# zbMATH — the first resource for mathematics

Multi-sequences with $$d$$-perfect property. (English) Zbl 1075.68024
Summary: Sequences with almost perfect linear complexity profile are defined by H. Niederreiter [in: Proceedings of the Salzburg Conference 1986, Teubner, Stuttgart, Contrib. Gen. Algebra 5, 221–233 (1987; Zbl 0641.65005)]. C. Xing and K. T. Lam [IEEE Trans. Inf. Theory 45, 1267–1270 (1999; Zbl 0943.94008)] and C. Xing [J. Complexity 16, 661–675 (2000; Zbl 1026.94006)] extended this concept from the case of single sequences to the case of multi-sequences and further proposed the concept of $$d$$-perfect multi-sequences. In this paper, based on the technique of $$m$$-continued fractions due to Dai et al., we investigate the property of $$d$$-perfect multi-sequences and obtain a sufficient and necessary condition for $$d$$-perfect multi-sequences. We show that $$d$$-perfect multi-sequences are not always strongly $$d$$-perfect. In particular, we give an example to disprove the conjecture, posed by Xing (2000), on $$d$$-perfect multi-sequences.

##### MSC:
 68P25 Data encryption (aspects in computer science)
Full Text:
##### References:
 [1] Dai, Z.; Wang, K.; Ye, D., m-continued fraction expansions of multi-Laurent series, Adv. math. (China), 33, 246-248, (2004) [2] Z. Dai, K. Wang, D. Ye, Multidimensional Continued Fraction and Rational Approximation, http://arxiv.org/abs/math.NT/0401141. [3] Lidl, R.; Niederreiter, H., Introduction to finite fields and their applications, (1986), Cambridge University Press Cambridge · Zbl 0629.12016 [4] H. Niederreiter, Continued fractions for formal power series, pseudorandom numbers, and linear complexity of sequences, Contributions to General Algebra, Proceedings of the Salzburg Conference 1986, Vol. 5, Teubner, Stuttgart, 1987, pp. 221-233. [5] H. Niederreiter, Sequences with almost perfect linear complexity profile, Advances in Cryptology-EUROCRYPT’ 87, Lecture Notes in Computer Science, Vol. 304, Springer, Berlin, 1988, pp. 37-51. [6] Rueppel, R.A., Analysis and design of stream ciphers, (1986), Springer Berlin · Zbl 0654.68044 [7] Stark, H., An introduction to number theory, (1979), MIT Press Cambridge, MA [8] Xing, C.; Lam, K.Y., Sequences with almost perfect linear complexity profiles and curves over finite fields, IEEE trans. inform. theory, 45, 1267-1270, (1999) · Zbl 0943.94008 [9] Xing, C., Multi-sequences with almost perfect linear complexity profile and function fields over finite fields, J. complexity, 16, 661-675, (2000) · Zbl 1026.94006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.