In all existing efficient proofs of knowledge of a solution to the infinity norm Inhomogeneous Small Integer Solution (ISIS~∞) problem, the knowledge extractor outputs a solution vector that is only guaranteed to be ?(n) times longer than the witness possessed by the prover. As a consequence, in many cryptographic schemes that use these proof systems as building blocks, there exists a gap between the hardness of solving the underlying ISIS~∞ problem and the hardness underlying the security reductions. In this paper, we generalize Stern’s protocol to obtain two statistical zero-knowledge proofs of knowledge for the ISIS~∞ problem that remove this gap. Our result yields the potential of relying on weaker security assumptions for various lattice-based cryptographic constructions. As applications of our proof system, we introduce a concurrently secure identity-based identification scheme based on the worstcase hardness of the SIVP _(ō(n~(1.5))) problem (in the ?_2 norm) in general lattices in the random oracle model, and an efficient statistical zeroknowledge proof of plaintext knowledge with small constant gap factor for Regev’s encryption scheme.
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