Let's look at number theory and codes. The thing that makes this new breed of code, technically called public key encryption algorithms, different is that you can give everyone the information necessary to send you coded messages without giving them the information to read them. (Think about it. With an old fashioned code, if I told you enough about the code you would not only be able to send me messages but would also be able to read any messages other people send me.) This is important in internet applications, for instance, because a company like Amazon.com wants to have lots of people send them encrypted messages (with credit card info, for example) but do not want all of those people also to be able to read those messages! That's where the difficulty of factoring huge numbers into their prime factors comes into play! In the RSA algorithm anyone who knows some really large number (an integer with thousands of digits) can encrypt a message just by turning it into a number, raising it to a power, and finding the remainder when you divide by the large number. However, in order to decode the message one needs to know the prime factors of that number. If the number is large enough, it could take years, decades or centuries to find those prime factors. Here's how the RSA encryption algorithm works:

• If I want to receive a coded message (whether I'm a person or a computer, it doesn't matter), I pick three numbers. Two prime numbers p and q and a third number r which has no factors in common with (p-1)(q-1). So, for example, if I want to get a message from you, I can pick p=11 and q=3. Then, I could pick r=3 because 3 and 20=(11-1)(3-1) have no common factors. In reality, these numbers are much too small, but for this example let's ignore that problem.
• Then I must also find the number d less than (p-1)(q-1) with the property that the remainder when you divide rd by (p-1)(q-1) is 1. In the example, d must be 7. Note that 3 X 7=21 which leaves a remainder of 1 when divided by 20.
• Now comes the interesting part. I publicly announce to you and everyone else both the product p X q and the number r. However, I don't tell anyone what p or q or d are! So, I would announce to you that the product of p and q is 33 and that the value of r is 3.
• Now comes your part. You want to send me a message and encode it using the knowledge of those two numbers. In fact, your message has to be a number too. However, that's not too hard. Everything in a computer is numbers in the end. However, as it has to be a number less than p X q, we are sort of restricted here. But, as a simple example, let's suppose the message you want to send me is the number ``18''. But first, you encode the message by raising it to the power of r dividing by the product p X q. In fact, the coded message is just the remainder in that division! Even though you want to tell me `18'', what you'll send me is the remainder when you divide 18³ by 33. That is, you'll send the message 24.
• Don't forget that anyone who was listening to our previous conversation knows the two numbers I sent you, and they know the number you sent me as the coded message. But, they cannot decode it unless they know the numbers that I've kept secret! (The factors of 33 and especially the number d...that's the secret, right?)
• Now, to turn the encoded message back to its original form, I need to raise the message you sent me to the power d and get the remainder when that is divided by p X q. In the example, this means I will compute 24⁷ and divide it by 33...or rather just find the remainder. And, guess what, this gives me back the message 18!
• Of course, there is something silly in the above example. The math is all correct, whatever message you start with I will be able to decode it back to your original just because of the way the number theory works. However, if I tell everyone my number 33 then there is no real secret what its factors are! 33=11 X 3 is the only way to factor it! The thing is, when this gets used for real, the numbers involved are super huge. They use numbers that are thousands of digits long, and the fact is that at the moment we have no way to factor those numbers in a short time. In twenty to one hundred years someone could factor it and break the code...but by then the information in the message will probably be obsolete.