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Feedback shift registers, 2adic span, and combiners with memory
 Journal of Cryptology
, 1997
"... 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 presen ..."
Abstract

Cited by 50 (7 self)
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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 BerlekampMassey 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 MarsagliaZaman 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, 2adic numbers, arithmetic code, 1/q sequence, linear span. 1
Algebraic feedback shift registers
 Theoretical Comp. Sci
, 1999
"... 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 underl ..."
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Cited by 8 (3 self)
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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; pseudorandom number generator. 1
Register synthesis for algebraic feedback shift registers based on nonprimes
 DESIGNS, CODES, AND CRYPTOGRAPHY
"... 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=(). W ..."
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Cited by 5 (2 self)
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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].
Periodicity, Correlation, and Distribution Properties of dFCSR sequences
 SIAM J. Comp
, 2000
"... A dfeedbackwithcarry shift register (dFCSR) is a finite state machine, similar to a linear feedback shift register, in which a small amount of memory and a delay (by dclock cycles) is used in the feedback algorithm (see [4, 5]). The output sequences of these simple devices may be described usi ..."
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Cited by 3 (1 self)
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A dfeedbackwithcarry shift register (dFCSR) is a finite state machine, similar to a linear feedback shift register, in which a small amount of memory and a delay (by dclock cycles) is used in the feedback algorithm (see [4, 5]). The output sequences of these simple devices may be described using arithmetic in a ramified extension field of the rational numbers. In this paper we show how many of these sequences may also be described using simple integer arithmetic, and consequently how to find such sequences with large periods. We also analyze the "arithmetic crosscorrelation" between pairs of these sequences and show that it often vanishes identically. Finally we study the distribution properties of short subsequences of a dFCSR sequence.
Periodicity and Correlation Properties of dFCSR Sequences
, 2001
"... Abstract. A dfeedbackwithcarry shift register (dFCSR) is a finite state machine, similar to a linear feedback shift register (LFSR), in which a small amount of memory and a delay (by dclock cycles) is used in the feedback algorithm (see Goresky and Klapper [4,5]). The output sequences of these ..."
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Cited by 1 (0 self)
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Abstract. A dfeedbackwithcarry shift register (dFCSR) is a finite state machine, similar to a linear feedback shift register (LFSR), in which a small amount of memory and a delay (by dclock cycles) is used in the feedback algorithm (see Goresky and Klapper [4,5]). The output sequences of these simple devices may be described using arithmetic in a ramified extension field of the rational numbers. In this paper we show how many of these sequences may also be described using simple integer arithmetic, and consequently how to find such sequences with large periods. We also analyze the ‘‘arithmetic crosscorrelation’’ between pairs of these sequences and show that it often vanishes identically.
Cryptanalysis Based on . . .
, 1995
"... This paper presents a new algorithm for cryptanalytically attacking stream ciphers. There is an associated measure of security, the 2adac 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 a ..."
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This paper presents a new algorithm for cryptanalytically attacking stream ciphers. There is an associated measure of security, the 2adac 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 2adic 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.