Paper 2023/785

Generation of two ''independent'' points on an elliptic curve of $j$-invariant $\ne 0,1728$

Abstract

This article is dedicated to a new generation method of two ``independent'' ${\mathbb{F}}_q$-points $P_0$, $P_1$ on almost any ordinary elliptic curve $E$ over a finite field ${\mathbb{F}}_q$ of large characteristic. In particular, the method is relevant for all standardized and real-world elliptic curves of $j$-invariants different from $0$, $1728$. The points $P_0$, $P_1$ are characterized by the fact that nobody (even a generator) knows the discrete logarithm ${\log }_{P_0}(P_1)$ in the group $E({\mathbb{F}}_q)$. Moreover, only one square root extraction in ${\mathbb{F}}_q$ (instead of two ones) is required in comparison with all previous generation methods.