On the cycle structure of some nonlinear shift register sequences

Mykkeltveit, Johannes; Siu, Man‐Keung; Tong, Po

doi:10.1016/s0019-9958(79)90708-3

Cited by 79 publications

(38 citation statements)

References 5 publications

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“…However, there are no general methods of designing maximum period NLFSRs. The construction of a special class of NLFSRs with maximum period has been given by Mykkeltveit et al [11]. Recently, maximum period NLFSRs of order up to n = 64 have been constructed in the paper of Mandal and Gong [9], but these NLFSRs have very complicated algebraic normal form (ANF ).…”

Section: Introductionmentioning

confidence: 99%

Searching for nonlinear feedback shift registers with parallel computing

Dbrowski

Labuzek

Rachwalik

et al. 2014

Information Processing Letters

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Abstract. Nonlinear feedback shift registers (NLFSRs) are used to construct pseudorandom generators for stream ciphers. Their theory is not so complete as that of linear feedback shift registers (LFSRs). In general, it is not known how to construct all NLFSRs with maximum period. The direct method is to search for such registers with suitable properties. Advanced technology of parallel computing has been applied both in software and hardware to search for maximum period NLFSRs having a fairly simple algebraic normal form.

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Section: Introductionmentioning

confidence: 99%

Searching for nonlinear feedback shift registers with parallel computing

Dbrowski

Labuzek

Rachwalik

et al. 2014

Information Processing Letters

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“…The unique cycle in a maximum-length FSR is called De Bruijn cycle or full cycle. The following lemma was proved in [16]. It can be verified that, we always have G( f ) ∩ G( f + 1) = ∅ for any characteristic function f .…”

Section: Feedback Shift Registersmentioning

confidence: 87%

“…However, when both of f and g are linear Boolean functions, we have f * g = g * f . We refer the reader to [6,16] for more details about the operation * . We say (x 0 + x 1 ) is a left * -factor of g, denote by (x 0 + x 1 ) L g, if g = (x 0 + x 1 ) * f for some Boolean function f .…”

Section: Boolean Functionsmentioning

confidence: 99%

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The adjacency graphs of some feedback shift registers

Jiang

Lin

2016

Des. Codes Cryptogr.

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In this paper, we consider the adjacency graphs of some feedback shift registers (FSRs), namely, the FSRs with characteristic functions of the form g = (x 0 + x 1 ) * f . Firstly, we give some properties about these FSRs. We prove that these FSRs generate only prime cycles, and these cycles can be divided into two sets such that each set contains no adjacent cycles. When f is a linear function, more properties about these FSRs are derived. For example, it is shown that, when f contains an odd number of terms, the adjacency graph of FSR((x 0 + x 1 ) * f ) can be determined directly from the adjacency graph of FSR( f ). Then, as an application of these results, we continue the work of Li et al. (IEEE Trans Inf Theory 60(5):3052-3061, 2014) to determine the adjacency graphs of FSR((1 + x) 4 p(x)) and FSR((1 + x) 5 p(x)), where p(x) is a primitive polynomial, and construct a large class of De Bruijn sequences from them.

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“…A systematic algorithm for NLFSR synthesis has not been discovered so far. Only solutions to some special cases have been presented [1], [21], [22], [23], [24], [25], [26], [27].…”

Section: Introductionmentioning

confidence: 99%

On analysis and synthesis of ( n, k )-non-linear feedback shift registers

Dubrova

Teslenko

Tenhunen

2008

Proceedings of the Conference on Design, Automation and Test in Europe

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Abstract-Non-Linear Feedback Shift Registers (NLFSRs) have been proposed as an alternative to Linear Feedback Shift Registers (LFSRs) for generating pseudo-random sequences for stream ciphers. In this paper, we introduce (n, k)-NLFSRs which can be considered a generalization of the Galois type of LFSR. In an (n, k)-NLFSR, the feedback can be taken from any of the n bits, and the next state functions can be any Boolean function of up to k variables. Our motivation for considering this type NLFSRs is that their Galois configuration makes it possible to compute each next state function in parallel, thus increasing the speed of output sequence generation. Thus, for stream cipher application where the encryption speed is important, (n, k)-NLFSRs may be a better alternative than the traditional Fibonacci ones. We derive a number of properties of (n, k)-NLFSRs. First, we demonstrate that they are capable of generating output sequences with good statistical properties which cannot be generated by the Fibonacci type of NLFSRs. Second, we show that the period of the output sequence of an (n, k)-NLFSR is not necessarily equal to the length of the largest cycle of its states. Third, we compute the period of an (n, k)-NLFSR constructed from several parallel NLFSRs whose outputs are XOR-ed and show how to maximize this period. We also present an algorithm for estimating the length of cycles of states of (n, k)-NLFSRs which uses Binary Decision Diagrams for representing the set of states and the transition relation on this set.

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On the cycle structure of some nonlinear shift register sequences

Cited by 79 publications

References 5 publications

Searching for nonlinear feedback shift registers with parallel computing

Searching for nonlinear feedback shift registers with parallel computing

The adjacency graphs of some feedback shift registers

On analysis and synthesis of ( n, k )-non-linear feedback shift registers

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