Mustafa Umut Kazancıoğlu

arXiv2026

Distortion maps for elliptic curves over finite fields

The Weil pairing on elliptic curves has deep links with discrete logarithm problems. In practice, to better suit the functionalities of cryptosystems, one often needs to modify the original Weil pairing via what is called a distortion map. We propose a study on the question of the existence of distortion maps for elliptic curves over finite fields. We revisit results from the literature and provide detailed proofs. We also propose new perspectives at times.

math.NT↗

arXiv2025

On Torsion Subgroups of Elliptic Curves over Quartic, Quintic and Sextic Number Fields

The list of all groups that can appear as torsion subgroups of elliptic curves over number fields of degree $d$, $d=4,5,6$, is not completely determined. However, the list of groups $Φ^{\infty}(d)$, $d=4,5,6$, that can be realized as torsion subgroups for infinitely many non-isomorphic elliptic curves over these fields are known. We address the question of which torsion subgroups can arise over a given number field of degree $d$. In fact, given $G\inΦ^{\infty}(d)$ and a number field $K$ of degree $d$, we give explicit criteria telling whether $G$ is realized finitely or infinitely often over $K$. We also give results on the field with the smallest absolute value of its discriminant such that there exists an elliptic curve with torsion $G$. Finally, we give examples of number fields $K$ of degree $d$, $d=4,5,6$, over which the Mordell-Weil rank of elliptic curves with prescribed torsion is bounded from above.

math.NT↗