Enhanced Flexibility for Homomorphic Encryption Schemes via CRT

The Chinese Remainder Theorem (CRT) has numerous applications including in cryptography. In a striking example of this utility, we demonstrate how the CRT facilitates making one additive homomorphic encryption scheme viable and making another more flexible. First we show that the CRT may be used to turn an intractable problem into a tractable one. Specifically, using the CRT to replace a single group element by a logarithmic number of elements in the same group, we lay the foundation for additively homomorphic encryption schemes using wellknown and previously deployed primitives. Our solution is shown to be secure and quite general in nature. We present a simple technique for ElGamal-type encryption schemes which facilitates encryption in an additively homomorphic manner. Secondly we apply the CRT to a previous encryption scheme proposed by Boneh, Goh and Nissim that supports efficient homomorphic evaluations of 2-DNF circuits [4]. One drawback mentioned in [4] was a restriction on the size of the output message space – prompting an open problem posed by the authors. Again employing the CRT, we devise an elegant modification in which we solve the problem, supporting arbitrary output sizes.