Fast Reduction of Ternary Quadratic Forms
- Friedrich EisenbrandAffiliated withMax-Planck-Institut für Informatik
- , Günter RoteAffiliated withInstitut für Informatik, Freie Universität Berlin
We show that a positive definite integral ternary form can be reduced with O(M(s) log2 s) bit operations, where s is the binary encoding length of the form and M(s) is the bit-complexity of s-bit integer multiplication.
This result is achieved in two steps. First we prove that the the classical Gaussian algorithm for ternary form reduction, in the variant of Lagarias, has this worst case running time. Then we show that, given a ternary form which is reduced in the Gaussian sense, it takes only a constant number of arithmetic operations and a constant number of binary-form reductions to fully reduce the form.
Finally we describe how this algorithm can be generalized to higher dimensions. Lattice basis reduction and shortest vector computation in fixed dimension d can be done with O(M(s) logd-1 s) bit-operations.
- Fast Reduction of Ternary Quadratic Forms
- Book Title
- Cryptography and Lattices
- Book Subtitle
- International Conference, CaLC 2001 Providence, RI, USA, March 29–30, 2001 Revised Papers
- pp 32-44
- Print ISBN
- Online ISBN
- Series Title
- Lecture Notes in Computer Science
- Series Volume
- Series ISSN
- Springer Berlin Heidelberg
- Copyright Holder
- Springer-Verlag Berlin Heidelberg
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- Editor Affiliations
- 4. Mathematics Department, Brown University
- Author Affiliations
- 5. Max-Planck-Institut für Informatik, Stuhlsatzenhausweg 85, 66123, Saarbrücken, Germany
- 6. Institut für Informatik, Freie Universität Berlin, Takustraβ e 9, 14195, Berlin, Germany
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