SVP$_p$ is Deterministically NP-Hard for all $p > 2$, Even to Approximate Within a Factor of $2^{\log^{1-\varepsilon} n}$
For cryptographers and complexity theorists, this resolves the deterministic NP-hardness of exact SVP_p for finite p and provides the first subexponential inapproximability result for SVP_p under standard reductions.
The paper proves that the Shortest Vector Problem (SVP) in ℓ_p norm for p > 2 is NP-hard to approximate within a factor of 2^{log^{1-ε} n} under deterministic Karp reductions, and also shows that exact SVP_p is NP-hard for finite p. This improves on prior results that required randomized reductions or only constant factor hardness.
We prove that SVP$_p$ is NP-hard to approximate within a factor of $2^{\log^{1 - \varepsilon} n}$, for all constants $\varepsilon > 0$ and $p > 2$, under standard deterministic Karp reductions. This result is also the first proof that \emph{exact} SVP$_p$ is NP-hard in a finite $\ell_p$ norm. Hardness for SVP$_p$ with $p$ finite was previously only known if NP $\not \subseteq$ RP, and under that assumption, hardness of approximation was only known for all constant factors. As a corollary to our main theorem, we show that under the Sliding Scale Conjecture, SVP$_p$ is NP-hard to approximate within a small polynomial factor, for all constants $p > 2$. Our proof techniques are surprisingly elementary; we reduce from a \emph{regularized} PCP instance directly to the shortest vector problem by using simple gadgets related to Vandermonde matrices and Hadamard matrices.