Segment LLL Reduction of Lattice Bases Using Modular Arithmetic
Abstract
:1. Introduction
1.1. Definitions of Reduced Lattice Bases
- D1.
- A basis is called size-reduced if The notion of a size reduced basis goes back to Hermite [12].
- D2.
- A basis is called (δ,η)-reduced if for , , , . For and it is called 2-reduced because the above inequality becomes . A basis is called δ-LLL reduced if it is size-reduced and δ-reduced. It is simply called LLL reduced if it is size-reduced and 2-reduced. The LLL reduced basis was introduced by Lenstra, Lenstra, and Lovász [7].
- D3.
- A basis is called semi-reduced if it is size-reduced and satisfies weaker conditions for .
- D4.
- A basis is called Korkine-Zolotarev basis if it is size-reduced and if for where is the orthogonal projection of L on the orthogonal complement of .
- D5.
- A basis is called Block KZ reduced basis if it is size-reduced and if the projections of all -blocks on the orthogonal complement of for are Korkine-Zolotarev reduced.
- D6.
- A basis is called k-segment LLL reduced if the following conditions hold.
- C1.
- It is size-reduced.
- C2.
- for , , i.e., vectors within each segment of the basis are δ-reduced, and
- C3.
- Letting , two successive segments of the basis are connected by the following two conditions.
- C3.1.
- for.
- C3.2.
- for.
1.2. Discussion on Various Reduced Bases
Algorithm | Lower Bounds on | Upper Bounds on | Arithmetic Steps | Precision |
---|---|---|---|---|
LLL reduced [7] | ||||
LLL reduced [13] | ||||
Modular LLL [14] | ||||
Semi-reduced [8] | ||||
Kannan [9] | ||||
Block KZ [10,15] 1 | ||||
Segment LLL [11] | ||||
Mod-Seg LLL | ||||
Mod-Seg LLL FMM | ||||
Nguyen and Stehle [16] | fl | |||
Schnorr [17] SLL | fl |
1.3. Paper Contribution and Organization
2. Methods for LLL-Reduced Lattice Bases
2.1. The LLL Basis Reduction Algorithm
Size Reduction of B
Swap of Two Adjacent Rows of B
3. Storjohann’s Improvements
3.1. The LLL-Reduction with Fraction Free Computations
3.2. The Modified LLL Algorithm with Modular Arithmetic
4. The Segment LLL Reduction of Lattice Bases
5. The Modular Segment LLL Reduction with Modular Arithmetic
5.1. Algorithm and Its Complexity
5.2. Correctness of the ModSegmentLLL Algorithm
Updating E
5.3. The Modular Segment LLL using Fast Matrix Multiplication
6. Concluding Remarks
Acknowledgement
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Mehrotra, S.; Li, Z. Segment LLL Reduction of Lattice Bases Using Modular Arithmetic. Algorithms 2010, 3, 224-243. https://doi.org/10.3390/a3030224
Mehrotra S, Li Z. Segment LLL Reduction of Lattice Bases Using Modular Arithmetic. Algorithms. 2010; 3(3):224-243. https://doi.org/10.3390/a3030224
Chicago/Turabian StyleMehrotra, Sanjay, and Zhifeng Li. 2010. "Segment LLL Reduction of Lattice Bases Using Modular Arithmetic" Algorithms 3, no. 3: 224-243. https://doi.org/10.3390/a3030224
APA StyleMehrotra, S., & Li, Z. (2010). Segment LLL Reduction of Lattice Bases Using Modular Arithmetic. Algorithms, 3(3), 224-243. https://doi.org/10.3390/a3030224