Abstract
This paper presents estimation results of the ratio of input to output sequence periods for combination schemes and the ratio of input to internal sequence periods for schemes with a memory element. A decrease in the length of output sequence period, which is possible in the case with circuits with a memory element, is not discussed. The following results are given:
(1) for any \({\varvec{m}}\) sequences with period lengths of \({{\varvec{T}}}_{1}\),…,\({{\varvec{T}}}_{{\varvec{m}}}\), \({\varvec{m}}>1\), a lower bound of their modification period length is given using the combination function \(\varvec{\upvarphi }({{\varvec{y}}}_{1},\dots ,{{\varvec{y}}}_{{\varvec{m}}})\) that is bijective for some variables;
(2) the modification period length for \({\varvec{m}}>1\) sequences generated by maximum-length linear feedback shift registers is proved to be maximal, using the combination function \(\varvec{\upvarphi }({{\varvec{y}}}_{1},\dots ,{{\varvec{y}}}_{{\varvec{m}}})\), that depends essentially on all variables, and the number of ones per period of the output sequence is estimated;
(3) for a certain class of non-autonomous automata, it is shown that the ratio of the period lengths of the internal sequence to the input sequence can be large, in particular, reach the number of automaton states.
The conditions under which the period lengths of resulting sequences are maximum are specified.
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Abbreviations
- \(N\) :
-
The set of natural numbers, \(n\in N\)
- \(X\times Y\) :
-
Cartesian product of sets \(X\) and \(Y\)
- \(\mathrm{gcd}\{{n}_{1},\dots ,{n}_{k}\}\) :
-
The greatest common divisor of numbers \({n}_{1},\dots ,{n}_{k }\in N\)
- \(\mathrm{lcm}\{{n}_{1},\dots ,{n}_{k}\}\) :
-
The least common multiple of numbers \({n}_{1},\dots ,{n}_{k }\in N\)
- \(\left.m\right|n\) :
-
\(m\) Divides \(n\) , where \(m\), \(n\in N\)
- \(Z\) :
-
The set of integers
- \({{\varvec{V}}}_{{\varvec{n}}} \) :
-
The set of binary \({\varvec{n}}\) -dimensional vectors
- \(\overrightarrow{X}\) :
-
The sequence \(\left\{{x}_{1},\dots ,{x}_{t},\dots \right\}\) of elements of the alphabet \(X\)
- \({X}^{*}\) :
-
The set of all sequences over the alphabet \(X\)
- \({\overrightarrow{X}}_{[i,j]}\) :
-
The sequence segment \(\left\{{x}_{i},\dots ,{x}_{j}\right\}\) of \(\overrightarrow{X}\), \(1\le i\le j\)
- \(LFS{R}_{L}\) :
-
The linear feedback shift register of length \(L\)
- \(\left\{{x}_{\uptheta +rt}, t=\mathrm{1,2},\dots \right\}\) :
-
The subsequence of the sequence \(\left\{{x}_{1},\dots ,{x}_{t},\dots \right\}\), called its \(\left(\uptheta ,r\right)\)—section, \(r\in N\), \(\uptheta \in \left\{1,\dots ,r\right\}\)
- \(\iff \) :
-
If and only if
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Acknowledgements
I am grateful for the reviewers for giving very perceptive comments and suggestions, which improved the quality of the paper significantly. I thank Anastasia Fomina for her help with translation. I also thank Dmitry Bobrovskiy for technical help.
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Fomichev, V.M. About long period sequences. J Comput Virol Hack Tech 18, 205–213 (2022). https://doi.org/10.1007/s11416-021-00408-9
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DOI: https://doi.org/10.1007/s11416-021-00408-9