This paper presents efficient protocols for private set intersection and approximate matching between two parties, using homomorphic encryption and balanced hashing. The protocols are secure against both semi-honest and malicious adversaries. For lists of length k, the protocols achieve O(k) communication and O(k ln ln k) computation. The semi-honest protocol is secure in the standard model, while the malicious protocol is secure in the random oracle model. The paper also considers the problem of approximating the size of the intersection, showing a linear lower bound on communication overhead and providing a secure protocol. Additionally, it investigates variants of the matching problem, including multi-party extensions and approximate matching. The protocols use homomorphic encryption to evaluate polynomials and hash functions to distribute inputs into bins, reducing computational overhead. The security of the protocols is analyzed under different adversary models, with the malicious case requiring additional techniques such as cut-and-choose and random oracle models. The paper also discusses the use of oblivious transfer and secure circuit evaluation for threshold matching and other functions. Finally, it presents a protocol for approximating intersection size, showing that any approximation protocol requires at least Θ(N) communication. The multi-party case is briefly discussed, with a protocol for secure intersection computation among n parties. The paper concludes with a discussion of fuzzy matching and search, where entries may not be exact matches.This paper presents efficient protocols for private set intersection and approximate matching between two parties, using homomorphic encryption and balanced hashing. The protocols are secure against both semi-honest and malicious adversaries. For lists of length k, the protocols achieve O(k) communication and O(k ln ln k) computation. The semi-honest protocol is secure in the standard model, while the malicious protocol is secure in the random oracle model. The paper also considers the problem of approximating the size of the intersection, showing a linear lower bound on communication overhead and providing a secure protocol. Additionally, it investigates variants of the matching problem, including multi-party extensions and approximate matching. The protocols use homomorphic encryption to evaluate polynomials and hash functions to distribute inputs into bins, reducing computational overhead. The security of the protocols is analyzed under different adversary models, with the malicious case requiring additional techniques such as cut-and-choose and random oracle models. The paper also discusses the use of oblivious transfer and secure circuit evaluation for threshold matching and other functions. Finally, it presents a protocol for approximating intersection size, showing that any approximation protocol requires at least Θ(N) communication. The multi-party case is briefly discussed, with a protocol for secure intersection computation among n parties. The paper concludes with a discussion of fuzzy matching and search, where entries may not be exact matches.