Parametric families of elliptic curves with cyclic \(\mathbb F_p\)-rational points groups
The author constructs some families of elliptic curves without complex multiplication over \(\mathbb{Q}\) such that for prime numbers of the form \(p=2^\alpha 3^\beta 5^\gamma q^\delta +1\) (\(q\) an odd prime) the groups \(\tilde{E}(\mathbb{F}_p)\) of the reductions of \(E\) modulo \(p\) are cyclic. Such families can be useful for cryptographic purposes. The constructions and proofs are based on a result by \textit{R. Gupta} and \textit{M. R. Murty} [Invent. Math. 101, No. 1, 225--235 (1990; Zbl 0731.14011)] which asserts, in particular, that for an elliptic curve over \(\mathbb{Q}\) with good reduction at prime \(p\) the group \(\tilde{E}(\mathbb{F}_p)\) is cyclic if and only if for any prime \(l\) \(p\) does not split completely in the field \(K_l(E)\) generated over \(\mathbb{Q}\) by all the \(l\)-division points of \(E\). Many relevant examples are included in the paper.
elliptic curves over number and finite fields
scientific article; zbMATH DE number 5015935