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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 ..."
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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
2adic shift registers
 In Fast Software Encryption  FSE’93, v. 809 of Lecture Notes in Computer Science
, 1993
"... Pseudorandom sequences, with a variety of statistical properties (such as high linear span, low autocorrelation and pairwise crosscorrelation values, and high pairwise hamming distance) are important in many areas of communications and computing (such as cryptography, spread spectrum communications ..."
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Cited by 18 (5 self)
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Pseudorandom sequences, with a variety of statistical properties (such as high linear span, low autocorrelation and pairwise crosscorrelation 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 msequences, 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 msequences) to error correcting codes (such as first order ReedMuller 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 2adic 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,
Feedback Registers Based on Ramified Extensions of the 2Adic Numbers (Extended Abstract)
 Advances in Cryptology  Eurocrypt 1994. Lecture Notes in Computer Science 718
, 1995
"... A new class of feedback register, based on ramified extensions of the 2adic numbers, is described. An algebraic framework for the analysis of these registers and the sequences they output is given. This framework parallels that of linear feedback shift registers. As one consequence of this, a metho ..."
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Cited by 11 (4 self)
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A new class of feedback register, based on ramified extensions of the 2adic numbers, is described. An algebraic framework for the analysis of these registers and the sequences they output is given. This framework parallels that of linear feedback shift registers. As one consequence of this, a method for cracking summation ciphers is given. These registers give rise to new measures of cryptologic security.
Distributional Properties of dFCSR Sequences
"... In this paper we study the distribution properties of dFCSR sequences. These sequences have ecient generators and have several good statistical properties. We show that for d = 2 the number of occurrences of an xed size subsequence diers from the average number of occurrences by at most a small ..."
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Cited by 3 (1 self)
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In this paper we study the distribution properties of dFCSR sequences. These sequences have ecient generators and have several good statistical properties. We show that for d = 2 the number of occurrences of an xed size subsequence diers from the average number of occurrences by at most a small constant times the square root of the average.
On Decimations of lSequences
, 2002
"... Maximal length Feedback with Carry Shift Register sequences have several remarkable statistical properties. Among them is the property that the arithmetic correlations between any two cyclically distinct decimations are precisely zero. It is open, however, whether all such pairs of decimations are i ..."
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Cited by 2 (0 self)
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Maximal length Feedback with Carry Shift Register sequences have several remarkable statistical properties. Among them is the property that the arithmetic correlations between any two cyclically distinct decimations are precisely zero. It is open, however, whether all such pairs of decimations are indeed cyclically distinct. In this paper we show that the set of distinct decimations is large and, in some cases, all decimations are distinct. 1
A Boolean function is a function
"... In this paper we introduce an arithmetic Walsh transform. It is a withcarry analog, based on modular arithmetic, of the usual Walsh transform of Boolean functions. We first develop some tools for analyzing arithmetic Walsh transforms. We then prove that the mapping from a Boolean function to its ar ..."
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In this paper we introduce an arithmetic Walsh transform. It is a withcarry analog, based on modular arithmetic, of the usual Walsh transform of Boolean functions. We first develop some tools for analyzing arithmetic Walsh transforms. We then prove that the mapping from a Boolean function to its arithmetic Walsh transform is injective. We then compute the average arithmetic Walsh transforms and the arithmetic Walsh transforms of affine functions.