Real-world applications of prime numbers? I am going through the problems from Project Euler and I notice a strong insistence on Primes and efficient algorithms to compute large primes efficiently. 
The problems are interesting per se, but I am still wondering what the real-world applications of primes would be. 
What real tasks require the use of prime numbers?

Edit: A bit more context to the question:
I am trying to improve myself as a programmer, and having learned a few good algorithms for calculating primes, I am trying to figure out where I could apply them. 
The explanations concerning cryptography are great, but is there nothing else that primes can be used for? 

You can use prime numbers to plot this fine pattern :)

Intensity of green colour for each pixel was calculated using a function, which can be described with this pseudocode snippet:
g_intensity = ((((y << 32) | x))^((x << 32) | y))) * 15731 + 1376312589) % 256

where x and y are a pixel coordinates in screen space, stored in a 64bit integer variables.
