Explicit Formula for Gram-Schmidt Vectors in LLL with Deep Insertions and Its Applications.

Lattice basis reduction algorithms have been used as a strong tool for cryptanalysis. The most famous one is LLL, and its typical improvements are BKZ and LLL with deep insertions (DeepLLL). In LLL and DeepLLL, at every time to replace a lattice basis, we need to recompute the Gram-Schmidt orthogonalization (GSO) for the new basis. Compared with LLL, the form of the new GSO vectors is complicated in DeepLLL, and no formula has been known. In this paper, we give an explicit formula for GSO in DeepLLL, and also propose an efficient method to update GSO in DeepLLL. As another work, we embed DeepLLL into BKZ as a subroutine instead of LLL, which we call "DeepBKZ", in order to find a more reduced basis. By using our DeepBKZ with blocksizes up to beta = 50, we have found a number of new solutions for the Darmstadt SVP challenge in dimensions from 102 to 123.