Why are huge prime numbers important in cryptography? I read an article the other day about the search for prime numbers.  According to the article and several online sources the biggest prime number is over 17 million digits!  This made me wonder why anyone would want a prime number that big.  The article went on to say that there is a $250k reward for who ever find the first billion+ digit prime.  The article said that primes are used in cryptography and explained briefly why.  But my question is why is there such a demand for these "super primes" which seem extremely bulky and unnecessarily big(a text file holding a billion digits would be over 1gb of data).

EDIT: I know that similar questions have been asked about primes place in cryptology but I'm wondering specifically about these massive primes that people are paying big bucks for.
Preposterously large primes are not useful for cryptography in and of themselves, but the tools and techniques developed to find them (such as massively parallel distributed computing, algorithms that can efficiently confirm primality, etc) are important for cryptography.  The prizes are meant to spur innovation in those areas.  See e.g. https://www.eff.org/awards/coop