Secure norm computation is becoming increasingly important in many real-world learning applications. However, existing cryptographic systems often lack a general framework for securely computing the $L^p$-norm over private inputs held by different parties. These systems often treat secure norm computation as a black-box process, neglecting to design tailored cryptographic protocols that optimize performance. Moreover, they predominantly focus on the $L^2$-norm, paying little attention to other popular $L^p$-norms, such as $L^1$ and $L^\infty$, which are commonly used in practice, such as machine learning tasks and location-based services. To our best knowledge, we propose the first comprehensive framework for secure two-party $L^p$-norm computations ($L^1$, $L^2$, and $L^\infty$), denoted as \mbox{Crypto-$L^p$}, designed to be versatile across various applications. We have designed, implemented, and thoroughly evaluated our framework across a wide range of benchmarking applications, state-of-the-art (SOTA) cryptographic protocols, and real-world datasets to validate its effectiveness and practical applicability. In summary, \mbox{Crypto-$L^p$} outperforms prior works on secure $L^p$-norm computation, achieving $82\times$, $271\times$, and $42\times$ improvements in runtime while reducing communication overhead by $36\times$, $4\times$, and $21\times$ for $p=1$, $2$, and $\infty$, respectively. Furthermore, we take the first step in adapting our Crypto-$L^p$ framework for secure machine learning inference, reducing communication costs by $3\times$ compared to SOTA systems while maintaining comparable runtime and accuracy.