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14
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 57 (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
Arithmetic Crosscorrelations of Feedback with Carry Shift Register Sequences
 IEEE Trans. Inform. Theory
, 1997
"... An arithmetic version of the crosscorrelation of two sequences is defined, generalizing Mandelbaum's arithmetic autocorrelations. Large families of sequences are constructed with ideal (vanishing) arithmetic crosscorrelations. These sequences are decimations of the 2 adic expansions of rationa ..."
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Cited by 13 (2 self)
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An arithmetic version of the crosscorrelation of two sequences is defined, generalizing Mandelbaum's arithmetic autocorrelations. Large families of sequences are constructed with ideal (vanishing) arithmetic crosscorrelations. These sequences are decimations of the 2 adic expansions of rational numbers p=q such that 2 is a primitive root modulo q.
Large Period Nearly deBruijn FCSR Sequences (Extended Abstract)
 In L.C. Guillou and J.J. Quisquater� editors� Advances in Cryptology � Eurocrypt �95
, 1995
"... Recently, a new class of feedback shift registers (FCSRs) was introduced, based on algebra over the 2adic 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 di ..."
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Cited by 12 (4 self)
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Recently, a new class of feedback shift registers (FCSRs) was introduced, based on algebra over the 2adic 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.
Periodicity and distribution properties of combined FCSR sequences
 Sequences and Their Applications  SETA 2006
, 2006
"... www.cs.uky.edu/~klapper Abstract. This is a study of some of the elementary statistical properties of the bitwise exclusive or of two maximum period feedback with carry shift register sequences. We obtain conditions under which the resulting sequences has the maximum possible period, and we obtain b ..."
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Cited by 5 (0 self)
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www.cs.uky.edu/~klapper Abstract. This is a study of some of the elementary statistical properties of the bitwise exclusive or of two maximum period feedback with carry shift register sequences. We obtain conditions under which the resulting sequences has the maximum possible period, and we obtain bounds on the variation in the distribution of blocks of a fixed length. This may lead to improved design of stream ciphers using FCSRs.
Efficient MultiplyWithCarry Random Number Generators With Optimal Distribution Properties
 ACM Transactions on Modeling and Computer Simulation
, 2003
"... Introduction 1.1. A pseudorox"q number gener ator (RNG) for high speed simulation and Monte CarS integrSqKx should have sever" pr" er"US : (1) it should haveenor""x perz d, (2) it should e hibitunifor distrqS""xI of dtuples(for all d), (3) it should exhibi ..."
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Cited by 5 (0 self)
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Introduction 1.1. A pseudorox"q number gener ator (RNG) for high speed simulation and Monte CarS integrSqKx should have sever" pr" er"US : (1) it should haveenor""x perz d, (2) it should e hibitunifor distrqS""xI of dtuples(for all d), (3) it should exhibit a good lattice str""Ezx in high dimensions, and (4) it should be e#ciently computable(prablexzF with a base b which is a power of 2). Typically the RNG is a member of a family ofsimilar generrxI withdi#erq tparU"xIEU and one hopes that parKq"qxI and seeds may be easily chosen so as toguarF tee pr" er"E" (1), (2), (3) and (4). Ther is no known family of RNG with all four pr" er"KS (see,for example, [M1]). 1.2. In [MZ], Mar aglia and Zaman showed that their addwithcarc (AWC) gener ator satisfy condition (1). By giving up on (4) and using an appr"FxIE" base b, they achieve good distrxSKEKx pr" er"Kq of dtuplesfor values d wh
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 (5 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].
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
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. ..."
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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.