A class of pseudorandom sequences From Function Fields

Motivated by the constructions of pseudorandom sequences over the cyclic elliptic function fields by Hu \textit{et al.} in \text{[IEEE Trans. Inf. Theory, 53(7), 2007]} and the constructions of low-correlation, large linear span binary sequences from function fields by Xing \textit{et al.} in \text{[IEEE Trans. Inf. Theory, 49(6), 2003]}, we utilize the bound derived by Weil \text{[Basic Number Theory, Grund. der Math. Wiss., Bd 144]} and Deligne \text{[ Lecture Notes in Mathematics, vol. 569 (Springer, Berlin, 1977)]} for the exponential sums over the general algebraic function fields and study the periods, linear complexities, linear complexity profiles, distributions of $r-$patterns, period correlation and nonlinear complexities for a class of $p-$ary sequences that generalize the constructions in \text{[IEEE Trans. Inf. Theory, 49(6), 2003]} and [IEEE Trans. Inf. Theory, 53(7), 2007].

💡 Research Summary

The paper investigates a broad class of p‑ary pseudorandom sequences constructed from algebraic function fields, extending earlier constructions based on finite fields, Artin‑Schreier extensions, and cyclic elliptic function fields. The authors’ main technical tool is the Weil–Deligne bound for exponential sums over a general function field F/𝔽_q, as formulated in Theorem II.6. For any non‑degenerate rational function f∈F, the bound states that

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