Learning with errors

Authors: Andrew Mendelsohn, Charles Grover, Cong Ling
Citation: Mendelsohn et al. (2025)

Abstract:
We propose a dimension-reducing transformation on Group Ring Learning with Errors (GRLWE) samples. We exhibit an efficiently computable isomorphism which takes samples defined over the group rings used in the construction of GRLWE to twice as many samples defined over matrix rings, in half the dimension. This is done by composing two maps: the first map is a transformation showing that the group rings used are orders in central simple algebras, and the second map takes the obtained central simple algebra to a matrix ring. When combined with lattice reduction on the resulting matrix samples, this gives an attack on the GRLWE problem. We extend this attack to other groups proposed for cryptographic use by the creators of GRLWE, and display some numerical results quantifying the effects of the transformation, using the ‘Lattice Estimator’. We then give a family of groups from which GRLWE-style group rings can be constructed which are immune to our attack, namely the generalized quaternion groups. Finally, we discuss the merits and vulnerabilities of a number of different forms of structured LWE.