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11
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 51 (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
Fibonacci and Galois Representations of FeedbackWithCarry Shift Registers
 IEEE Trans. Inform. Theory
, 2002
"... A feedbackwithcarry shift register (FCSR) with "Fibonacci" architecture is a shift register provided with a small amount of memory which is used in the feedback algorithm. Like the linear feedback shift register (LFSR), the FCSR provides a simple and predictable method for the fast gener ..."
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Cited by 22 (2 self)
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A feedbackwithcarry shift register (FCSR) with "Fibonacci" architecture is a shift register provided with a small amount of memory which is used in the feedback algorithm. Like the linear feedback shift register (LFSR), the FCSR provides a simple and predictable method for the fast generation of pseudorandom sequences with good statistical properties and large periods. In this paper, we describe and analyze an alternative architecture for the FCSR which is similar to the "Galois" architecture for the LFSR. The Galois architecture is more efficient than the Fibonacci architecture because the feedback computations are performed in parallel. We also describe the output sequences generated by theFCSR, a slight modification of the (Fibonacci) FCSR architecture in which the feedback bit is delayed for clock cycles before being returned to the first cell of the shift register. We explain how these devices may be configured so as to generate sequences with large periods. We show that the FCSR also admits a more efficient "Galois" architecture.
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 10 (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.
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 9 (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 6 (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].
Polynomial pseudonoise sequences based on algebraic feedback shift registers
 IEEE Trans. Info. Theory
"... Abstract—Over the past half century, various statistical properties of pseudorandom sequences have played important roles in a variety of applications. Among these properties are Golomb’s randomness conditions: (R1) balance, (R2) run property, and (R3) ideal autocorrelations, as well as the closely ..."
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Cited by 3 (0 self)
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Abstract—Over the past half century, various statistical properties of pseudorandom sequences have played important roles in a variety of applications. Among these properties are Golomb’s randomness conditions: (R1) balance, (R2) run property, and (R3) ideal autocorrelations, as well as the closely related properties (R4) shift and add, and (R5) de Bruin (uniform distribution of subblocks). The purpose of this paper is to describe the relationships among these conditions, and to introduce a new method for generating sequences with all these properties, using algebraic feedback shift registers. Index Terms—De Bruijn sequences, feedback shift registers, function fields, ideal autocorrelation, pseudorandom sequences. I.
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.
Register Synthesis for Algebraic Feedback Shift Registers Based on NonPrimes
"... Abstract 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 ssadic numbers, where ss is an element in a ring R, and produce sequences of elemen ..."
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Abstract 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 ssadic numbers, where ss is an element in a ring R, and produce sequences of elements in R=(ss). We give several cases where the register synthesis problem can be solved by an efficient algorithm. Consequently, any keystreams over R=(ss) 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].