Abstract
Non-linear feedback shift registers (NLFSRs) are a generalization of linear feedback shift registers in which a current state is a non-linear function of the previous state. The interest in NLFSRs is motivated by their ability to generate pseudo-random sequences which are typically hard to break with existing cryptanalytic methods. However, it is still not known how to construct large \(n\)-stage NLFSRs which generate full cycles of \(2^n\) possible states. This paper presents a method for generating full cycles by a composition of NLFSRs. First, we show that an \(n*k\)-stage register with period \(O(2^{2n})\) can be constructed from \(k\) NLFSRs with \(n\)-stages by adding to their feedback functions a logic block of size \(O(nk)\), for \(k > 1\). This logic block implements Boolean functions representing pairs of states whose successors have to be exchanged in order to join cycles. Then, we show how to join all cycles into one by using one more logic block of size \(O(nk)\).
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Notes
Note that the period of a shift register is traditionally defined as the length of the longest cyclic output sequence it produces [18]. For shift registers, both definitions are equivalent. However, for general registers, in which each stage can be updated by its own function, the length of the longest state cycle can be a multiple of the length of the longest cyclic output sequence. For example, the 2-stage register with the state cycle \(((00), (11), (01), (10))\) of length 4 generates the cyclic output sequence \((0,1)\) of length 2.
A 2-input gate implements a binary Boolean operation of type \(\{0,1\}^2 \rightarrow \{0,1\}\).
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Acknowledgments
The author is indebted to the anonymous reviewers for their valuable comments over the manuscript. This work was supported in part the research Grant No 621-2010-4388 from the Swedish Research Council and in part by the research Grant No SM12-0005 from the Swedish Foundation for Strategic Research.
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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.
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Dubrova, E. Generation of full cycles by a composition of NLFSRs. Des. Codes Cryptogr. 73, 469–486 (2014). https://doi.org/10.1007/s10623-014-9947-3
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DOI: https://doi.org/10.1007/s10623-014-9947-3