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This is the algorithm which is very widely used for encrypting all kinds of private messages:

• Find two large prime numbers P and Q;
  
• Choose a number E less than PQ, which has no prime factors in common with (P−1)(Q−1);
  
• Find E, the multiplicative inverse of D mod (P−1)(Q−1). This means that DE=1 (mod(P−1)(Q−1)), i.e. (DE−1) is divisible by (P-1)(Q-1);
  
• Now the function to encrypt a message represented by a positive integer T, is f(T)=T^E(mod(PQ)).
  
• The function to decrypt an encrypted message represented by C, is g(C)=C^D(mod(PQ)).
  

The public key is the pair of numbers (PQ, E). This can be published freely. Your private key is the number D, and must be kept secret. This means that anyone can encrypt messages to me using my public key, but only I can read them using my private key. This works because there is no known way to work out D, P or Q given (PQ, E), except to factorise PQ. If P and Q each have around 1024 digits, in binary, this factorisation would take billions of years using present-day computers.