# Alternating Step Generators Controlled by De Bruijn Sequences

- First Online:

## Abstract

The alternating step generator (ASG) is a new generator of pseudorandom sequences which is closely related to the stop-and-go generator. It shares all the good properties of this latter generator without Posessing its weaknesses. The ASG consists of three subgenerators *k, m*, and Open image in new window. The main characteristic of its structure is that the output of one of the subgenerators, *k*, controls the clock of the two others, *m* and Open image in new window. In the present contribution, we determine the period, the distribution of short patterns and a lower bound for the linear complexity of the sequences generated by an ASG. The proof of the lower bound is greatly simplified by assuming that *k* generates a de Bruijn sequence. Under this and other not very restrictive assumptions the period and the linear complexity are found to be proportional to the period of the de Bruijn sequence. Furthermore the frequency of all short patterns as well as the autocorrelations turn out to be ideal. This means that the sequences generated by the ASG are provably secure against the standard attacks.

### Selected References

- [1]S.A. Tretter, “Properties of PN
^{2}sequences”, IEEE Trans. Inform. Theory, vol. IT-20, pp. 295–297, March 1974.CrossRefMathSciNetGoogle Scholar - [2]K. Kjeldsen and E. Andresen, “Some randomness properties of cascaded sequences”, IEEE Trans. Inform. Theory, vol. IT-26, pp. 227–232, March 1980.CrossRefMathSciNetGoogle Scholar
- [3]T. Beth and F. Piper, “The stop-and-go-generator”, in Proc. of EUROCRYPT 84, Springer Lect. Notes in Comp. Science, vol. 209, pp. 88–92.Google Scholar
- [4]R. Vogel, “On the linear complexity of cascaded sequences”, in Proc. of EUROCRYPT 84, Springer Lect. Notes in Comp. Science, vol. 209, pp. 99–109.Google Scholar
- [5]D. Gollman, “Pseudo random properties of cascade connections of clock controlled shift registers”, in Proc. of EUROCRYPT 84, Springer Lect. Notes in Comp. Science, vol. 209, pp. 93–98.Google Scholar
- [6]W.G. Chambers and S.M. Jennings, “Linear equivalence of certain BRM shiftregister sequences”, Electronics Letters, vol. 20, pp. 1018–1019, Nov. 1984.CrossRefGoogle Scholar
- [7]N.G. de Bruijn, “A combinatorial problem”, Proc. K. Ned. Akad. Wet., vol. 49, pp 758–764, 1946.Google Scholar
- [8]C.G. Günther, “Alternating step generators”, submitted to IEEE Trans, on Inform. Theory.Google Scholar
- [9]T. Siegenthaler, “Correlation-immunity of non-linear combining functions for cryptographic applications”, IEEE Trans, on Inform. Theory, vol. IT-30, pp. 776–780, Sept. 1984.CrossRefMathSciNetGoogle Scholar
- [10]N. Zierler, “Linear recurring sequences”, J. Soc. Indust. Appl. Math., vol. 7, pp. 31–48, March 1959.MATHCrossRefMathSciNetGoogle Scholar
- [11]E.S. Selmer, Linear Recurrence Relations Over Finite Fields, Department of Mathematics, University of Bergen, Norway 1966.Google Scholar
- [12]A.H. Chan, R.A. Games and E.L. Key, “On the complexities of de Bruijn sequences”, J. of Comb. Theory, Series A, vol. 33, pp. 233–246, 1982.MATHCrossRefMathSciNetGoogle Scholar