Across social science disciplines, empirical studies related to social networks have become the most popular research subjects in recent years. A frequently examined topic within these studies is the estimation of peer influence while controlling for homophily effects. However, although researchers may have access to all observable homophily variables, there is scarce literature addressing latent homophily effects stemming from unobservable features. Recent endeavors have demonstrated the efficacy of node embeddings derived from network structure in controlling latent homophily. Inspired by the network embedding research, this study introduces two methods that integrate node embeddings to better control latent homophily, particularly the nonlinear latent homophily effect. The first method uses double machine learning in the partially linear regression literature to alleviate estimation bias. The second method estimates peer influence effects directly by a novel neural network model. Our experimentation results show that our approaches outperform existing estimators in reducing the omitted variable bias due to homophily effects in network regression models. Theoretical analysis of two new estimation methods is also provided in this paper. History: Accepted by Ram Ramesh, Area Editor for Data Science & Machine Learning. Funding: This research is supported by the National Research Foundation, Singapore under its Industry Alignment Fund - Pre-positioning (IAF-PP) Funding Initiative [Grant A-0003504-02-00]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0287 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0287 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Additively Homomorphic Encryption (AHE) has been widely used in various applications, such as federated learning, blockchain, and online auctions. Elliptic Curve (EC) based AHE has the advantages of efficient encryption, homomorphic addition, scalar multiplication algorithms, and short ciphertext length. However, EC-based AHE schemes require solving a small exponential Elliptic Curve Discrete Logarithm Problem (ECDLP) when running the decryption algorithm, i.e., recovering the plaintext$m\in \{0,1\}^{\ell} $from$m \ast G$. Therefore, the decryption of EC-based AHE schemes is inefficient when the plaintext length$\ell > 32$. This leads to people being more inclined to use RSA-based AHE schemes rather than EC-based ones. This paper proposes an efficient algorithm called$\mathsf {FastECDLP}$for solving the small exponential ECDLP at 128-bit security level. We perform a series of deep optimizations from two points: computation and memory overhead. These optimizations ensure efficient decryption when the plaintext length$\ell $is as long as possible in practice. Moreover, we also provide a concrete implementation and apply$\mathsf {FastECDLP}$to some specific applications. Experimental results show that$\mathsf {FastECDLP}$is far faster than the previous works. For example, the decryption can be done in 0.35 ms with a single thread when$\ell = 40$, which is about 30 times faster than that of Paillier. Furthermore, we experiment with$\ell $from 27 to 54, and the existing works generally only consider$\ell \leq 32$. The decryption only requires 1 second with 16 threads when$\ell = 54$. In the practical applications, we can speed up model training of existing vertical federated learning frameworks by 4 to 14 times. At the same time, the decryption efficiency is accelerated by about 140 times in a blockchain financial system (ESORICS 2021) with the same memory overhead.