Abstract This paper describes the formal verification of NP-hardness reduction functions two key problems relevant in algebraic lattice theory: closest vector problem and shortest problem, both infinity norm. The formalization uncovered a number with existing proofs literature. how these were corrected formalization. work was carried out proof assistant Isabelle.

Vector quantization (VQ) is an efficient technique for image compression. However, one of the most serious problems of VQ is the heavily computational load for searching the closest codeword to the input vector in the codebook. To overcome this problem, we propose a simple and fast partial distance search algorithm for VQ codebook by using the sum-norm pyramid structure of codewords. The propos...

In this work we consider the closest vector problem (CVP) —a problem also known as maximum-likelihood decoding— in the tensor of two root lattices of type A (Am⊗An), as well as in their duals (Am⊗An). This problem is mainly motivated by lattice based cryptography, where the cyclotomic rings Z[ζc] (resp. its co-different Z[ζc]) play a central role, and turn out to be isomorphic as lattices to te...

We present a new efficient algorithm for the search version of the approximate Closest Vector Problem with Preprocessing (CVPP). This is the problem of finding a lattice vector whose distance from the target point is within some factor γ of the closest lattice vector, where the algorithm is allowed to take polynomial-length advice about the lattice from an unbounded preprocessing algorithm. Our...

We show that the Closest Vector Problem with Preprocessing over `∞ norm (CVPP∞) is NP-hard to approximate to within a factor of (log n)1/2− , unless NP⊆ DTIME (2polylog(n)). The result is the same as that in [19] by Regev and Rosen, but our proof methods are different from theirs. Their reductions are based on norm embeddings. However, our reductions are based on the reduction of [2] and the pr...

Improving on the Voronoi cell based techniques of [28, 24], we give a Las Vegas Õ(2n) expected time and space algorithm for CVPP (the preprocessing version of the Closest Vector Problem, CVP). This improves on the Õ(4n) deterministic runtime of the Micciancio Voulgaris algorithm [24] (henceforth MV) for CVPP 1 at the cost of a polynomial amount of randomness (which only affects runtime, not cor...

W e provide a brief history and overview of lattice based cryptography and cryptanalysis: shortest vector problems, closest vector problems, subset sum problem and knapsack systems, GGH, Ajtai-Dwork and NTRU. A detailed discussion of the algorithms NTRUEncrypt and NTRUSign follows. These algorithms have attractive operating speed and keysize and are based on hard problems that are seemingly int...