Homomorphic Signatures with Efficient Verification for Polynomial Functions?

Abstract. A homomorphic signature scheme for a class of functions C allows a client to sign and upload elements of some data set D on a server. At any later point, the server can derive a (publicly verifiable) signature that certifies that some y is the result computing some f ∈ C on the basic data set D. This primitive has been formalized by Boneh and Freeman (Eurocrypt 2011) who also proposed the only known con-struction for the class of multivariate polynomials of fixed degree d ≥ 1. In this paper we construct new homomorphic signature schemes for such functions. Our schemes provide the first alternatives to the one of Boneh-Freeman, and improve over their solution in three main aspects. First, our schemes do not rely on random oracles. Second, we obtain security in a stronger fully-adaptive model: while the solution of Boneh-Freeman requires the adversary to query messages in a given data set all at once, our schemes can tolerate adversaries that query one message at a time, in a fully-adaptive way. Third, signature verification is more efficient (in an amortized sense) than computing the function from scratch. The latter property opens the way to using homomorphic signatures for publicly-verifiable computation on outsourced data. Our schemes rely on a new assumption on leveled graded encodings which we show to hold in a generic model.