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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.
This is based on the fact that a reciprocal lattice vector (the vector indicating a reciprocal lattice point from the reciprocal lattice origin) is the wavevector of a plane wave in the Fourier series of a spatial function (e.g., electronic density function) which periodicity follows the original Bravais lattice, so wavefronts of the plane wave ...
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 ...
Corner transfer matrices are related to the partition function in a simple way. In our simplified example, we construct the full lattice from four rotated copies of the lattice quadrant, where the inner boundary spin sets σ, σ', σ" and σ'" are allowed to differ: The partition function is then written in terms of the corner transfer matrix A as
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 coefficients. In cryptography, the Coppersmith method is mainly used in attacks on RSA when parts of the secret key are known and forms a base for Coppersmith's attack.
The private key is a basis of a lattice with good properties (such as short nearly orthogonal vectors) and a unimodular matrix. The public key is another basis of the lattice L {\displaystyle L} of the form B ′ = U B {\displaystyle B'=UB} .
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 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 ...