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Books > Reference & Interdisciplinary > Communication studies > Coding theory & cryptology
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Complexity of Lattice Problems - A Cryptographic Perspective (Hardcover, 2002 ed.)
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Complexity of Lattice Problems - A Cryptographic Perspective (Hardcover, 2002 ed.)
Series: The Springer International Series in Engineering and Computer Science, 671
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Lattices are geometric objects that can be pictorially described as
the set of intersection points of an infinite, regular
n-dimensional grid. De spite their apparent simplicity, lattices
hide a rich combinatorial struc ture, which has attracted the
attention of great mathematicians over the last two centuries. Not
surprisingly, lattices have found numerous ap plications in
mathematics and computer science, ranging from number theory and
Diophantine approximation, to combinatorial optimization and
cryptography. The study of lattices, specifically from a
computational point of view, was marked by two major breakthroughs:
the development of the LLL lattice reduction algorithm by Lenstra,
Lenstra and Lovasz in the early 80's, and Ajtai's discovery of a
connection between the worst-case and average-case hardness of
certain lattice problems in the late 90's. The LLL algorithm,
despite the relatively poor quality of the solution it gives in the
worst case, allowed to devise polynomial time solutions to many
classical problems in computer science. These include, solving
integer programs in a fixed number of variables, factoring
polynomials over the rationals, breaking knapsack based
cryptosystems, and finding solutions to many other Diophantine and
cryptanalysis problems."
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