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Lattice reduction in two dimensions: the black vectors are the given basis for the lattice (represented by blue dots), the red vectors are the reduced basis. In mathematics, the goal of lattice basis reduction is to find a basis with short, nearly orthogonal vectors when given an integer lattice basis as input. This is realized using different ...
However, an LLL-reduced basis is nearly as short as possible, in the sense that there are absolute bounds > such that the first basis vector is no more than times as long as a shortest vector in the lattice, the second basis vector is likewise within of the second successive minimum, and so on.
The Korkine–Zolotarev (KZ) lattice basis reduction algorithm or Hermite–Korkine–Zolotarev (HKZ) algorithm is a lattice reduction algorithm. For lattices in R n {\displaystyle \mathbb {R} ^{n}} it yields a lattice basis with orthogonality defect at most n n {\displaystyle n^{n}} , unlike the 2 n 2 / 2 {\displaystyle 2^{n^{2}/2}} bound of ...
Lattice reduction algorithms aim, given a basis for a lattice, to output a new basis consisting of relatively short, nearly orthogonal vectors. The Lenstra–Lenstra–Lovász lattice basis reduction algorithm (LLL) was an early efficient algorithm for this problem which could output an almost reduced lattice basis in polynomial time. [33]
For example, the Lenstra–Lenstra–Lovász lattice basis reduction algorithm (LLL) has been used in the cryptanalysis of many public-key encryption schemes, [2] and many lattice-based cryptographic schemes are known to be secure under the assumption that certain lattice problems are computationally difficult. [3]
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Thus, by running the LLL-lattice basis reduction algorithm with = /, we obtain a decomposition of as a sum of squares. Note that because every vector in L {\textstyle L} has norm squared a multiple of p {\textstyle p} , the vector returned by the LLL-algorithm in this case is in fact a shortest vector.
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related to: lattice basis reduction