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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.

Abstract. The critical points of the length function on the free loop space Λ(M ) of a compact Riemannian manifold M are the closed geodesics on M. The length function gives a filtration of the homology of Λ(M ) and we show that the Chas-Sullivan product

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