literka,

Could you please present the number of operations needed to carry on all your calculation (1) - (5), etc, compared to a simple multiplication of two factors.

It appears that to assert these:

Code:

Several equalities will be needed:
(1) p=208648999^2+126945596^2 = 512 * q+1
(2) s = 208648999*52542249 + 126945596*31967597
(3) 126945596*52542249 - 208648999*31967597 = 1
(4) 309*q + r = 2^55
(5) r = (s div 512)+1

one will have to do much more than to simply assert

Code:

59649589127497217 *
5704689200685129054721
= 340282366920938463463374607431768211457
= 2^128+1

And even less work

*in octal* (with an octal multiplication table at your side)

Code:

2^128+1 = 4000000000000000000000000000000000000000001
(note: this equality you will have for free!)
and then
3237257607274243001
*
1152401672664431414535001
=
4000000000000000000000000000000000000000001