**Authors:** Miroslav Kureš

**Keywords:** Imaginary quadratic field, ring of integers, non-elementary matrices, special linear group, public-key cryptography, lattice-based cryptosystems.

### Abstract

All cube roots of the identity in the special linear group of $2\times 2$-matrices
with entries in the ring of integers in $\mathbb Q[\sqrt{d}]$ are described. These matrices generate subgroups of the third order; it is shown that such subgroups may contain non-elementary matrices in the sense of P. M. Cohn.
All this is viewed with respect to possible applications in lattice cryptography.

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