The Discrete Logarithm Problem (DLP) for elliptic curves has been extensively studied since, for instance, it is the core of the security of cryptosystems like Elliptic Curve Cryptography (ECC). In this paper, we present an attack to the DLP for elliptic curves based on its connection to the problem of lifting, by using the exponential map for elliptic curves and its inverse over $ \mathbb{Z} / p^k \mathbb{Z} $. Additionally, we show that hyperelliptic curves are resistant to this attack, meaning that these latter curves offer a higher level of security compared to the classic elliptic curves used in cryptography.

翻译：椭圆曲线上的离散对数问题（DLP）自提出以来已被广泛研究，例如它是椭圆曲线密码学（ECC）等密码系统安全性的核心。本文基于椭圆曲线的指数映射及其在 \( \mathbb{Z} / p^k \mathbb{Z} \) 上的逆映射，利用提升问题与DLP之间的联系，提出了一种针对椭圆曲线DLP的攻击方法。此外，我们证明超椭圆曲线对此攻击具有抵抗性，这意味着与密码学中使用的经典椭圆曲线相比，超椭圆曲线能提供更高的安全级别。