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June 3rd, 2014, 03:18 PM   #1
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Congruences and diophantine equations

Hi everybody,

Let n positive integer > 1
p and q odd distinct prime numbers

Find n, p and q such as :

(p^n mod pq) + (q^n mod pq) = p + q

I know that there are an infinite number of solutions.
Is there some theorems to build?
Is there a way to factorize some odd semi-prime numbers?
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June 6th, 2014, 04:00 PM   #2
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Hi,

No answer yet
Is there something wrong?
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June 6th, 2014, 06:13 PM   #3
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Well, no, you can't factor semiprimes quickly, otherwise the RSA algorithm would all break down, and cryptography would be utterly ruined.

As for the original equation, I'm not sure whether it can be simplified.
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June 7th, 2014, 05:40 AM   #4
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Quote:
Originally Posted by eddybob123 View Post
Well, no, you can't factor semiprimes quickly, otherwise the RSA algorithm would all break down, and cryptography would be utterly ruined.

As for the original equation, I'm not sure whether it can be simplified.
The discrete logarithm which is labeled as stronger than RSA was broken few days ago.
So why are presuming so?
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June 7th, 2014, 07:24 AM   #5
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Here you can read in french the article about the discrete logarithm

Actualité > En bref : un nouvel algorithme déjoue les systèmes de cryptographie

or here in english :

A Heuristic Quasi-Polynomial Algorithm for Discrete Logarithm in Finite Fields of Small Characteristic - Springer

The breach discovery is very recent.
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