Past exam of the computer science course of the University of Oxford / 2015 / Computer security / 1 / b / i

For distinct prime numbers $p,q$, compute the Euler totient functionThe RSA cryptosystem private exponent is a modular multiplicative inverse of the public exponent:Use the extended Euclidean algorithm on $3$ and $\phi (n)$. It produces integers $u,v$ with $3u+\phi (n)v=1$, and reducing $u$ modulo $\phi (n)$ gives a suitable positive $d$. Searching through possible exponents is unnecessary.

To see correctness even for a plaintext sharing a factor with $n$, write $3d=1+t\phi (n)$. Modulo $p$, either $m=0$ or Fermat's little theorem gives $m^{3d}=m$; the same holds modulo $q$. The Chinese remainder theorem then gives $c^d=m(\mathrm{mod}n)$. One may alternatively invert $3$ modulo $\mathrm{lcm}(p-1,q-1)$, but inversion modulo $\phi (n)$ is sufficient for this paper.