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Most of the cryptography is limited to use of real integer fields. Complex numbers are seldom used. Is there any reason for it? Wouldn't complex numbers provide more structure?

If there is already some work on this area, I would be happy to learn about it.

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  • $\begingroup$ With "complex numbers" he means "imaginary numbers" like $i^2 = -1$ $\endgroup$ – mentallurg Jun 8 at 11:20
  • $\begingroup$ @mentallurg OP can tell us themselves what they mean. Please don't edit other people's posts based on what you think they mean. $\endgroup$ – fkraiem Jun 8 at 12:20
  • $\begingroup$ @fkraiem: See the answer from "djao" below: 'I don't know what you mean by "complex number fields"'. He clearly says he does not understand the OP. $\endgroup$ – mentallurg Jun 8 at 13:15
  • $\begingroup$ @mentallurg It isn't clear whether the OP meant merely an analogue of the complex numbers as in a field extension with a root of $i^2 + 1$, as the phrases ‘real integer’ (in contrast, perhaps, to Gaussian integer) and ‘complex numbers’ suggest, or whether the OP meant to discuss number fields, as the phrase ‘complex number fields’ suggests. I interpreted it the first way; djao (reasonably) interpreted it the second way. Don't speak for the OP in deciding which interpretation they meant—they can speak for themselves. $\endgroup$ – Squeamish Ossifrage Jun 8 at 13:35
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    $\begingroup$ @girisha-shankar: Please help us :) Explain what you mean by "complex numbers". $\endgroup$ – mentallurg Jun 8 at 13:39
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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.)

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  • $\begingroup$ Thanks for your inputs. What I meant was use of complex numbers and their algebraic structures in cryptography. From what you have described, looks like they are in use. $\endgroup$ – Girisha Shankar Jun 9 at 13:29
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I don't know what you mean by "complex number fields" or "real integer fields"; neither term registers on Google, and the integers in particular are not a field (nor are reals integers, in general). Assuming by "complex number field" you mean a number field which is not totally real, there are significant applications of such number fields to cryptography, provided that you know where to look. For example, the canonical embedding in lattice-based cryptography (see top of page 29 of Peikert's notes) uses number fields in the complex numbers. CSIDH and class group cryptosystems are examples of cryptosystems that use imaginary (i.e. non-real) quadratic number fields.

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