Public Key Generation

• To receive encrypted messages a receiver must determine parameters e, d, n.
• The receiver publishes e and n, it keeps d secret.
• Instead of sending plain text information P, transmitters compute the remainder C when P^e is divided by n.
• The receiver recovers the unencrypted message P by computing the remainder of C^d divided by n.
• This works only if d, e, n are chosen carefully.
• The following algorithm can be used to generate public key parameters d, e, n.
• Algorithm:
  • Select two prime numbers p and q.
  • Let $n=p \cdot q$.
  • Let $z=(p-1) \cdot (q-1)$.
  • Choose a number d that does not divide z.
  • Choose a number e such that $e \cdot d$ is a multiple of z plus 1.
• Example:
  • p=3, q=11
  • $\rightarrow$ n=33, z=20
  • Choose d=7
  • Choose e=3, $e \cdot d =21$, i.e., z+1
• Example:
  • p=11, q=17
  • $\rightarrow$ n=187, z=160
  • Choose d=51
  • Choose e=91, $e \cdot d = 4641 = 29 \cdot 160 +1$
• The strength of this method for generating d, e, n is based on the fact that is very difficult to find p and q from n if p and q are large prime numbers.