Homomorphic Encryption — Theory and Application
Jaydip Sen · InTech eBooks · 2013
The chapter is organized as follows.In Section 2, we provide some basic and fundamental information on cryptography and various types of encryption schemes.Section 3 presents a formal discussion on homomorphic encryption schemes and discusses their various features.In Section 4, we discuss some of the most well-known and classical homomorphic encryption schemes in the literature.Section 5 provides a brief presentation on various properties and applications of homomorphic cryptosystems.Section 6 presents a discussion on fully homomorphic encryption schemes which are the most powerful encryption schemes for providing a framework for computing over encrypted data.Finally, Section 7 concludes the chapter while outlining a number of research directions and emerging trends in this exciting field of computation which has a tremendous potential of finding applications in the real-world deployments. Fundamentals of cryptographyIn this Section, we will recall some important concepts on encryption schemes.For more detailed information, the reader may refer to (Menezes et al., 1997; Van Tilborg, 2011).Encryption schemes are designed to preserve confidentiality.The security of encryption schemes must not rely on the obfuscation of their codes, but it should only be based on the secrecy of the key used in the encryption process.Encryption schemes are broadly of two types: symmetric and asymmetric encryption schemes.In the following, we present a very brief discussion on each of these schemes. Symmetric encryption schemes:In these schemes, the sender and the receiver agree on the key they will use before establishing any secure communication session.Therefore, it is not possible for two persons who never met before to use such schemes directly.This also implies that in order to communicate with different persons, we must have a different key for each people.Requirement of large number of keys in these schemes make their key generation and management relatively more complex operations.However, symmetric schemes present the advantage of being very fast and they are used in applications where speed of execution is a paramount requirement.Among the existing symmetric encryption systems, AES (Daemen & Rijmen, 2000;Daemen & Rijmen, 2002), One-Time Pad (Vernam, 1926) and Snow (Ekdahl & Johansson, 2002) are very popular. Asymmetric encryption schemes:In these schemes, every participant has a pair of keysprivate and public.While the private key of a person is known to only her, the public key of each participant is known to everyone in the group.Such schemes are more secure than their symmetric counterparts and they don't need any prior agreement between the communicating parties on a common key before establishing a session of communication.RSA (Rivest et al., 1978b) and ElGamal (ElGamal, 1985) are two most popular asymmetric encryption systems.Security of encryption schemes: Security of encryption schemes was first formalized by Shannon (Shannon, 1949).In his seminal paper, Shannon first introduced the notion of perfect secrecy/unconditional secrecy, which characterizes encryption schemes for which the knowledge of a ciphertext does not give any information about the corresponding plaintext and the