Improved Progressive BKZ Algorithms and their Precise Cost Estimation by Sharp Simulator

In this paper, we investigate a variant of the BKZ algorithm, called progressive BKZ, which performs BKZ reductions by starting with a small blocksize and gradually switching to larger blocks as the process continues. We discuss techniques to accelerate the speed of the progressive BKZ algorithm by optimizing the following parameters: blocksize, searching radius and probability for pruning of the local enumeration algorithm, and the constant in the geometric series assumption (GSA). We then propose a simulator for predicting the length of the Gram-Schmidt basis obtained from the BKZ reduction. We also present a model for estimating the computational cost of the proposed progressive BKZ by considering the efficient implementation of the local enumeration algorithm and the LLL algorithm. Finally, we compare the cost of the proposed progressive BKZ with that of other algorithms using instances from the Darmstadt SVP Challenge. The proposed algorithm is approximately 50 times faster than BKZ 2.0 (proposed by Chen-Nguyen) for solving the SVP Challenge up to 160 dimensions

The Euclidean Distortion of Flat Tori

We show that for every n-dimensional lattice L the torus R n /L can be embedded with distortion O(n · √ log n) into a Hilbert space. This improves the exponential upper bound of O(n 3n/2) due to Khot and Naor (FOCS 2005, Math. Annal. 2006) and gets close to their lower bound of Ω ( √ n). We also obtain tight bounds for certain families of lattices. Our main new ingredient is an embedding that maps any point u ∈ R n /L to a Gaussian function centered at u in the Hilbert space L2(R n /L). The proofs involve Gaussian measures on lattices, the smoothing parameter of lattices and Korkine-Zolotarev bases.