A 3-Dimensional Lattice Reduction Algorithm
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Abstract
The aim of this paper is a reduction algorithm for a basis b1,b2, b3 of a 3-dimensional lattice in ℝn for fixed n ≥ 3. We give a definition of the reduced basis which is equivalent to that of the Minkowski reduced basis of a 3-dimensional lattice. We prove that for b 1 , b2, b3 ε ℤn, n ≥ 3 and |b1|, |b2|, |b3 | ≤ M, our algorithm takes O(log2 M) binary operations, without using fast integer arithmetic, to reduce this basis and so to find the shortest vector in the lattice. The definition and the algorithm can be extended to any dimension. Elementary steps of our algorithm are rather different from those of the LLL-algorithm, which works in O(log3 M) binary operations without using fast integer arithmetic.
Keywords
3-dimensional lattice lattice reduction problem shortest vector in a lattice Gaussian algorithm LLL-algorithmPreview
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References
- 1.J.W.S. Cassels, Rational quadratic forms, Academic Press, London, New York, 1978.zbMATHGoogle Scholar
- 2.H. Cohen, A course in computational algebraic number theory, Springer-Verlag, Berlin, Heidelberg, 1993.zbMATHGoogle Scholar
- 3.F. Eisenbrand, Fast reduction of ternary quadratic forms, this volume.Google Scholar
- 4.A.K. Lenstra, H.W. Lenstra, and L. Lovasz, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 515–534.zbMATHMathSciNetCrossRefGoogle Scholar
- 5.B. Vallee, An affine point of view on minima finding in integer lattices of lower dimensions, Proc. of EUROCAL’87, LNCS 378, Springer-Verlag, Berlin, 1989, 376–378.Google Scholar