DNA fingerprinting and matching for identifying suspects has been a common practice in criminal investigation. Such proceedings involve multiple parties such as investigating agencies, suspects and forensic labs. A major challenge in such settings is to carry out the matching process between the suspects' DNA samples and the samples obtained from the crime scene without compromising the privacy of the suspects' DNA profiles. Additionally, it is necessary that sensitive details pertaining to the investigation such as the identities of the suspects and evidence obtained from the crime scene must be kept private to the investigating agency. We present a novel DNA matching scheme, termed as PrivaMatch, which addresses multiple concerns about privacy of the suspects' DNA profiles and the crime scene evidence. In the proposed scheme, the investigating agencies oblivious transfer and zero-knowledge proofs to privately obtain the DNA profiles of the suspects from the forensic lab's database.In addition, we present a clever data obfuscation technique using homomorphic encryption and modular arithmetic for the investigating agency to privately obtain the DNA profile of the crime scene's sample, keeping the profile oblivious from the forensic lab. The DNA profile of the crime scene sample is operated on using a homomorphic cryptosystem such that neither of the parties (e.g., the investigation agency, forensic labs, DNA database owners) learns about the private data of the other parties. The proposed scheme is analysed formally and the practicality of its security strengths is verified using simulations under standard assumptions.