week06 rsa - d leaves upon division by N ; this is M....

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RSA Coding Receiver chooses 2 distinct large prime p and q , and lets N = pq . Note: N ± N ) = ( p -1 ) ( q - 1) Receiver also chooses an e relatively prime to N ± N ). Receiver announces the pair ( N , e ) to all who wish to send messages. Message must be a natural number, M, that is less than N . Sender sends the remainder, that M e leaves upon division by N . Thus M e R (mod N ). To de-code, we need a Lemma Lemma: If N = pq , p , q relatively prime, and k an int. then a k N ²±1 ³ a a (mod N ) a. Proof: If a relatively prime to N , then a N ± N ) 1 (mod N ) (by Euler's Theorem), so the result follows. Also it is true if N | a (both sides 0) Suppose a = pl with ( b , q ) = 1 a N ± N ) k a – a divisible by p ; must show div by q . But N ²± N ) = ( p – 1 ) ( q – 1), so, a N ± N ) (a q -1 ) k ( p -1) (mod q ) 1 (mod q ) a N ± N ) a a (mod q ) This finishes the proof of the Lemma.
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Now, Find d so that de + k N ± N ) = 1 for some k ( can do, since e , N ²± N ) relatively prime). Then you decode by finding the remainder that R
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Unformatted text preview: d leaves upon division by N ; this is M. Proof: R d ≡ (M e ) d (mod N ) ≡ M ed (mod N ) ≡ M k N ± N ) + 1 (mod N) ≡ M (mod N ) Thus the original message M is recovered. Example of RSA message sending: p = 11, q = 7, N = 77, N ± N ) = 60. Let e = 13, d = 37 (37.13 = 481) 13.37 = 481 – 1 + 8.60 (see how to find 37 later) Let M = 71 M e = (71) 13 71 ≡- 6 mod 77 So M e ≡- 6 13 (mod 77) 6 3 ≡ 216 ≡-15 (mod 77) 6 6 ≡ (-15) 2 ≡ 225 (mod 77 ) ≡ 6 (mod 77) ∴ 6 12 ≡ (-15) 2 ≡ 36 (mod 77 ) 6 13 ≡ 216 ≡-15 ∴ M e ≡- 6 13 (mod 77) ≡ 15 (mod 77) So, send 15 as the message. De-code 15 37 ≡ x (mod 77) 15 2 ≡-6 (mod 77) ∴ 15 26 ≡ (- 6 ) 13 (mod 77) ≡ 15 (mod 77) 15 4 ≡ 36 (mod 77) 15 8 ≡ 36 2 ≡ 1296 15 37 ≡ 15 26 36 8 36 3 ≡ 71 (mod 77) © P. Rosenthal , University of Toronto, Department of Mathematics typed by A. Ku Ong...
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This note was uploaded on 04/26/2011 for the course MATH 246 taught by Professor Applebaugh during the Spring '10 term at University of Toronto.

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week06 rsa - d leaves upon division by N ; this is M....

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