Find private exponent in RSA

Question

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?

Answer 1

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.