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

A feedback-with-carry shift register (FCSR) with "Fibonacci" architecture is a shift register provided with a small amount of memory which is used in the feedback algorithm. Like the linear feedback shift register (LFSR), the FCSR provides a simple and predictable method for the fast generation of pseudorandom sequences with good statistical properties and large periods. In this paper, we describe and analyze an alternative architecture for the FCSR which is similar to the "Galois" architecture for the LFSR. The Galois architecture is more efficient than the Fibonacci architecture because the feedback computations are performed in parallel. We also describe the output sequences generated by the-FCSR, a slight modification of the (Fibonacci) FCSR architecture in which the feedback bit is delayed for clock cycles before being returned to the first cell of the shift register. We explain how these devices may be configured so as to generate sequences with large periods. We show that the -FCSR also admits a more efficient "Galois" architecture.

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A general framework for the design of feedback registers based on algebra over complete rings is described. These registers generalize linear feedback shift registers and feedback with carry shift registers. Basic properties of the output sequences are studied: relations to the algebra of the underlying ring; synthesis of the register from the sequence (which has implications for cryptanalysis); and basic statistical properties. These considerations lead to security measures for stream ciphers, analogous to the notion of linear complexity that arises from linear feedback shift registers. We also show that when the underlying ring is a polynomial ring over a finite field, the new registers can be simulated by linear feedback shift registers with small nonlinear filters. Key words: cryptography; feedback shift register; complete ring; stream cipher; pseudo-random number generator. 1 Introduction Linear Feedback Shift Registers (LFSRs) [3] have long been the basis of most research on ...

Recently, a new class of feedback shift registers (FCSRs) was introduced, based on algebra over the 2-adic numbers. The sequences generated by these registers have many algebraic properties similar to those generated by linear feedback shift registers. However, it appears to be significantly more difficult to find maximal period FCSR sequences. In this paper we exhibit a technique for easily finding FCSRs that generate nearly maximal period sequences. We further show that these sequence have excellent distributional properties. They are balanced, and nearly have the deBruijn property for distributions of subsequences.

In this paper, we describe a solution to the register synthesis problem for a class of sequence generators known as Algebraic Feedback Shift Registers (or AFSRs). These registers are based on the algebra of -adic numbers, where is an element in a ring R, and produce sequences of elements in R=(). We give several cases where the register synthesis problem can be solved by an ecient algorithm. Consequently, any keystreams over R=() used in stream ciphers must be unable to be generated by a small register in these classes. This paper extends the analyses of feedback with carry shift registers and algebraic feedback shift registers by Goresky, Klapper, and Xu [4, 5, 11].

models for answering questions on the existence of secure families of sequence generators. 5. Design and analysis of families of sequences for secure spread-spectrum communications. These sequences include geometric sequences and d-form sequences (the latter invented by me).

This paper presents a new algorithm for cryptanalytically attacking stream ciphers. There is an associated measure of security, the 2-adac 8pan. In order for a stream cipher to be secure, its Zadic span must be large. This attack exposes a weakness of Rueppel and Massey's summation combiner. The algorithm, based on De Weger and Mahler's rational approximation theory for 2-adic numbers, synthesizes a shortest feedback with cam shaft qwter that outputs a particular key stream, given a small number of bits of the key stream. It is adaptive in that it does not neeed to know the number of available bits beforehand.