Homomorphic Encryption and Applications pp 27-46 | Cite as

# Homomorphic Encryption

## Abstract

Homomorphic encryption is a form of encryption which allows specific types of computations to be carried out on ciphertexts and generate an encrypted result which, when decrypted, matches the result of operations performed on the plaintexts. This is a desirable feature in modern communication system architectures. RSA is the first public-key encryption scheme with a homomorphic property. However, for security, RSA has to pad a message with random bits before encryption to achieve semantic security. The padding results in RSA losing the homomorphic property. To avoid padding messages, many public-key encryption schemes with various homomorphic properties have been proposed in last three decades. In this chapter, we introduce basic homomorphic encryption techniques. It begins with a formal definition of homomorphic encryption, followed by some well-known homomorphic encryption schemes.

## Keywords

Encryption Scheme Quadratic Residue Homomorphic Encryption Homomorphic Property Semantic Security## References

- 1.M. Abdalla, M. Bellare, P. Rogaway, DHAES: an encryption scheme based on the Diffie–Hellman problem. Submission to IEEE P1363a, 1998. http://www.di.ens.fr/~mabdalla/papers/dhes.pdf
- 2.D. Boneh, E. Goh, K. Nissim, Evaluating 2-DNF formulas on ciphertexts, in
*Proceedings of Theory of Cryptography, TCC’05*, 2005, pp. 325–341Google Scholar - 3.R. Cramer, V. Shoup, A practical public key cryptosystem provably secure against adaptive chosen ciphertext attack, in
*Proceedings of Advances in Cryptology, CRYPTO’98*, 1998, pp. 13–25Google Scholar - 4.T. ElGamal, A public-key cryptosystem and a signature scheme based on discrete logarithms. IEEE Trans. Inf. Theory
**31**(4), 469–472 (1985)CrossRefzbMATHMathSciNetGoogle Scholar - 5.D.M. Freeman, Converting pairing-based cryptosystems from composite-order groups to prime-order groups, in
*Proceedings of Advances in Cryptology, EUROCRYPT’10*, 2010, pp. 44–61Google Scholar - 6.K. Gjosteen, Subgroup membership problems and public key cryptosystems, Dissertation, Norwegian University of Science and Technology, 2004Google Scholar
- 7.S. Goldwasser, S. Micali, Probabilistic encryption and how to play mental poker keeping secret all partial information, in
*Proceedings of 14th Symposium on Theory of Computing*, 1982, pp. 365–377Google Scholar - 8.S. Goldwasser, S. Micali, Probabilistic encryption. J. Comput. Syst. Sci.
**28**(2), 270–299 (1984)CrossRefzbMATHMathSciNetGoogle Scholar - 9.A. Menezes, P. van Oorschot, S. Vanstone,
*Handbook of Applied Cryptography*. CRC Press, 1996Google Scholar - 10.T. Okamoto, S. Uchiyama, A new public-key cryptosystem as secure as factoring, in
*Proceedings of Advances in Cryptology, EUROCRYPT’98*, 1998, pp. 308–318Google Scholar - 11.P. Paillier, Public key cryptosystems based on composite degree residue classes,
*Proceedings of Advances in Cryptology, EUROCRYPT’99*, 1999, pp. 223–238Google Scholar - 12.P. Paillier, D. Pointcheval, Efficient public-key cryptosystems provably secure against active adversaries, in
*Proceedings of Advances in Cryptology, ASIACRYPT’99*, 1999, pp. 165–179Google Scholar