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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?

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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.

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    $\begingroup$ Koblitz curves are also over the field $\mathbb{F}^{*}_{2^n}$. $\endgroup$ – puzzlepalace Jan 17 at 21:58

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