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 claw-free, but also collision-resistant. And the maximum noise comparing to Gorbunov-Vaikuntanathan-Wichs' HTDF roughly reduces from $$Om^d\beta $$Omdβ to $$O4^dm\beta $$O4dmβ, which will result in polynomial modulus $$q=\mathrm {poly}\lambda $$q=polyλ when $$d=O\log \lambda $$d=Ologλ, where $$\lambda $$λ 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.