can someone tell me exactly how key generators work?

Answer 1

http://en.wikipedia.org/wiki/Key_generator <----- Here you go babe. More related articles in the link ;)

Answer 2

A keygen (a portmanteau of "key generator") is a small program that will generate a CD key or serial/registration number for a piece of software or cryptographic algorithm.

How they work
Typically software requests a serial number only during installation. The installation software applies an algebraic expression to the key entered to validate the key. As an example, the algorithm may define that a key entered must contain 5 numbers that have a checksum of 25, and that the key must also contain 3 letters that, when mapped into their numerical equivalents, have a sum of 42.

In order to write a keygen, the author typically uses a disassembler to look at the raw assembly code of the program they are writing the keygen for, checking either the software itself or the installer. Once they have access to the program's code, they can locate the subroutine(s) responsible for verifying that the key entered is valid. Using this knowledge, they can reverse engineer the algorithm used to generate valid keys, which is then incorporated into the keygen.

Answer 3

Software programs validate keys by having a secret, complex mathematical formula that runs on the key you enter and, if the answer is one of the expected values, the key is valid.

A key generator is simply a program created by someone who figured out what the mathematical formula is, and can create new keys that will produce the correct answer.

A very simple example could be:

(a+b)%2=0

That means that the sum of A and B must be an even number.

So if your key is 12 (position A is 1, position B is 2), then the sum is 3 - not an even number, not a valid key.