Cyclic Groups Other than $\mathbb{Z}^*_n$ or Elliptic Curves

I see two types of cyclic groups are most commonly used in cryptography:

• modulo multiplicative group of integers with prime order
• elliptic curves

Are there any other cyclic groups used in cryptography?

The multiplicative group $$\mathbb{F}_{2^n}^{\ast}$$ (which is cyclic) is used in defining LFSR sequences and nonlinearly filtered sequences typically as building blocks of stream ciphers.

This group is part of the picture for block ciphers; most prominently, the field $$\mathbb{F}_{2^n}$$ is used (for $$n=8$$) for defining the AES. Boolean functions and vector boolean functions defined over $$\mathbb{F}_{2^n}$$ also appear.

• Koblitz curves are also over the field $\mathbb{F}^{*}_{2^n}$. Jan 17 '19 at 21:58