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

in flag

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 $N$?

kelalaka avatar
in flag
What is the origion of this Q? How much this difference?
Manc avatar
in flag
diff/n is about 3n/5,I thought it looks like some special trick in Cryptanalysis of RSA with two decryption exponents
kodlu avatar
sa flag
you mean absolute difference not difference divided by n clearly
fgrieu avatar
ng flag
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 $d$](

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