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.