Distortion maps for elliptic curves over finite fields
Nikita Andrusov, Sevag Büyüksimkeşyan, Dimitrios Noulas, Fabien Pazuki, Mustafa Umut Kazancıoğlu, Jordi Vilà-Casadevall
TL;DR
This work surveys the existence and construction of distortion maps for elliptic curves over finite fields, linking the End$(E)$ structure and Frobenius action to the embedding degree and the modified Weil pairing $e_{m,\phi}$. It develops a unifying framework encompassing the trace map, Legendre-symbol criteria, isogeny transfer, and explicit algorithmic methods, and it distinguishes between ordinary and supersingular cases. Key contributions include precise criteria via the discriminant $D_\phi$ and embedding degree, transfer rules along isogenies, and a comprehensive catalog of known distortion maps and algorithmic approaches (supersingular via Deuring/Tate- module methods; ordinary via volcano and CM-based techniques). The findings inform secure curve selection for DDH-based schemes and enable efficient construction of pairings for pairing-based cryptography, while highlighting fundamental limitations such as the nonexistence of a universal distortion map and the dependence on embedding degree and endomorphism structure.
Abstract
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.