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

TL;DR

This paper investigates the pseudorandom properties of sequences generated by filtered Legendre symbols using polynomial constructions, showing their measures are smaller than theoretical bounds and suggesting near orthogonality among different polynomial-based sequences.

Contribution

It introduces a polynomial-based construction of Legendre symbol sequences and demonstrates their pseudorandom measures are significantly below theoretical bounds.

Findings

Pseudorandom measures are smaller than Weil theorem bounds.

Sequences from different polynomials are nearly orthogonal.

Construction allows for a large variety of sequences with low correlation.

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.