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"Public key crypto algorithms utilize prime numbers extensively; indeed, prime numbers are an essential part of the major public key systems" http://www.sans.org/reading_room/whitepapers/vpns/prime-numb...

but they cannot use "biggest" primes because, typically, security relies on the prime being unknown.

Yes. But they employ techniques that were at the forefront of the search for big prime numbers in earlier times.

Is that how cracking begins? They start with "famously" prime numbers? I've always wondered about this.