How to make a simple public-key cryptographic algorithm?

Question

I want to make a simple public-key(asymmetric) encryption. It doesn't have the be secure, I just want to understand the concepts behind them. For instance, I know simple symmetric ciphers can be made with an XOR. I saw in a thread on stackexchange that you need to use trapdoor functions, but I can't find much about them. I want to say, take a group of bytes, and be able to split them someway to get a public/private key. I get the ideas of a shared secret. Say, I generate the random number of 256(not random at all :P), and I split it into 200 and 56. If I do an XOR with 200, I can only decrypt with 200. I want to be able to split numbers random and such to be able to do it asymmetrically.

Answer 1

OK, just a simple demo-idea, based on adding/modulo operation.

1. Lets say we have a modulo value, for our example 256. This is a public-known, common value.

2. Let's say you generate a random secret private key in the interval [1-255], for example, pri=133. Keep secret key in the pocket.

3. Generate a public key, pub = 256 - pri = 123. This public key (123) you can share to the world. Imagine, 3rd party does not know, how to compute the private key from a public. So, they know only public key (123).

4. Someone from the public wants to send you an encrypted ASCII-byte. He gets his byte, and adds to it the public key by modulo 256 operation:

   encrypted = (input_value + pub) % modulto;
   

For example, I want to send you the letter "X", ASCII code = 88 in encrypted form. So, I compute:

(88 + 123) % 256 = 211;

5. I am sending you the value 211 - encrypted byte.

6. You decrypt it by the same scheme with your private key:

   decrypted = (input_value + pri) % 256 = (211 + 133) % 256 = 88;
   

Of course, using the simple generation pair in this example is weak, because of the well-known algorithm for generating the private key from the public, and anybody can easily recover the private using the modulo and public. But, in real cryptography, this algorithm is not known. But, theoretically, it can be discovered in future.