Before anyone suggest it, I\'ve yet to pick up \"Applied Cryptography\" I\'m pla

Question

Before anyone suggest it, I've yet to pick up "Applied Cryptography" I'm planning on picking it up the next time I visit amazon.

I know that the keys are somehow mathematically related, and I know that they're somehow generated via prime numbers, but how exactly are they related? I realize that it'd probably be best for me to go read about it inside of the book, if it has it, but this is something that I've been wondering about for a little while and I figure that this is the perfect site to ask it on.

Also the final keysize, is that the size of the prime number? For example, if I set it to 8192bits does that mean that the prime numbers are that long?

Explanation / Answer

There are several kinds of asymmetric cryptographic algorithms. All use some sort of mathematical structure, but not the same, and not all involve prime integers.

RSA is the most well-known asymmetric algorithm, which includes several variants (e.g. for asymmetric encryption or for digital signature). In a RSA public key, there is a big integer called the modulus, which is the product of two (or more) prime integers; the key size is traditionally the size, in bits, of that modulus. Usually, two prime integers of the same size are used, so for a 8192-bit RSA key, the two primes should have size 4096 or 4097 bits each. (A RSA public key contains more information than just the modulus.)

For the ElGamal asymmetric encryption, there is again a modulus, but this time it is a prime integer, not a product of several primes; the traditional "size" of an ElGamal public key is the size of that prime.

For anything with elliptic curves, there are prime integers at some point, but quite some additional mathematics on top of that.

For the whole mathematical details, have a look at the Handbook of Applied Cryptography, which is downloadable for free. This book contains less smooth talk than Schneier's "Applied Cryptography", but it is also more precise and thorough.

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