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MIT6_042JS10_lec23

MIT6_042JS10_lec23 - Euler Mathematics for Computer Science...

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1 Albert R Meyer, April 2, 2010 lec 8F.1 Euler’s Theorem RSA encryption Mathematics for Computer Science MIT 6.042J/18.062J Albert R Meyer, April 2, 2010 lec 8F.2 Euler function (n) ::= # k 0,1,…,n-1 s.t. k rel. prime to n has a (mod n) inverse Albert R Meyer, April 2, 2010 lec 8F.3 (n) ::= # k 0,1,…,n-1 s.t. k rel. prime to n (7) = 6 ( 12) = 4 0, 1, 2,3,4 ,5, 6 ,7, 8,9,10 ,11 1,2,3,4,5,6 Euler function Albert R Meyer, April 2, 2010 lec 8F.4 Calculating If p prime, everything from 1 to p-1 is rel. prime to p , so (p) = p – 1 Albert R Meyer, April 2, 2010 lec 8F.5 (9)? so, (9) = 9 - (9/ 3 ) = 6 k rel. prime to 9 iff k rel. prime to 3 3 divides every 3 rd number 0,1,2,3,4,5,6,7,8 0 ,1,2, 3 ,4,5, 6 ,7,8 Euler function Albert R Meyer, April 2, 2010 0 , 1 , ... , p , ... , 2p, ... ..,p k -p , ... , p k -1 lec 8F.6 Calculating ( p k ) p divides every p th number 2p , . . . . ., p p k /p of these numbers are not rel. prime to p k
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2 Albert R Meyer, April 2, 2010 (p k ) = p k p k /p – p k-1 lec 8F.8 Calculating ( p k ) so Albert R Meyer, April 2, 2010 lec 8F.9 Calculating ( a ! b ) Lemma : For a, b relatively prime , (a ! b) = (a) ! (b) pf: Pset 8. Another way in 2 weeks.
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