# zbMATH — the first resource for mathematics

Multisequences with almost perfect linear complexity profile and function fields over finite fields. (English) Zbl 1026.94006
Let $$A=\{a_{1}, a_{2},\cdots,a_{m}\}$$ be a multisequence over the finite field $$\mathbb{F}_{q}$$ where $$a_{i}=\{a_{i1},a_{i2},\cdots \}$$ is an infinite sequence, and let $$\{l_{n}(A)\}_{n=1}^{\infty}$$ be the linear complexity profile of $$A$$. If $$l_{n}(A)\geq\lceil \frac{m(n+1)-d}{m+1}\rceil$$ for all $$n$$, $$A$$ is called $$d$$-perfect for a positive integer $$d$$. $$A$$ is called perfect if $$A$$ is $$m$$-perfect. It is proved in this paper that $$A$$ is perfect if and only if $$l_{n}(A)=\lceil \frac{mn}{m+1}\rceil$$. A construction of $$d$$-perfect multisequences is given in this paper by using a function field over a finite field. Let $$\mathbf{F}$$ be a global function field with the full constant field $$\mathbb{F}_{q}$$. For a place $$\mathbb{Q}$$ of degree $$m$$ of $$\mathbf{F}$$, let $$\mathbf{F}_{\mathbb{Q}}=\mathcal{O}_{\mathbb{Q}}/\mathcal{P}_{\mathbb{Q}}$$ be the residue field of $$\mathbb{Q}$$, where $$\mathcal{O}_{\mathbb{Q}}$$, $$\mathcal{P}_{\mathbb{Q}}$$ are the integral ring and maximal ideal at $$\mathbb{Q}$$ respectively. Assume that $$t$$ is a local parameter of $$\mathbb{Q}$$ with $$\text{deg}(t)_{\infty}=m+1$$. Choose $$m$$ elements $$x_{1}, x_{2},\cdots, x_{m} \in \mathcal{O}_{\mathbb{Q}}$$ such that $$x_{1}(\mathbb{Q}), x_{2}(\mathbb{Q}),\cdots, x_{m}(\mathbb{Q})$$ form an $$\mathbb{F}_{q}$$-basis of $$\mathbf{F}_{\mathbb{Q}}$$. For an element $$y\in \mathcal{O}_{\mathbb{Q}}$$, it can be expressed by a formal series $y=\sum_{j=0}^{\infty}\Biggl(\sum_{i=1}^{m}a_{ij}x_{i}\Biggr)t^{j}.$ Put $$a_{i}(y)=(a_{i1}, a_{i2}, \cdots)$$ ($$1\leq i\leq m$$). Then it is proved that $$A=(a_{1}, a_{2},\cdots,a_{m})$$ is $$d$$-perfect where $$d=\text{deg} ((y)_{\infty}\vee (x_{1})_{\infty}\vee\cdots\vee(x_{m})_{\infty}).$$ Some examples of this construction are given in the paper.

##### MSC:
 94A55 Shift register sequences and sequences over finite alphabets in information and communication theory 11T71 Algebraic coding theory; cryptography (number-theoretic aspects) 94A60 Cryptography
Full Text:
##### References:
 [1] Kohel, D.R.; Ling, S.; Xing, C.P., Explicit sequence expansions, (), 308-317 · Zbl 1005.11064 [2] Niederreiter, H., Continued fractions for formal power series, pseudorandom numbers, and linear complexity of sequences, Contributions to general algebra 5 (proc. Salzburg conf., 1986), (1987), Teubner Stuttgart, p. 221-233 [3] Niederreiter, H., Sequences with almost perfect linear complexity profile, (), 37-51 [4] Niederreiter, H.; Vielhaber, M., Linear complexity profiles: Hausdorff dimensions for almost perfect profiles and measures for general profiles, J. complexity, 13, 353-383, (1997) · Zbl 0934.94013 [5] Rueppel, R.A., Analysis and design of stream ciphers, (1986), Springer-Verlag Berlin · Zbl 0654.68044 [6] Rueppel, R.A., Stream ciphers, (), 65-134 [7] Stichtenoth, H., Algebraic function fields and codes, (1993), Springer-Verlag Berlin · Zbl 0816.14011 [8] Xing, C.P.; 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.P.; Niederreiter, H.; Lam, K.Y.; Ding, C.S., Constructions of sequences with almost perfect linear complexity profile from curves over finite fields, Finite fields appl., 5, 301-313, (1999) · Zbl 0943.94005
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.