Worst-Case to Average-Case Reductions Based on Gaussian Measures

13 years 8 months ago

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We show that finding small solutions to random modular linear equations is at least as hard as approximating several lattice problems in the worst case within a factor almost linear in the dimension of the lattice. The lattice problems we consider are the shortest vector problem, the shortest independent vectors problem, the covering radius problem, and the guaranteed distance decoding problem (a variant of the well known closest vector problem). The approximation factor we obtain is n logO(1) n for all four problems. This greatly improves on all previous work on the subject starting from Ajtai's seminal paper (STOC, 1996), up to the strongest previously known results by Micciancio (SIAM J. on Computing, 2004). Our results also bring us closer to the limit where the problems are no longer known to be in NP intersect coNP. Our main tools are Gaussian measures on lattices and the high-dimensional Fourier transform. We start by defining a new lattice parameter which determines the a...

Daniele Micciancio, Oded Regev

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