This paper introduces a public key encryption with keyword search (PEKS) scheme, which allows a user to encrypt data and enable a third party to search for specific keywords without learning anything else about the encrypted data. The scheme is designed to allow a user, such as Alice, to encrypt her emails and provide a key to a gateway or server that can search for specific keywords, such as "urgent," without revealing the content of the email. The scheme ensures that the gateway or server learns nothing about the email beyond the presence of the keyword. The paper defines the concept of PEKS and provides two constructions. The first construction is based on bilinear maps and assumes the random oracle model. It is proven to be semantically secure against chosen keyword attacks under the assumption that the Bilinear Diffie-Hellman problem is intractable. The second construction is based on general trapdoor permutations and is less efficient but does not require the random oracle model. The paper also discusses the relationship between PEKS and Identity-Based Encryption (IBE), showing that PEKS implies IBE, but the converse is not necessarily true. The constructions presented are based on recent IBE constructions and leverage their properties to prove security. The paper concludes that constructing a PEKS is a challenging problem, and the proposed constructions provide efficient and secure solutions under certain assumptions. The work highlights the importance of privacy in encrypted data and the need for secure mechanisms that allow for efficient keyword searches without compromising data confidentiality.This paper introduces a public key encryption with keyword search (PEKS) scheme, which allows a user to encrypt data and enable a third party to search for specific keywords without learning anything else about the encrypted data. The scheme is designed to allow a user, such as Alice, to encrypt her emails and provide a key to a gateway or server that can search for specific keywords, such as "urgent," without revealing the content of the email. The scheme ensures that the gateway or server learns nothing about the email beyond the presence of the keyword. The paper defines the concept of PEKS and provides two constructions. The first construction is based on bilinear maps and assumes the random oracle model. It is proven to be semantically secure against chosen keyword attacks under the assumption that the Bilinear Diffie-Hellman problem is intractable. The second construction is based on general trapdoor permutations and is less efficient but does not require the random oracle model. The paper also discusses the relationship between PEKS and Identity-Based Encryption (IBE), showing that PEKS implies IBE, but the converse is not necessarily true. The constructions presented are based on recent IBE constructions and leverage their properties to prove security. The paper concludes that constructing a PEKS is a challenging problem, and the proposed constructions provide efficient and secure solutions under certain assumptions. The work highlights the importance of privacy in encrypted data and the need for secure mechanisms that allow for efficient keyword searches without compromising data confidentiality.