Simultaneous reduction of a lattice basis and its reciprocal basis

@article{Seysen1993SimultaneousRO,
  title={Simultaneous reduction of a lattice basis and its reciprocal basis},
  author={Martin Seysen},
  journal={Combinatorica},
  year={1993},
  volume={13},
  pages={363-376}
}
  • M. Seysen
  • Published 1993
  • Mathematics, Computer Science
  • Combinatorica
AbstractGiven a latticeL we are looking for a basisB=[b1, ...bn] ofL with the property that bothB and the associated basisB*=[b1*, ...,bn*] of the reciprocal latticeL* consist of short vectors. For any such basisB with reciprocal basisB* let $$S(B) = \mathop {\max }\limits_{1 \leqslant i \leqslant n} (|b_i | \cdot |b_i^ * |)$$ . Håstad and Lagarias [7] show that each latticeL of full rank has a basisB withS(B)≤exp(c1·n1/3) for a constantc1 independent ofn. We improve this upper bound toS(B)≤exp… Expand
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References

SHOWING 1-10 OF 23 REFERENCES
Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice
TLDR
It is proved that and, where andγj denotes Hermite's constant, are lower bounds for polynomial time computable quantities λ1(L) andΜ(x,L), where Μ( x,L) is the Euclidean distance fromx to the closest vector inL. Expand
A Hierarchy of Polynomial Time Lattice Basis Reduction Algorithms
  • C. Schnorr
  • Computer Science, Mathematics
  • Theor. Comput. Sci.
  • 1987
TLDR
An algorithm which for k?N finds a nonzero lattice vector b so that |b|2?(6k2)nk?(L)2, and successively applies Korkine?Zolotareff reduction to blocks of length k of the lattice basis. Expand
A More Efficient Algorithm for Lattice Basis Reduction
  • C. Schnorr
  • Mathematics, Computer Science
  • J. Algorithms
  • 1988
TLDR
A new algorithm simulates the Lovasz algorithm through approximate arithmetic, which uses at most O ( n 4 log B ) arithmetic operations on O-bit integers to transform a given integer lattice basis into a reduced basis. Expand
Simultaneously good bases of a lattice and its reciprocal lattice
It is well-known that every lattice L has a basis consisting of relatively short vectors. To state a quantitative version of this fact, let 21(L) denote the i-th successive minimum of L, which is theExpand
How to calculate shortest vectors in a lattice
A method for calculating vectors of smallest norm in a given lattice is outlined. The norm is defined by means of a convex, compact, and symmetric subset of the given space. The main tool is theExpand
Improved algorithms for integer programming and related lattice problems
  • R. Kannan
  • Computer Science, Mathematics
  • STOC
  • 1983
TLDR
The proposed algorithm first finds a “more orthogonal” basis for a lattice than those of Lenstra (1981) and Lenstra, Lenstra and Lovasz (1982), but in time 0(ndn poly (length of input)). Expand
Solving low density subset sum problems
  • J. Lagarias, A. Odlyzko
  • Computer Science, Mathematics
  • 24th Annual Symposium on Foundations of Computer Science (sfcs 1983)
  • 1983
TLDR
This method gives a polynomial time attack on knapsack public key cryptosystems that can be expected to break them if they transmit information at rates below dc (n), as n → ∞. Expand
Basis Reduction Algorithms and Subset Sum Problems
This thesis investigates a new approach to lattice basis reduction suggested by M. Seysen. Seysen''s algorithm attempts to globally reduce a lattice basis, whereas the Lenstra, Lenstra, Lovasz (LLL)Expand
Factoring Integers and Computing Discrete Logarithms via Diophantine Approximations
  • C. Schnorr
  • Mathematics, Computer Science
  • EUROCRYPT
  • 1991
TLDR
It is shown, under the assumption that the smooth integers distribute "uniformly", that there are Ne+o(1) many solutions (e1,...,et) if c > 1 and if Ɛ := c - 1 - (2c - 1) log log N / log pt > 0. Expand
Lattice Basis Reduction: Improved Practical Algorithms and Solving Subset Sum Problems
TLDR
Empirical tests show that the strongest of these algorithms solves almost all subset sum problems with up to 58 random weights of arbitrary bit length within at most a few hours on a UNISYS 6000/70 or within a couple of minutes on a SPARC 2 computer. Expand
...
1
2
3
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