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The definition of a KZ-reduced basis was given by Aleksandr Korkin and Yegor Ivanovich Zolotarev in 1877, a strengthened version of Hermite reduction. The first algorithm for constructing a KZ-reduced basis was given in 1983 by Kannan. [2] The block Korkine-Zolotarev (BKZ) algorithm was introduced in 1987. [3]
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 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 ...
The role of the deoxyriboside form of 8-oxoguanine, 8-oxo-2'-deoxyguanosine (abbreviated 8-oxo-dG or 8-OHdG) in cancer and aging also applies to 8-oxoguanine. Oxoguanine glycosylase is employed in the removal of 8-oxoguanine from DNA by the process of base excision repair.
It is sometimes called a normal form of f by G. In general this form is not uniquely defined because there are, in general, several elements of G that can be used for reducing f; this non-uniqueness is the starting point of Gröbner basis theory. The definition of the reduction shows immediately that, if h is a normal form of f by G, one has
The doubly deprotonated (In 2-) phenolate form (the anion form of phenol) gives the familiar pink color. In strongly basic solutions, phenolphthalein is converted to its In(OH) 3− form, and its pink color undergoes a rather slow fading reaction [ 6 ] and becomes completely colorless when pH is greater than 13.
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[8] Since isotopy classes are disjoint, the number of reduced Latin squares gives an upper bound on the number of isotopy classes. Also, the total number of Latin squares is n!(n − 1)! times the number of reduced squares. [9] One can normalize a Cayley table of a quasigroup in the same manner as a reduced Latin square.