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Why are very large prime numbers important in cryptography?
I'm interested in how the algorithms for creating key pairs to be used in dual key encryption work. I have looked them up and they are a little too much for me to parse at my level but I could really use a conceptual explanation of how prime numbers fit into the picture (they seem to the the key to it all).
Another way of saying this is I know what dual key encryption is and I know what prime numbers are. I would like to understand what properties of the one connects it to the other.
A related curiosity is this: if prime numbers are spread out and hard to find, how is there a seemingly unlimited number of key pairs and how can programs generate them so quickly?