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Shortest Vector Problem

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Closest Vector Problem; Lattice; Lattice Reduction; Lattice-Based Cryptography; NTRU


Given k linearly independent (typically integer) vectors \(\mathbf{ B} = [{\mathbf{b}}_{1},\ldots,{\mathbf{b}}_{k}]\) in n-dimensional Euclidean space \({\mathbb{R}}^{n}\), the Shortest Vector Problem (SVP) asks to find a nonzero integer linear combination \(\mathbf{{B}{x}} ={ \sum \nolimits }_{i=1}^{k}{\mathbf{b}}_{i}{x}_{i}\) (with \({x} \in {\mathbb{Z}}^{k}\setminus \{{0}\}\)) such that the norm \(\|\mathbf{{B}{x}}\|\) is as small as possible. The problem can be defined with respect to any norm, but the Euclidean norm \(\|\mathbf{v}\| = \sqrt{{\sum \nolimits }_{i=1}^{n}{v}_{i}^{2}}\) is the most common. The set of all integer linear combinations \(\mathcal{L}(\mathbf{B}) =\{\mathbf{{B}{x}:{x}} \in {\mathbb{Z}}^{k}\}\) is a lattice. So, SVP can be concisely defined as the problem of finding the shortest nonzero vector in the lattice represented by \(\mathbf{B}\). (Refer...

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  1. Dinur I (2002) Approximating SVP to within almost-polynomial factors is NP-hard. Theor Comput Sci 285(1):55–71

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Micciancio, D. (2011). Shortest Vector Problem. In: van Tilborg, H.C.A., Jajodia, S. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA.

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