College Algebra Exam Review 73

College Algebra Exam Review 73 - c : m :.p 1;q 1/ . Use the...

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1.12. AN APPLICATION TO CRYPTOGRAPHY 83 You send me b in an e–mail message. I recover a by raising b to the s th power and reducing modulo n . I then recover the list of character codes by extracting the base 256 digits of a and finally I use an ASCII character code table to restore your message “ALGEBRA IS REALLY INTERESTING”. It goes without saying that one does not do such computations by hand. The Mathematica notebook RSA.nb on my Web site contains programs to automate all the steps of finding large primes, and encoding, encrypting, decrypting, and decoding a text message. Exercises 1.12 1.12.1. (a) Let G Š H ± K be a direct product of finite groups. Show that every element in G has order dividing l : c : m :. j H j ; j K j / . (b) Let n D pq the product of two primes, and let m D l :
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Unformatted text preview: c : m :.p 1;q 1/ . Use the isomorphism .pq/ .p/ .q/ to show that a m 1 . mod n/ whenever a is relatively prime to n . 1.12.2. Show that if a snoop were able to nd '.n/ , then he could also nd p and q . 1.12.3. Suppose that Andre needs to send a message to Bernice in such a way that Bernice will know that the message comes from Andre and not from some impostor. The issue here is not the secrecy of the message but rather its authenticity. How can the RSA method be adapted to solve the problem of message authentication? 1.12.4. How can Andre and Bernice adapt the RSA method so that they can exchange messages that are both secure and authenticated?...
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