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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
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.
JOURNAL OF NUMBER THEORY 24. 7C88 ( 1986) Approximation Lattices of padic Numbers
, 1984
"... Approximation lattices occur in a natural way in the study of rational approximations to padic numbers. Periodicity of a sequence of approximation lattices is shown to occur for rational and quadratic padic numbers. and for those only, thus establishing a padic analogue of Lagrange’s theorem on p ..."
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Approximation lattices occur in a natural way in the study of rational approximations to padic numbers. Periodicity of a sequence of approximation lattices is shown to occur for rational and quadratic padic numbers. and for those only, thus establishing a padic analogue of Lagrange’s theorem on periodic continued fractions. Using approximation lattices we derive upper and lower bounds for the best approximations to a padic number, thus establishing the padic analogue of a theorem of Hurwitz. ‘q1 ’ 19X6 Academic Press, Inc 1.
THE METRIC THEORY OF p−ADIC APPROXIMATION
, 907
"... Abstract. Metric Diophantine approximation in its classical form is the study of how well almost all real numbers can be approximated by rationals. There is a long history of results which give partial answers to this problem, but there are still questions which remain unknown. The DuffinSchaeffer ..."
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Abstract. Metric Diophantine approximation in its classical form is the study of how well almost all real numbers can be approximated by rationals. There is a long history of results which give partial answers to this problem, but there are still questions which remain unknown. The DuffinSchaeffer Conjecture is an attempt to answer all of these questions in full, and it has withstood more than fifty years of mathematical investigation. In this paper we establish a strong connection between the DuffinSchaeffer Conjecture and its p−adic analogue. Our main theorems are transfer principles which allow us to go back and forth between these two problems. We prove that if the variance method from probability theory can be used to solve the p−adic DuffinSchaeffer Conjecture for even one prime p, then almost the entire classical DuffinSchaeffer Conjecture would follow. Conversely if the variance method can be used to prove the classical conjecture then the p−adic conjecture is true for all primes. Furthermore we are able to unconditionally and completely establish the higher dimensional analogue of this conjecture in which we allow simultaneous approximation in any finite number and combination of real and p−adic fields, as long as the total number of fields involved is greater than one. Finally by using a mass transference principle for Hausdorff measures we are able to extend all of our results to their corresponding analogues with Haar measures replaced by the Hausdorff measures associated with arbitrary dimension functions. 1.