Homomorphic Signatures for Correct Computation of Group Elements

We present two homomorphic signature schemes for group elements. The first construction is capable of computing exponentiations on signed data and the second one is designed for polynomial functions. Given the signed data and the public key, we can evaluate the specified functions on the signed data and produce a signature of the result of computations on the original data. Using the signature, any other party can verify the correctness of the computation result in a public way (the computation result can be verified by any other party without using the secret key). Previous works about homomorphic signature schemes almost can only handle linear functions. Boneh and Freeman first proposed homomorphic signatures for polynomial functions, but the privacy property of not leaking information about the original data can only be achieved for linear functions. Compared to previous solutions, our schemes are homomorphic respect to multiply operation in a group and the privacy property can be achieved for exponentiations and polynomial functions. Furthermore our work first put forward the notion of reusing the bilinear maps.