  Group ring NTRU (GR-NTRU) provides a general structure to design different
variants of NTRU-like schemes by employing different groups. Although, most of
the schemes in literature are built over cyclic groups, nonabelian groups can
also be used. Coppersmith and Shamir in 1997 have suggested that
noncommutativity may result in better security against some lattice attacks for
some groups. Lattice attacks on the public key of NTRU-like cryptosystems try
to retrieve the private key by solving the shortest vector problem (SVP) or its
approximation in a lattice of a certain dimension, assuming the knowledge of
the public key only. This paper shows that dihedral groups do not guarantee
better security against this class of attacks. We prove that retrieving the
private key is possible by solving the SVP in two lattices with half the
dimension of the original lattice generated for GR-NTRU based on dihedral
groups. The possibility of such an attack was mentioned by Yasuda et
al.(IACR/2015/1170). In contrast to their proposed approach, we explicitly
provide the lattice reduction without any structure theorem from the
representation theory for finite groups. Furthermore, we demonstrate the
effectiveness of our technique with experimental results.
