On discrepancy estimates for pseudorandom vectors constructed by the elliptic curve congruential generator

TL;DR

This paper provides new discrepancy bounds for pseudorandom vectors generated by elliptic curve congruential generators, especially in non-translational cases, using Fourier analysis and period conditions.

Contribution

It introduces novel bounds for discrepancy measures in full-coset regimes and reduces general bounds to Fourier mass estimations, highlighting arithmetic bottlenecks.

Findings

Bounds of type $q^{1/2}/t$ for discrepancy and serial discrepancy in full-coset regime.

Discrepancy bounds in sub-period regime linked to Fourier $ ext{l}^1$ masses of index sets.

Identifies arithmetic bottleneck for improving discrepancy bounds.

Abstract

This paper studies the problem of discrepancy estimates for pseudorandom vectors constructed by the elliptic curve congruential generator, particularly in the non-translational case. Two families of results are obtained. First, in a full-coset regime characterized by a relative maximal period condition (RMPC) on an induced one-dimensional linear congruential generator, one proves bounds of type $q^{1/2}/t$ for the discrepancy $D$, the serial discrepancy $D_s$, and, under the corresponding derived RMPC, the non-overlapping discrepancy $\widetilde D_s$. Second, in the general sub-period regime, one reduces bounds for $D$, $D_s$, and $\widetilde D_s$ to estimation of Fourier $\ell^1$ masses of admissible index sets attached to one-dimensional linear congruential generators. This isolates the arithmetic bottleneck for further improvement.