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College Algebra Exam Review 71

College Algebra Exam Review 71 - 1.12 AN APPLICATION TO...

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1.12. AN APPLICATION TO CRYPTOGRAPHY 81 Similarly, p divides a h a . But then a h a is divisible by both p and q , and hence by n D pq D l : c : m :.p; q/ . n Let r be any natural number relatively prime to m , and let s be an inverse of r modulo m , rs 1 . mod m/ . We can encrypt and decrypt natural numbers a that are relatively prime to n by first raising them to the r th power modulo n , and then raising the result to the s th power modulo n . Lemma 1.12.2. For all integers a , if b a r . mod n/; then b s a . mod n/: Proof. Write rs D 1 C tm Then b s a rs D a . mod n/ , by Lemma 1.12.1 . n Here is how these observations can be used to encrypt and decrypt information: I pick two very large primes p and q , and I compute the quantities n , m , r , and s . I publish n and r (or I send them to you privately, but I don’t worry very much if some snoop intercepts them). I keep p , q , m , and s secret. Any message that you might like to send me is first encoded as a natural number by a standard procedure; for example, a text message is
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