New orthogonality criterion for shortest vector of lattices and its applications

The security of most lattice based cryptography relies on the hardness of computing a shortest nonzero vector of lattices. We say that a lattice basis is SV-reduced if it contains a shortest nonzero vector of the lattice. In this paper, we prove that, π∕6 orthogonality between the shortest vector of the basis and the vector space spanned by other vectors of the basis is enough to be SV-reduced under the assumption that a plausible condition Cn holds. By using the π∕6 orthogonality under C2, we prove a new complexity bound log3∥x∥23detL+1 for Gauss–Lagrange algorithm which clarifies why the currently known complexity is so far fall short to expose the efficiency of the algorithm we experience in practice. Our experiments suggest that our complexity bound of Gauss–Lagrange algorithm is somewhat close to actual efficiency of the algorithm. We also show that LLL(δ) algorithm outputs a SV-reduced basis if δ≥1∕3 for two dimensional lattice. We present an efficient three dimensional SV-reduction algorithm by using the condition C3 and π∕6 orthogonality and how to generalize the algorithm for higher dimension.