Cube-root-subgroups of SL2 over imaginary quadratic integers

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.

Keywords

• imaginary quadratic field
• ring of integers
• non-elementary matrices
• special linear group
• public-key cryptography
• lattice-based cryptosystems