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What does this information mean Of course we know, thanks to Euclid, that there are in nitely many primes So we can nd prime numbers with as many digits as we wish Each number n in the RSA encryption book is the product of two 75digit primes p and q: Thus n = p q Each number e is chosen to be a number with at least 100 digits that is relatively prime to (n) = ( p 1) (q 1) Of course we do not publish the prime factorization of the number n; we also do not publish (n) All that we publish is n and e for each individual Now an important point to understand is that Bush does not need to understand any mathematics or any of the theory of RSA encryption in order to encode his message (Well, it would be nice if he understood modular arithmetic But he is, after all, the President of the United States) All he does is this: 1 First he breaks the message into units of ve letters We call these words , even though they may not be English language words For the message from Bush to Rumsefeld, the words would be YOURT IMEIS UPHAS TALAV ISTAB ABY 2 He transliterates each word into a sequence of numerical digits, using our usual scheme of translation 3 Then he encodes each transliterated word w with the rule w w e mod n

The Pages tab not only gives you a visual representation of each page in your document, but can also be used to edit the document by dragging-and-dropping thumbnails You can use the Pages tab to change the order in which pages display and to import pages from other documents

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Bush will send to Rumsfeld this sequence of encrypted words That is all there is to it The real question now is: What does it take to decrypt the encoded message How can Rumsfeld read the message This is where some mathematics comes into the picture We must use Fermat s theorem, and we must use our ideas about relatively prime integers But the short answer to the question is this If w is a word encrypted according to the simple scheme described above, then we decrypt it with this algorithm: We nd integers x and y so that xe + y (n) = 1 and then we calculate w x mod n That will give the decrypted word w with which we began [We shall provide the mathematical details of this assertion in the next section] Since w has only ve characters, and n has 150 characters, we know that w mod n = w so there is no ambiguity arising from modular arithmetic We can translate w back into roman characters, and we recover our message Now here is the most important point in our development thus far: In order to encrypt a message, we need only look up n and e in the public record RSA encryption book But, in order to decrypt the message, we must know x Calculating x necessitates knowing (n), and that necessitates knowing the prime factorization of n It is a theorem that calculating the prime factorization of an integer with k digits is a problem of exponential complexity in k For an integer with 150 digits, using a reasonably fast computer, it would take several years to nd the prime factorization2

You can use thumbnails to change the order in which pages appear in a PDF document To move a single page or a selection of pages to a new position in the document, select them and then dragand-drop them to a new location To change the order of document pages, follow these steps: 1 Open the Pages tab and select the thumbnail that corresponds to the page you want to move You can move more than one page at a time by selecting contiguous or noncontiguous thumbnails, as discussed previously To view several thumbnails at once, choose Reduce Page Thumbnails from the Pages tab Options menu or the Pages tab Context menu You can also display more thumbnails by clicking the vertical border between the Navigation pane and the Document pane and dragging it to the right

In fact, with all the preliminary setup that we have in place, it is a simple matter to explain the RSA encryption system

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