## Cryptology ePrint Archive: Report 2016/906

On Basing Search SIVP on NP-Hardness

Tianren Liu

Abstract: The possibility of basing cryptography on the minimal assumption NP$\nsubseteq$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 $\text{SIVP}_{\tilde O(n)}$.

Although SIVP is NP-hard in its exact version, Guruswami et al (CCC 2004) showed that $\text{gapSIVP}_{\sqrt{n/\log n}}$ is in NP$\cap$coAM and thus unlikely to be NP-hard. Indeed, any language that can be reduced to $\text{gapSIVP}_{\tilde O(\sqrt n)}$ (under general probabilistic polynomial-time adaptive reductions) is in AM$\cap$coAM by the results of Peikert and Vaikuntanathan (CRYPTO 2008) and Mahmoody and Xiao (CCC 2010). However, none of these results apply to reductions to search problems, still leaving open a ray of hope: can NP be reduced to solving search SIVP with approximation factor $\tilde O(n)$?

We eliminate such possibility, by showing that any language that can be reduced to solving search $\text{SIVP}_{\gamma}$ with any approximation factor $\gamma(n) = \omega(n\log n)$ lies in AM intersect coAM. As a side product, we show that any language that can be reduced to discrete Gaussian sampling with parameter $\tilde O(\sqrt n)\cdot\lambda_n$ lies in AM intersect coAM.

Category / Keywords: separation, lattice-based cryptography

Original Publication (with minor differences): IACR-TCC-2018

Date: received 16 Sep 2016, last revised 18 Oct 2018

Contact author: liutr at mit edu

Available format(s): PDF | BibTeX Citation

Note: The intro is rewritten for eurocrypt

Short URL: ia.cr/2016/906

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