# Re: Quadratic Field Cryptography

• From: Colin Percival <cperciva@xxxxxx>
• Date: Wed, 8 Mar 2006 22:12:17 +0000 (UTC)

tomstdenis@xxxxxxxxx wrote:
bobic wrote:
Recently, I have been told that there were Quadractic Field
Cryprography, and I have no idea about this. However, I cannot find
any reference on it. Can you give me the introduction on Quadractic
Field Cryptography, or some good references? Thanks in advance!

What the hell is a quadractic field?
^

My first impression when reading this was "sounds like quackery to me".

Proceeding on the basis of a second reading and the hope that this is
merely a typographical error for "quadratic" field...

A finite field cannot be created with a quadratic unless you mean
extension fields e.g. GF(p^2)[x] which is really modulo some second
degree polynomial which is iireducible in GF(p^k)[x].

Not all fields are finite fields; for example, Q(sqrt(2)) -- that is,
the field formed by the rationals plus rational multiples of the square
root of 2 -- is a quadratic field.

In the case of finite fields, of course, you're quite correct:
Z_p(sqrt(k)) is the same as Z_p[x]/(x^2 - k), which is the same as
GF(p^2) providing that x^2 - k is irreducible modulo p. Personally I
find the polynomial representation to be far more intuitive than the
algebraic representation, but I know mathematicians who feel oppositely.

Colin Percival
.