # # # . (1) So the parameter n denotes the size of the matrix and the 2n 1 successive elements c k , c k+1 , . . . , c k+2n-2 occur in the diagonals of the Hankel matrix. We shall further denote the determinant of a Hankel matrix by n ). Hankel matrices occur in the Berlekamp - Massey algorithm for the decoding of BCH - codes and they found recent applications in Combinatorics motivated by the proof of the refined alternating sign matrix conjecture on the one hand and by the derivation of combinatorial identities for their determinants on the other hand. One such identity concerns the Catalan numbers