2-Adic shift registers

Abstract

Pseudorandom sequences, with a variety of statistical properties (such as high linear span, low autocorrelation and pairwise cross-correlation values, and high pairwise hamming distance) are important in many areas of communications and computing (such as cryptography, spread spectrum communications, error correcting codes, and Monte Carlo integration). Binary sequences~ such as m-sequences, more general nonlinear feedback shift register sequences, and summation combiner sequences, have been widely studied by many researchers. Linear feedback shift register hardware can be used to relate certain of these sequences (such as m-sequences) to error correcting codes (such as first order Reed-Muller codes). In this paper a new type of feedback register, feedback with carry shift registers (or FCSRs), will be presented. These relatively simple devices can be used to relate summation combiner sequences, arithmetic codes, and 1/q sequences. We describe an algebraic framework, based on algebra over the numbers, in which the sequences generated by FCSRs can be analyzed, in much the same way that algebra over finite fields can be used to analyze LFSR sequences. As a consequence of this analysis, we present a method for cracking the summation combiner [9] which has been suggested for generating cryptographicaily secure binary sequences. In general, one must consider this 2-adic span as a measure of security along with ordinary linear span. At the same time, FCSRs are a new, general, and therefore exciting, mechanism for generating sequences with enough structure for analysis. Many of the methods of nonlinearization that have been applied to linear feed back shift registers (LFSRs) can be applied to FCSRs, and some of these possibilities are be described here. Hopefully, they will result in sequences with greater cryptologic security. The many threads that are brought together by our analysis have analogues in the theory of LFSRs. In an LFSR, certain register cells are tapped, their contents are added modulo 2 (using exclusive OR gates) and the sum is returned to the first cell of the shift register. Any periodic binary sequence may be realized as the output sequence from some LFSlZ with appropriate taps. Recall some of the well known concepts and consequences which are derived from this point of view.