# The Maximum Order Complexity of Sequence Ensembles

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## Abstract

In this paper we extend the theory of maximum order complexity from a single sequence to an ensemble of sequences. In particular, the maximum order complexity of an ensemble of sequences is defined and its properties discussed. Also, an algorithm is given to determine the maximum order complexity of an ensemble of sequences linear in time and memory. It is also shown how to determine the maximum order feedback shift register equivalent of a given ensmble of sequences, i.e. including a feedback function. Hence, the problem of finding the absolutely shortest (possibly nonlinear) feedback shift register, that can generate two or more given sequences with characters from some arbitrary finite alphabet, is solved. Finally, the consequences for sequence prediction based on the minimum number of observations are discussed.

## Keywords

Truth Table Maximum Order Single Sequence Periodic Sequence Feedback Function## References

- [1]R. Arratia and M. S. Waterman. “Critical Phenomena in Sequence Matching”,
*The Annals of Probability*, vol. 13, no. 4, pp. 1236–1249, 1985.zbMATHCrossRefMathSciNetGoogle Scholar - [2]R. Arratia and M. S. Waterman. “An Erdös-Rényi Law with Shifts”,
*Adv. in Math.*, vol. 55, pp. 13–23, 1985.zbMATHCrossRefMathSciNetGoogle Scholar - [3]R. Arratia, L. Gordon and M. S. Waterman. “An Extreme Value Theory for Sequence Matching”,
*The Annals of Statistics*, vol. 14, no. 3, pp. 971–993, 1986.zbMATHCrossRefMathSciNetGoogle Scholar - [4]A. Blumer, J. Blumer, A. Ehrenfeucht, D. Haussler and R. McConnell. “Linear Size Finite Automata for the Set of all Subwords of a Word: An Outline of Results”,
*Bul. Eur. Assoc. Theor. Comp. Sci.*, no. 21, pp. 12–20, 1983.Google Scholar - [5]A. Blumer, J. Blumer, D. Haussler, R. McConnell and A. Ehrenfeucht. “Complete Inverted Files for Efficient Text Retrieval and Analysis”,
*JACM*, vol. 34, no. 3, pp. 578–595, July 1987.CrossRefMathSciNetGoogle Scholar - [6]H. Fredricksen. “A survey of full-length nonlinear shift register cycle algorithms”,
*SIAM Rev.*, vol. 24, pp. 195–221, April 1982.zbMATHCrossRefMathSciNetGoogle Scholar - [7]C. J. A. Jansen and D. E. Boekee. “The Algebraic Normal Form of Arbitrary Functions of Finite Fields”,
*Proceedings of the Eighth Symposium on Information Theory in the Benelux, Deventer, The Netherlands*, pp. 69–76, May 1987.Google Scholar - [8]C. J. A. Jansen.
*Investigations On Nonlinear Streamcipher Systems: Construction and Evaluation Methods*, PhD. Thesis, Technical University of Delft, Delft, 1989.Google Scholar - [9]C. J. A. Jansen and D. E. Boekee. “The Shortest Feedback Shift Register That Can Generate A Given Sequence”,
*Proceedings of Crypto’ 89, Santa Barbara, USA*.Google Scholar - [10]C. J. A. Jansen and D. E. Boekee. “On the Significance of the Directed Acyclic Word Graph in Cryptology”,
*Proceedings of Auscrypt’ 90, Sydney, Australia*.Google Scholar - [11]C. J. A. Jansen “On the Construction of Run Permuted Sequences”,
*proceedings of Eurocrypt’ 90, Århus, Denmark*.Google Scholar