Generation of two “independent” points on an elliptic curve of 𝑗-invariant ≠0,1728

This article is dedicated to a new generation method of two “independent” F q \mathbb {F}_{\!q} -points P 0 P_0 , P 1 P_1 on almost any ordinary elliptic curve E E over a finite field F q \mathbb {F}_{\!q} of large characteristic. In particular, the method is relevant for all standardized and real-world elliptic curves of j j -invariants different from 0 0 , 1728 1728 . The points P 0 P_0 , P 1 P_1 are characterized by the fact that nobody (even a generator) knows the discrete logarithm log P 0 ⁡ ( P 1 ) \log _{P_0}(P_1) in the group E ( F q ) E(\mathbb {F}_{\!q}) . Moreover, only one square root extraction in F q