# 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

.

**References**:**Quadratic Field Cryptography***From:*bobic

**Re: Quadratic Field Cryptography***From:*tomstdenis

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