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

Question

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?

Answer 1

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.