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A class of pseudorandom sequences From Function Fields

Xiaofeng Liu, Jun Zhang, Fang-Wei Fu

TL;DR

This work develops a general framework for constructing $p$-ary pseudorandom sequences from algebraic function fields, leveraging exponential-sum bounds to analyze key PR properties. By extending prior approaches based on cyclic fields and elliptic curves to arbitrary function fields, it derives rigorous bounds on periods, linear complexity, linear-complexity profiles, $r$-pattern distributions, period correlations, and nonlinear complexities. Two explicit instances—the rational function field and a cyclic elliptic function field—are worked out, yielding large periods, strong linear and nonlinear complexity, and low correlations under precise degree and non-degeneracy conditions, with provisions for perfect sequences. The results unify and extend earlier work (Hu et al.; Xing et al.) to broad function-field settings, offering new $p$-ary sequence families with potential applications in CDMA, stream ciphers, and broadband communications. The analysis hinges on Weil–Deligne–Bombieri bounds for exponential sums and Riemann–Roch machinery to control pole/order structures and their impact on sequence metrics.

Abstract

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].

A class of pseudorandom sequences From Function Fields

TL;DR

This work develops a general framework for constructing -ary pseudorandom sequences from algebraic function fields, leveraging exponential-sum bounds to analyze key PR properties. By extending prior approaches based on cyclic fields and elliptic curves to arbitrary function fields, it derives rigorous bounds on periods, linear complexity, linear-complexity profiles, -pattern distributions, period correlations, and nonlinear complexities. Two explicit instances—the rational function field and a cyclic elliptic function field—are worked out, yielding large periods, strong linear and nonlinear complexity, and low correlations under precise degree and non-degeneracy conditions, with provisions for perfect sequences. The results unify and extend earlier work (Hu et al.; Xing et al.) to broad function-field settings, offering new -ary sequence families with potential applications in CDMA, stream ciphers, and broadband communications. The analysis hinges on Weil–Deligne–Bombieri bounds for exponential sums and Riemann–Roch machinery to control pole/order structures and their impact on sequence metrics.

Abstract

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 patterns, period correlation and nonlinear complexities for a class of ary sequences that generalize the constructions in \text{[IEEE Trans. Inf. Theory, 49(6), 2003]} and [IEEE Trans. Inf. Theory, 53(7), 2007].
Paper Structure (9 sections, 18 theorems, 91 equations)

This paper contains 9 sections, 18 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The linear complexity and correlation
  4. Function fields and exponential sums
  5. Elliptic Function Fields
  6. The general construction of $p-$ary sequences
  7. Example construction over the rational function fields
  8. Example construction over the cyclic elliptic function fields
  9. Conclusion

Key Result

Lemma 2.4

If there is a place $Q$ of $F$ for which $v_{Q}(z)<0$ and this integer is relatively prime to $p$, then $z$ is non-degenerate.

Theorems & Definitions (33)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4: 26
  • Lemma 2.5
  • Theorem 2.6: see 313233
  • Remark 2.7
  • Lemma 2.8
  • Lemma 2.9
  • Proposition 3.1
  • ...and 23 more