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An early successful application of the LLL algorithm was its use by Andrew Odlyzko and Herman te Riele in disproving Mertens conjecture. [5]The LLL algorithm has found numerous other applications in MIMO detection algorithms [6] and cryptanalysis of public-key encryption schemes: knapsack cryptosystems, RSA with particular settings, NTRUEncrypt, and so forth.
Lattice reduction algorithms are used in a number of modern number theoretical applications, including in the discovery of a spigot algorithm for . Although determining the shortest basis is possibly an NP-complete problem, algorithms such as the LLL algorithm [ 2 ] can find a short (not necessarily shortest) basis in polynomial time with ...
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]
The Coppersmith method, proposed by Don Coppersmith, is a method to find small integer zeroes of univariate or bivariate polynomials, or small zeros modulo a given integer. The method uses the Lenstra–Lenstra–Lovász lattice basis reduction algorithm (LLL) to find a polynomial that has the same zeroes as the target polynomial but smaller ...
In linear algebra, the Hermite normal form is an analogue of reduced echelon form for matrices over the integers Z.Just as reduced echelon form can be used to solve problems about the solution to the linear system Ax=b where x is in R n, the Hermite normal form can solve problems about the solution to the linear system Ax=b where this time x is restricted to have integer coordinates only.
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 ...
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.
Many lattice-based cryptographic schemes are known to be secure assuming the worst-case hardness of certain lattice problems. [3] [6] [7] I.e., if there exists an algorithm that can efficiently break the cryptographic scheme with non-negligible probability, then there exists an efficient algorithm that solves a certain lattice problem on any ...
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