Cryptosystems defined over the real numbers
The complex roots of unity form a finite group under multiplication

$$z^k = \exp\bigg(\frac{2k{\pi}i}{n}\bigg) = \cos\frac{2k{\pi}}{n} + i\sin\frac{2k{\pi}}{n},$$

where $z$ is a complex generator of the order-$n$ multiplicative group. This is a cyclic group and is isomorphic to any other cyclic group of the same order, for example the integers modulo 2 under addition, which is equivalent to Boolean XOR

$$a \oplus b \iff z^a z^b = z^{a+b}$$

This equivalence gives rise to the group Fourier transform, in which vector convolution in the complex domain correspond to point-wise multiplication in the additive group. This fact is used by Joan Daemen in his thesis to derive the DC/LC properties for the AES S-BOX.

The obvious practical problem with computing in this group is the error associated with floating-point arithmetic. As the number of arithmetic operations increases, so does the error. This kind of round-off error may be more appropriately called quantization noise in the signal processing context.