Lower bounds of shortest vector lengths in random knapsack lattices and random NTRU lattices

Finding the shortest vector of a lattice is one of the most important problems in computational lattice theory. For a random lattice, one can estimate the length of the shortest vector using the Gaussian heuristic. However, no rigorous proof can be provided for some classes of lattices, as the Gaussian heuristic may not hold for them. In the paper we study two types of random lattices in cryptography: the knapsack lattices and the NTRU lattices. For random knapsack lattices, we prove lower bounds of shortest vector lengths, which are very close to lengths predicted by the Gaussian heuristic. For a random NTRU lattice, we prove that with a overwhelming probability, the ratio between the length of the shortest vector and the length of the target vector, which corresponds to the secret key, is at least a constant, independent of the dimension of the lattice. The main technique we use is the incompressibility method from the theory of Kolmogorov complexity.