# RSA Private Exponent

Is it possible to calculate the private exponent when only the RSA public key (e;N) = (9292162750094637473537; 13029506445953503759481) is given?

• That would only be possible for definitions of RSA that uniquely define the private exponent, such as FIPS 186-4; not for definitions of RSA that allow several private exponents, such as PKCS#1.
– fgrieu
Nov 27 '16 at 4:22

## 1 Answer

Yes, since $N=p \times q$, so we have $13029506445953503759481 = 109217267039×119298960679$, also the private and public exponent $e$ and $d$ are related by the equation $ed\equiv1\text{ mod }\phi(N)\equiv1\text{ mod }((109217267039-1)×(119298960679-1))$. You should then be able to find $d$, the private exponent.

• How were you able to find the primes? Nov 26 '16 at 17:32
• @ArsalanAmjid probably by handing the modulus to a factoring tool?
– SEJPM
Nov 26 '16 at 17:47
• @Arsalan Wolfram Alpha factored it in about 3 seconds. Note that in real life, N (and hence p and q) would be much, much larger...
– Dan
Nov 26 '16 at 20:11
• Yes, @Dan is right, now the standard for the modulus $N$ is at least $1024$ bits. Nov 27 '16 at 3:54