Assessing the influence of the algebraic structure of q-ary lattices on the complexity of cryptanalysis of problems on lattices

Abstract

The work is devoted to the influence of the algebraic structure of q-ary lattices on the complexity of cryptanalysis of cryptographic problems on lattices. The root-mean-square errors for existing lattice reduction models are obtained. It is shown that the deterministic Albrecht-Lee simulator shows the smallest mean square deviations on small lattices. Collapsibility estimates for nesting and decoding attacks on the algebraic structure of q-ary lattices have been found. A new method for selecting attack parameters has been proposed for the decoding attack. It is shown that a decoding attack with such a choice of parameters can be at the expense of nesting attacks. To distribute the power of the secret vector in decoding attacks, which differ from the normal one, the approximation method is used by the normal division, which minimizes the Kolmogorov-Smirnov equation and calculates the optimal values for some distributions. There are no parameters for any divisions. For the SIS problem, a new approach to safety assessment has been proposed, which takes into account the possibility of various Euclidean norms. Based on the new approach, estimates of the SIS problem for Crystals-Dilithium electronic signature schemes have been calculated.