A Homomorphic Signature Scheme for Quadratic Polynomials

Homomorphic signatures can provide a credential of a result which is indeed computed with a given function on a data set by an untrusted third party like a cloud server, when the input data are stored with the signatures beforehand. Boneh and Freeman in EUROCRYPT2011 proposed a homomorphic signature scheme for polynomial functions of any degree, however the scheme is not based on the normal short integer solution (SIS) problems as its security assumption. In this paper, we show a homomorphic signature scheme for quadratic polynomial functions those security assumption is based on the normal SIS problems. Our scheme constructs the signatures of multiplication as tensor products of the original signature vectors of input data so that homomorphism holds. Moreover, security of our scheme is reduced to the hardness of the SIS problems respect to the moduli such that one modulus is the power of the other modulus. We show the reduction by constructing solvers of the SIS problems respect to either of the moduli from any forger of our scheme.