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The calculations continue, and the end result is the original plaintext message THIS HEART OR THAT HEADX
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Cryptography
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As you can see, an X is af xed to the end to force the message to have an even number of characters (counting blank spaces) so that the digraph method will work One important point that the last example illustrates is that cryptography will always entail a certain amount of (organized) guesswork
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1071 BASICS AND BACKGROUND
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Modern security considerations make it desirable for us to have new types of encryption schemes It is no longer enough to render a message so that only the intended recipient can read it (and outsiders cannot) In today s complex world, and with the advent of high-speed digital computers, there are new demands on the technology of cryptography The present section will discuss some of these considerations In the old days (beginning even with Julius Caesar), it was enough to have a method for disguising the message that we were sending For example, imagine that the alphabet is turned into numeric symbols by way of the scheme A 0 B 1 C 2 and so forth Then use an encryption like n n + 3 mod 26 (103)
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And now convert these numbers back to roman letters We have discussed such encryption schemes in the preceding sections Today life is more complex One can imagine that there would be scenarios in which 1 You wish to have a means that a minimum-wage security guard (whom you don t necessarily trust) can check that people entering a facility know a password but you don t want him to know the password
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2 You wish to have a technology that allows anyone to encrypt a message using a standard, published methodology but only someone with special additional information can decrypt it 3 You wish to have a method to be able to convince someone else that you can perform a procedure, or solve a problem, or prove a theorem, without actually revealing the details of the process This may all sound rather dreamy, but in fact thanks to the efforts and ideas of R Rivest, A Shamir, and L Adleman it is now possible The so-called RSA encryption scheme is now widely used For example, the e-mail messages that I receive on my cell phone are encrypted using RSA Banks, secure industrial sites, high-tech government agences (for example, the National Security Agency), and many other parts of our society routinely use RSA to send messages securely In this discussion, we shall describe how RSA encryption works, and we shall encrypt a message using the methodology We shall describe all the mathematics behind RSA encryption, and shall proved the results necessary to esh out the theory behind RSA We shall also describe how to convince someone that you can prove the Riemann hypothesis without revealing any details of the proof This is a fascinating idea something like convincing your mother that you have cleaned your room without letting her have a look at the room But in fact the idea has profound and far-reaching applications
1072 PREPARATION FOR RSA
Background Ideas
We now sketch the background ideas for RSA These are all elementary ideas from basic mathematics It is remarkable that these are all that are needed to make this profound new idea work
Computational Complexity
Suppose that you have a deck of N playing cards and you toss them in the air Now you want to put them back into their standard order How many steps will this take (We want to answer this question in such a manner that a machine could follow the instructions) First we look through all N cards and nd the rst card in the ordering Then we look through the remaining N 1 cards and nd the second card in the ordering And so forth
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