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 ² 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 ² 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
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This note was uploaded on 05/27/2011 for the course CS 6.042J taught by Professor Prof.albertr.meyer during the Spring '11 term at MIT.

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MIT6_042JS10_lec23 - Euler Mathematics for Computer Science...

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