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.

翻译：椭圆曲线上的Weil配对与离散对数问题有着深刻的联系。在实际应用中，为了更适配密码系统的功能需求，通常需要通过所谓的扭曲映射对原始Weil配对进行修正。本文针对有限域上椭圆曲线扭曲映射的存在性问题展开研究。我们重新审视了文献中的已有结果，并提供了详细的证明过程。在某些方面，我们也提出了新的研究视角。