Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the pseudorandom properties of filtered Legendre symbol sequences using three polynomials

Katalin Gyarmati, Károly Müllner

TL;DR

This work analyzes pseudorandom properties of filtered Legendre symbol sequences constructed from three polynomials. By defining $E_{f,g,h}$ with $f,g,h$ of degree at most $k$ and suitable nondegeneracy conditions, the authors derive Weil-type upper bounds showing $W(E_{f,g,h}) \le 10 k p^{1/2} \log p$ and $C_\ell(E_{f,g,h}) \le 2^{\ell+3} \ell k p^{1/2} \log p$. They corroborate these theoretical bounds with numerical calculations, which indicate the actual pseudorandom measures are significantly smaller and that $E_{f,g,h}$ retains strong pseudorandomness comparable to single-polynomial constructions, with security advantages due to triple-uncertainty. The cross-correlation analysis connects the performance of $E_{f,g,h}$ to the cross-correlation measure $\Phi_k(\mathcal{F})$ of a large family of base sequences, underscoring the practical cryptographic potential of these constructions. Overall, the paper contributes concrete bounds, numerical validation, and a robust security rationale for multi-polynomial Legendre-sequence filters in pseudorandomness applications.

Abstract

The primary objective of this section is to demonstrate that the actual pseudorandom measures of our construction are significantly smaller than the theoretical upper bounds derived from the Weil theorem. Regarding the family of sequences, we note that the construction $E_{f,g,h}$ allows for a large variety of sequences by choosing different triples of polynomials. While the detailed analysis of the cross-correlation measure of such a family is a challenging problem and lies beyond the scope of the present paper, the structure of the construction suggests that sequences generated by different polynomials will remain nearly orthogonal. Indeed, since each sequence is built from distinct Legendre symbol sequences with proven low correlation, their combinations are expected to maintain the same level of independence.

On the pseudorandom properties of filtered Legendre symbol sequences using three polynomials

TL;DR

This work analyzes pseudorandom properties of filtered Legendre symbol sequences constructed from three polynomials. By defining with of degree at most and suitable nondegeneracy conditions, the authors derive Weil-type upper bounds showing and . They corroborate these theoretical bounds with numerical calculations, which indicate the actual pseudorandom measures are significantly smaller and that retains strong pseudorandomness comparable to single-polynomial constructions, with security advantages due to triple-uncertainty. The cross-correlation analysis connects the performance of to the cross-correlation measure of a large family of base sequences, underscoring the practical cryptographic potential of these constructions. Overall, the paper contributes concrete bounds, numerical validation, and a robust security rationale for multi-polynomial Legendre-sequence filters in pseudorandomness applications.

Abstract

The primary objective of this section is to demonstrate that the actual pseudorandom measures of our construction are significantly smaller than the theoretical upper bounds derived from the Weil theorem. Regarding the family of sequences, we note that the construction allows for a large variety of sequences by choosing different triples of polynomials. While the detailed analysis of the cross-correlation measure of such a family is a challenging problem and lies beyond the scope of the present paper, the structure of the construction suggests that sequences generated by different polynomials will remain nearly orthogonal. Indeed, since each sequence is built from distinct Legendre symbol sequences with proven low correlation, their combinations are expected to maintain the same level of independence.
Paper Structure (3 sections, 5 theorems, 64 equations, 4 tables)

This paper contains 3 sections, 5 theorems, 64 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Numerical Calculations
  3. Proofs

Key Result

Theorem 1

Let $p$ be an odd prime and $f(x),g(x),h(x)\in\mathbb F_p[x]$ be three polynomials of degrees between $1$ and $k$, that have no multiple roots. Also assume that Define the sequence $E_{f,g,h} = \{e_1,\dots,e_p\}\in\{-1,+1\}^p$ by Construction cns02. Then, Assume that one of the following three conditions for $\ell$, which is the order of the correlation, holds: (i) $\ell=2$; (ii) $\ell<p$ and 2

Theorems & Definitions (7)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Definition 2
  • Theorem 3
  • Lemma 1
  • Lemma 2