Primes With Complex Factors
Frank Shute
frank at shute.org.uk
Sun May 11 23:23:29 BST 2008
On Sun, May 11, 2008 at 08:58:40PM +0100, Lee Brotherston wrote:
>
> On Sun, May 11, 2008 at 08:50:42PM +0100, Ceri Davies wrote:
>
> > Of course there are; that was generally AS or degree level stuff when I
> > was doing my degree though. That's what "complex numbers" are.
>
> Ok, but I thought those were more an abstract concept rather than a
> whole number that you could class as a factor?
They are an abstract concept the same way as integers are :)
But as you point out, factors are generally taken to be the positive
integers not complex numbers.
In a non-abstract setting they are used in some areas of electronics
where i is denoted by j, I believe.
I first learnt about them doing mech/man engineering (but never used
them in an engineering setting!).
Wrote a program in turbo pascal to x,/,+,- them. The first program I
wrote :)
> But I never did maths at degree level so I'm totally up for being
> wrong :)
I think they used to come up at A-level. I've done some degree
level maths and they come up fairly early. IIRC they fall under the
remit of number theory, which then progresses to groups, and crypto is
a sort of offshoot.
It's a few years ago that I learnt the mathematical theory of how
primes fit into crypto and I've all but forgotten it :(
IIRC, the product of two large primes is not possible to factorise
other than by iterative means (which takes a *long* time), and this
forms the basis of modern crypto.
Another thing I remember in "The Road Ahead", author: Bill Gates:
"In the future with increased computing power, we may be able to
factor large primes."
Or words to that effect :)
As to the original postulation that the Chinese are getting good at
cracking crypto by means of factorisation of primes with complex
numbers. I don't believe it.
It has been mathematically proven that you can't crack strong
prime-based crypto. They'd have to overthrow Gauss, Pascal, Hardy,
Galois, Mersenne and all the rest of them. That doesn't happen in
maths.
The only way they could do it, is by improving the iterative process
ie. by building *much* faster computers.
>
> Cheers
>
> Lee
Regards,
--
Frank
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