Abstract
The shortest vector problem is a central computational problem in the classical area of geometry of numbers. The approximation algorithm presented below has many applications in computational number theory and cryptography. Two of its most prominent applications are the derivation of polynomial time algorithms for factoring polynomials over the rationals and for simultaneous diophantine approximation.
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A.K. Lenstra, H.W. Lenstra, Jr., and L. Lovâsz. Factoring polynomials with rational coefficients. Math. Ann., 261:513–534, 1982. (Cited on p. 292)
C.P. Schnorr. A hierarchy of polynomial time lattice basis reduction algorithms. Theoretical Computer Science,53:201–224, 1987. (Cited on p. 292)
C.F. Gauss. Disquisitiones Arithmeticae. English edition translated by A.A. Clarke. Springer-Verlag, New York, NY, 1986. (Cited on p. 292)
J. Lagarias. Worst case complexity bounds for algorithms in the the theory of integral quadratic forms. Journal of Algorithms,1:142–186, 1980. (Cited on p. 292)
R. Kannan. Algorithmic geometry of numbers. In Annual Review of Computer Science, Vol. 2,pages 231–267. Annual Reviews, Palo Alto, CA, 1987. (Cited on p. 293)
R. Kannan. Minkowski’s convex body theorem and integer programming. Mathematics of Operations Research,12(3):415–440, 1987. (Cited on p. 293)
J. Lagarias, H.W. Lenstra, Jr., and C.-P. Schnorr. Korkin—Zolotarev bases and successive minima of a lattice and its reciprocal lattice. Combinatorica,10:333–348, 1990. (Cited on p. 293)
M. Ajtai. The shortest vector problem in.f2 is NP-hard for randomized reductions. In Proc. 30th ACM Symposium on the Theory of Computing, pages 10–19, 1998.
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Vazirani, V.V. (2003). Shortest Vector. In: Approximation Algorithms. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04565-7_27
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DOI: https://doi.org/10.1007/978-3-662-04565-7_27
Publisher Name: Springer, Berlin, Heidelberg
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