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The identity-based homomorphic signature (IBHS) enables an untrusted server to run some computation over the outsourced data and derive a short signature, vouching for the correctness of the output of the computation, while greatly simplifying key management. To our knowledge, constructions of IBHS have been few and far between. However, the existing IBHS schemes, which either handle only linear functions or has a large public key parameter and satisfies only the artificial notion of selective security. In this work, we construct the first leveled adaptively secure identity-based fully homomorphic signature (IBFHS) schemes without additional public parameters, which can be used to sign many different datasets. Thereby positively answering the open question of constructing a leveled IBFHS scheme with short public parameters, proposed by Wang et al., (ISC, 2015, Springer). We achieve the stronger security and better parameters by using the trapdoor vanishing and vector encoding technique. In our scheme, the size of every evaluated signature depends only logarithmically on the size of the input dataset, and the complexity of verifying a signature for a computation can be amortized when verifying the same computation on many different datasets. Furthermore, we prove that our construction is strongly-unforgeable against adaptively chosen identity and message attacks under the small integer solution (SIS) assumption in standard lattices.

Leveled Adaptively Strong-Unforgeable Identity-Based Fully Homomorphic Signatures