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

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 2-adic 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,