LLL Lattice Basis Reduction

A lattice is a discrete subgroup of \(\mathbb{R}^n\): the set of all integer linear combinations of a basis \(B = \{b_1, \ldots, b_k\}\). The same lattice has infinitely many bases; some bases consist of short, nearly orthogonal vectors, while others are long and nearly parallel. The quality of the basis determines how efficiently lattice problems can be solved.

Lattice basis reduction is the task of finding a “good” (short, nearly orthogonal) basis for a given lattice. The LLL algorithm (Lenstra, Lenstra, Lovász, 1982) does this in polynomial time.

The LLL algorithm

LLL produces a basis \((b_1^*, \ldots, b_k^*)\) satisfying two conditions:

1. Size reduction: $$

   \mu_{ij}

   \leq \frac{1}{2}\(for all\)i > j\(, where\)\mu_{ij} = \langle b_i, b_j^* \rangle / |b_j^*|^2$$ are the Gram–Schmidt coefficients.

2. Lovász condition: \(\|b_i^*\|^2 \geq \left(\frac{3}{4} - \mu_{i,i-1}^2\right)\|b_{i-1}^*\|^2\).

The first output vector \(b_1\) satisfies \(\|b_1\| \leq 2^{(k-1)/2} \lambda_1\), where \(\lambda_1\) is the length of the shortest nonzero lattice vector. LLL thus gives an exponential approximation to the shortest vector problem (SVP).

Applications

Factoring polynomials over \(\mathbb{Q}\) (the original application): reduce polynomial factoring to finding short vectors in a lattice derived from the polynomial. LLL was the key tool in the first polynomial-time factoring algorithm.

Integer programming in fixed dimension: short vectors in the dual lattice correspond to cutting planes that are tight for the integer hull.

Cryptanalysis: LLL (and its stronger variant BKZ) breaks many lattice-based cryptographic schemes when parameters are too small.

Diophantine approximation: find rational approximations to real numbers with controlled denominator size.

Goal

Implement LLL over \(\mathbb{Q}\) (exact arithmetic), verify the size-reduction and Lovász conditions after each step, and apply it to at least one of: polynomial factoring over \(\mathbb{Q}\), or finding short vectors in random lattices.

Milestones

Tasks

Deliverables

• Julia implementation with exact arithmetic
• Correctness tests and approximation factor measurements
• Short report (6–8 pages)

References

1. A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász. Factoring polynomials with rational coefficients. Mathematische Annalen, 261(4):515–534, 1982. DOI 10.1007/bf01457454