I`m solving one crypto problem on rsa.

p^e - q^e = C1 (mod n)
(p-q)^e   = C2 (mod n)

n = p*q*r; p,q,r are prime numbers
e = 2 * 65537

We have e, n, C1, C2.

It's impossible to find p, q, r from this system of equations, since there are 3 unknowns in the system of 2 equations. But is there any way to reduce the possible options?

  • 2
    $\begingroup$ The argument about the number of variables and equations only holds for real valued variables. With just n as the product of the primes, over the integers it is uniquely defined (ignoring relabeling). You could try: Expand the C2 equation and see what you get. And also make use of the fact that e is even. But I don't have an intuition if C1 and C2 reveal the factorization or not. $\endgroup$
    – tylo
    Nov 17, 2022 at 9:45
  • 2
    $\begingroup$ See if you can find a combination of $C_1$ and $C_2$ that is divisible by $p$ and ditto for $q$. $\endgroup$
    – Daniel S
    Nov 17, 2022 at 11:05


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