Efficient Two-level Homomorphic Encryption in Prime-order Bilinear Groups and A Fast Implementation in WebAssembly.

We construct an efficient two-level homomorphic public-key encryption in prime-order bilinear groups. Such a scheme supports polynomially many homomorphic additions and one multiplication over encrypted data, similar to the cryptosystem of Boneh, Goh, and Nissim (BGN, presented at TCC 2005), which was constructed in composite-order bilinear groups. Prior to our work, the state-of-the-art for two-level homomorphic public-key encryption is the Freeman scheme (presented at Eurocrypt 2010), which is indeed the prime-order realization of the BGN scheme. Our proposed scheme significantly improves efficiency for almost all the aspects of the Freeman scheme, while retains the same ciphertext sizes. Our scheme is surprisingly simple as it is indeed (a concatenation of two copies of) the ElGamal encryption "in the exponent'' resided in an asymmetric bilinear groups. We provide a software implementation of our scheme in the x86 architecture. Besides this usual implementation, we also implement our scheme in WebAssembly (wasm), which is a portable low-level bytecode format; this allows our scheme to be run (very fast) on any popular web browser, without any plugins required.