Leveled Strongly-Unforgeable Identity-Based Fully Homomorphic Signatures

Recently, Gorbunov, Vaikuntanathan and Wichs proposed a new powerful primitive: (fully) homomorphic trapdoor function (HTDF) based on small integer solution (SIS) problem in standard lattices, from which they constructed the first leveled existentially-unforgeable fully homomorphic signature (FHS) schemes. In this paper, we first extend the notion of HTDF to identity-based setting with stronger security and better parameters. The stronger security requires that the identity-based HTDF (IBHTDF) is not only clawfree, but also collision-resistant. And the maximum noise comparing to Gorbunov-Vaikuntanathan-Wichs' HTDF roughly reduces from O(4(d)m beta) to O(4(d)m beta), which will result in polynomial modulus q = poly(lambda) when d = O(log lambda), where lambda is the security parameter and d is the depth bound of circuit. We then define and construct the first leveled strongly-unforgeable identity-based fully homomorphic signature (IBFHS) schemes.