Why aren't Complex number fields used in cryptography?
Why aren't complex number fields used in cryptography?


If you mean fields of the form $k[i]/(i^2 + 1)$ where $i^2 + 1$ is irreducible over the base field $k$, well, they are!  FourQ is a twisted Edwards curve $-x^2 + y^2 = 1 + d x^2 y^2$ over the quadratic extension field $\mathbb F_{p^2} = \mathbb F_p[i]/(i^2 + 1)$ of Mersenne prime characteristic $p = 2^{127} - 1$, where $$d = 25317048443780598345676279555970305165i + 4205857648805777768770$$ is nonsquare.  FourQ is appealing because it takes advantage of cheap arithmetic modulo a Mersenne prime on a curve together with a cheap endomorphism.  The coordinate field is in a sense the ‘complex’ extension of $\mathbb F_{2^{127} - 1}$ arising by adjoining a square root of $-1$.

(FourQ is not universally appealing: it has a relatively high cofactor, $8\cdot7^2 = 392$, and it does not provide modern standards of twist security, so it's not a drop-in replacement for X25519 or Ed25519.  But it is implemented in practice by Microsoft's FourQlib and probably deployed somewhere.)