Prime numbers are those divisible only by themselves and 1. While small primes like 5 or 17 are easy to spot, for very large numbers, those hundreds of digits long, there never had been a formula of "primality testing" that didn't have a slight chance of error.
That encryption system takes two big, and secret, prime numbers and multiplies them. For a bad guy to decrypt your message, he'd need to take the product of that multiplication and figure out the two prime numbers used to generate it. It's called the "factoring problem," and fortunately it's something no one on Earth knows how to do quickly. A speedy method of factoring would make existing Internet security useless, not a pleasant thought in this Internet age.
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