# How to factorize RSA modulus while given two Public Exponents and the difference between two Private Exponents?

The RSA modulus is the product of two $$2048$$-bit primes.

And the two Public Exponents are both $$16$$-bit.

I also got the difference between two Private Exponents $$\left | d_1-d_2 \right |.$$

Is there any way to factorize the Modulus $$N$$?

• What is the origion of this Q? How much this difference? Commented Oct 9, 2022 at 21:51
• diff/n is about 3n/5,I thought it looks like some special trick in Cryptanalysis of RSA with two decryption exponents
– Manc
Commented Oct 9, 2022 at 22:00
• you mean absolute difference not difference divided by n clearly Commented Oct 9, 2022 at 22:32
• So you know $N,e_1,e_2, |d_1-d_2|$ with small $e_1$ and $e_2$, and want to factor $N$. Hint: write a relation that must exist between $e_1$ and $d_1$, same between $e_2$ and $d_2$. If these contain$\bmod$, apply the definition of that to remove it. And proceed to adapt the usual method to factor $N$ given $e$ and $d$.
– fgrieu
Commented Oct 10, 2022 at 5:47