RSA works based on Fermat’s theorem, all the formula are down here

$$ a^{p-1}\ mod\ p = 1\\

$$

$$ x=a^m\\ x^{-1}=a^{p-1-m} $$

$$ q^x\ mod\ p=a^{(p-1)*p+r}\ mod\ p\\ =a^r\ mod\ p\\ q^x\ mod\ p=q^{x\ mod\ (p-1)}\ mod\ p\\ gcd(x,p)=1\\

$$

$$ \phi(n) = (p-1)(q-1)\\ a^{\phi(n)} = a^{(p-1)(q-1)} $$

$$ a^x\ mod\ n = a^{x\ mod\ (p-1)(q-1)}\ mod\ n $$

$$ e*d=1\ mod\ (p-1)(q-1) $$

$$ \sqrt[7]{2+n*q} $$

• Note