# Distribution of the Sequence [m]P in Elliptic Curves

## TL;DR

This paper investigates the distribution properties of sequences generated by elliptic curve points over complex and real fields, analyzing their equidistribution and randomness characteristics.

## Contribution

It introduces a detailed analysis of the distribution of sequences on elliptic curves over characteristic zero fields, extending understanding of their randomness and equidistribution behavior.

## Key findings

- Sequences in complex elliptic curves can be characterized by specific measures.
- Certain sequences are not equidistributed modulo 1 despite being equidistributed with respect to a measure.
- The study provides conditions under which polynomial sequences on elliptic curves deviate from uniform distribution.

## Abstract

Major controversy surrounds the use of Elliptic Curves in finite fields as Random Number Generators. There is little information however concerning the "randomness" of different procedures on Elliptic Curves defined over fields of characteristic $0$. The aim of this paper is to investigate the behaviour of the sequence $\psi_m=[m]P$ and then generalize to polynomial seuences of the form $\phi_m=[p(m)]P$. We examine the behaviour of this sequence in different domains and attempt to realize for which points it is not equidistributed in $\mathbb{C}/\Lambda$. We will first study the sequence in the space of Elliptic Curves $E(\mathbb{C})$ defined over the complex numbers and then reconsider our approach to tackle real valued Elliptic Curves. In the process we obtain the measure with respect to which the sequence $\psi$ is equidistributed in $E(\mathbb{R})$. In Section 4 we prove that every sequence of points $P_n=(x_n,y_n,1)$ equidistributed w.r.t. that measure is not equidistributed$\mod(1)$ with the obvious map $x_n\to\{x_n\}$.