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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
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 6 (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
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.
A New Class of Pseudonoise Sequences
, 2003
"... We apply the framework of adic algebra and algebraic feedback shift registers to polynomial rings over nite elds. We give a construction of new pseudorandom sequences over a nonprime nite eld that satisfy Golomb's randomness criteria. ..."
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We apply the framework of adic algebra and algebraic feedback shift registers to polynomial rings over nite elds. We give a construction of new pseudorandom sequences over a nonprime nite eld that satisfy Golomb's randomness criteria.