Find private exponent in RSA The other answers for this confuse me, I know the principle of d in terms of the equation it has to satisfy. However I'm not sure how I would go about finding the private exponent without trial and error. I want to implement this in a program, but I'm not quite sure the algorithm I would use to find d. Can someone explain the equation or algorithm for finding d?
If you know the prime factors (p,q) of the modulus, then it would be easy to find the private exponent.
You'll proceed as follow:


Compute the Carmichael function $\lambda(n)=\operatorname{lcm}(p-1,q-1)=\frac{(p-1)\times(q-1)}{\operatorname{gcd}(p-1,q-1)}$
Choose a number e such that $\operatorname{gcd}(e,\lambda(n))=1$,
Apply the Extended Euclid Algorithm to compute the inverse of $d=\frac{1}{e}\bmod\lambda(n)$,    Cf Donald Knuth:Semi-Numerical Algorithms II (or any other reference book),


Then the public Key is $(n, e)$, while the private Key is $(n,d)$. This correspond to the straightforward parameters of RSA Keys.
You can also generate with a similar way, the parameters associated to the CRT algorithm with allow a substantial gain when executing Modular exponentiation with the private Key.
If required I could depicted it.