Abstract
Lattice reduction is a powerful concept for solving diverse problems involving point lattices. Lattice reduction has been successfully utilizing in Number Theory, Linear algebra and Cryptology. Not only the existence of lattice based cryptosystems of hard in nature, but also has vulnerabilities by lattice reduction techniques. In this survey paper, we are focusing on point lattices and then describing an introduction to the theoretical and practical aspects of lattice reduction. Finally, we describe the applications of lattice reduction in Number theory, Linear algebra.
Highlights
Lattices are periodic arrangements of discrete points
Lattice reduction is concerned with finding improved representations of a given lattice using algorithms like LLL (Lenstra, Lenstra, Lovasz) reduction .There are some versions for lattice reduction, but people are using the LLL algorithm for theoretical and practical purposes
Lattice reduction techniques have a long tradition in mathematics in the field of number theory
Summary
Lattices are periodic arrangements of discrete points. Apart from their wide-spread use in pure mathematics, lattices have found applications in numerous other fields as diverse as cryptography/cryptanalysis, the geometry of numbers, factorization of integer polynomials, subset sum and knapsack problems, integer relations and Diophantine approximations, coding theory. Lattice reduction is concerned with finding improved representations of a given lattice using algorithms like LLL (Lenstra, Lenstra, Lovasz) reduction .There are some versions for lattice reduction, but people are using the LLL algorithm for theoretical and practical purposes It is a polynomial time algorithm and the vectors are nearly orthogonal. This technique, in turn can be applied to break knapsack cryptosystems like Merkle-Hellman. This technique, in turn can be applied to check vulnerabilities of RSA cryptosystem.
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