On Basing Search SIVP on NP-Hardness

The possibility of basing cryptography on the minimal assumption \(\mathbf{NP }\nsubseteq \mathbf{BPP }\) is at the very heart of complexity-theoretic cryptography. The closest we have gotten so far is lattice-based cryptography whose average-case security is based on the worst-case hardness of approximate shortest vector problems on integer lattices. The state-of-the-art is the construction of a one-way function (and collision-resistant hash function) based on the hardness of the \(\tilde{O}(n)\)-approximate shortest independent vector problem \({\textsf {SIVP}}_{\tilde{O}(n)}\).

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