This paper addresses the problem of computing the intersection of private datasets from two parties, where the datasets contain elements from a large domain. The authors present protocols based on homomorphic encryption and balanced hashing for both semi-honest and malicious environments. For lists of length \( k \), the communication overhead is \( O(k) \) and the computation overhead is \( O(k \ln \ln k) \). The semi-honest protocol is secure in the standard model, while the malicious protocol is secure in the random oracle model. The paper also discusses variants of the private matching protocol, including computing the intersection size and deciding if it exceeds a threshold. Additionally, it introduces a protocol for approximating the intersection size and extends the protocol to a multi-party setting. The authors provide a linear lower bound for the communication overhead of approximating the intersection size and present a secure protocol. They also investigate the problem of secure approximate matching and search.This paper addresses the problem of computing the intersection of private datasets from two parties, where the datasets contain elements from a large domain. The authors present protocols based on homomorphic encryption and balanced hashing for both semi-honest and malicious environments. For lists of length \( k \), the communication overhead is \( O(k) \) and the computation overhead is \( O(k \ln \ln k) \). The semi-honest protocol is secure in the standard model, while the malicious protocol is secure in the random oracle model. The paper also discusses variants of the private matching protocol, including computing the intersection size and deciding if it exceeds a threshold. Additionally, it introduces a protocol for approximating the intersection size and extends the protocol to a multi-party setting. The authors provide a linear lower bound for the communication overhead of approximating the intersection size and present a secure protocol. They also investigate the problem of secure approximate matching and search.