The unique Shortest vector problem (uSVP) in lattice theory plays a crucial role in many public-key cryptosystems. The security of those cryptosystems bases on the hardness of uSVP. However, so far there is no proof for the proper hardness of uSVP even in its exact version. In this paper, we show that the exact version of uSVP for `∞ norm is NP-hard. Furthermore, many other lattice problems inc...

We give a polynomial time Turing reduction from the γ √ napproximate closest vector problem on a lattice of dimension n to a γapproximate oracle for the shortest vector problem. This is an improvement over a reduction by Kannan, which achieved γn 3

We show that given oracle access to a subroutine which returns approximate closest vectors in a lattice, one may find in polynomial time approximate shortest vectors in a lattice. The level of approximation is maintained; that is, for any function f , the following holds: Suppose that the subroutine, on input a lattice L and a target vector w (not necessarily in the lattice), outputs v ∈ L such...

We consider the problem of finding the closest match for a given target vector in a codebook of codeword vectors. We present an approach which makes it possible to obtain quick full-search equivalent methods for finding the closest match with respect to some distortion measure. Our algorithms assume that the distortion measure obeys the triangle inequality of metric spaces. Results indicate tha...

In this paper we consider the problem of finding a vector that can be written as a nonnegative, integer and linear combination of given 0-1 vectors, the generators, such that the l1-distance between this vector and a given target vector is minimized. We prove that this closest vector problem is NP-hard to approximate within an additive error of (ln 2− ǫ)d ≈ (0, 693 − ǫ) d for all ǫ > 0 where d ...

This paper presents data selection procedures for support vector machines (SVM). The purpose of data selection is to reduce the dataset by eliminating as many non support vectors (non-SVs) as possible. Based on the fact that support vectors (SVs) are those vectors close to the decision boundary, data selection keeps only the closest pair vectors of opposite classes. The selected dataset will re...

of n linearly independent vectors b1, . . . ,bn ∈ Rm in m-dimensional Euclidean space. For computational purposes, the lattice vectors b1, . . . ,bn are often assumed to have integer (or rational) entries, so that the lattice can be represented by an integer matrix B = [b1, . . . ,bn] ∈ Zm×n (called basis) having the generating vectors as columns. Using matrix notation, lattice points in L(B) c...

We give a method for approximating any n-dimensional lattice with a lattice Λ whose factor group Z/Λ has n− 1 cycles of equal length with arbitrary precision. We also show that a direct consequence of this is that the Shortest Vector Problem and the Closest Vector Problem cannot be easier for this type of lattices than for general lattices.

In this paper, we introduce a method of threshold secret sharing scheme (TSSS) in which secret reconstruction is based on Babai's nearest plane algorithm. In order to supply secure public channels for transmitting shares to parties, we need to ensure that there are no quantum threats to these channels. A solution to this problem can be utilization of lattice-based cryptosystems for these channe...