A new fully homomorphic encryption over the integers using smaller public key

Abstract

Fully homomorphic encryption scheme with practical time complexity is a widely acknowledged research problem in cryptography. In this work, a new somewhat homomorphic encryption with practical time complexities is proposed, from which fully homomorphic encryption is obtained using the optimisations suggested in the contemporary works. The central idea behind the proposition in achieving such reasonable time complexities lies in employing a small public key containing only two big integers with consequent reduction in the message expansion. The scheme may be considered as a variant of the DGHV's integers-based scheme, which was one of the earlier attempts in devising a conceptually simpler fully homomorphic encryption. The semantic security of the proposition is proved in the standard model by reducing the same to the hard problem of solving partial approximate greatest common divisor of two integers.