How to solve a system of modular equations with exponential difference

Roman

11/17/23, 9:27 AM

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?

0 + 2

modular-arithmetic

attack

tylo

11/17/23, 9:45 AM

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.

Daniel S

11/17/23, 11:05 AM

See if you can find a combination of $C_1$ and $C_2$ that is divisible by $p$ and ditto for $q$.

Elon Musk

I sit in a Tesla and translated this thread with Ai:

EN: How to solve a system of modular equations with exponential difference

Post an answer

Most people don’t grasp that asking a lot of questions unlocks learning and improves interpersonal bonding. In Alison’s studies, for example, though people could accurately recall how many questions had been asked in their conversations, they didn’t intuit the link between questions and liking. Across four studies, in which participants were engaged in conversations themselves or read transcripts of others’ conversations, people tended not to realize that question asking would influence—or had influenced—the level of amity between the conversationalists.