Abstract
The algorithm of Lenstra, Lenstra, and Lovász (LLL) transforms a given integer lattice basis into a reduced basis. Storjohann improved the worst case complexity of LLL algorithms by a factor of O(n) using modular arithmetic. Koy and Schnorr developed a segment-LLL basis reduction algorithm that generates lattice basis satisfying a weaker condition than the LLL reduced basis with O(n) improvement than the LLL algorithm. In this paper we combine Storjohann’s modular arithmetic approach with the segment-LLL approach to further improve the worst case complexity of the segment-LLL algorithms by a factor of n0.5.
Highlights
Given row vectors b1, . . . , bn ∈ Zd an integer lattice L is defined as ( ) n X d d L := v ∈ Z |v =zi bi, zi ∈ Z, bi ∈ Z i=1Several important theoretical and practical problems benefit from studying lattices
This is a difficult problem to solve. It is shown by Ajtai [4] that the problem of finding the shortest non-zero lattice vector under l2 norm is NP-hard under randomized reduction [4]
In this paper we show that the modular arithmetic computation approach of [14] can be combined with the segment concept in [20] to develop a modular segment reduction algorithm
Summary
Several important theoretical and practical problems benefit from studying lattices. These include problems in geometry [1], cryptography [2], and integer programming [3]. The shortest lattice vector problem is a special case of finding the shortest lattice vector only. This is a difficult problem to solve. It is shown by Ajtai [4] that the problem of finding the shortest non-zero lattice vector under l2 norm is NP-hard under randomized reduction [4]. Micciancio [5] showed that an
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
