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lec_sept_26 - CS 2800 Discrete Math Sept 26 2011 Lecture...

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CS 2800: Discrete Math Sept. 26, 2011 Lecture Lecturer: John Hopcroft Scribe: June Andrews Review See previous lecture notes for the Fundamental Theorem of Arithmetic, Euler’s φ function, and Euler’s Theorem. The cool part of Euler’s Theorem was that we got 2 very useful corollaries from it: Fermat’s Little Theorem and a test of primality. And a small note, the exam will only cover today’s lecture and previous lectures. Module Tricks We present yet another corollary from Euler’s Theorem (useful theorem!). Corollary 1. If gcd ( a, n ) = 1 , then a x a x mod φ ( n ) mod n . This may seem a bit of a surprise. How did we get the mod φ ( n ) in the exponent? Let us remember our mod tricks and look at the proof. Proof. We need two tools to prove the corollary: 1. Euler’s theorem: if gcd ( a, n ) = 1, then a φ ( n ) 1 mod n 2. The module of a product is the product of the modules: z 1 y 1 mod n z 2 y 2 mod n z 1 z 2 y 1 y 2 mod n. So if x = ( x mod φ ( n )) + ( n ), then a x a x mod φ ( n ) a ( n ) mod n a x mod φ ( n ) mod n a ( n ) mod n mod n by product of modules a x mod φ ( n ) mod n (1 mod n ) mod n by Euler’s Theorem a x mod φ ( n ) mod n Done! Now how does this help us? Example 2. Let’s say we wanted to calculate 2 999 mod 21 . It would be nice if we could avoid calculating 2 999 . To use the previous corollary, we need to check a = 2 and n = 21 are relatively prime, gcd ( a, n ) = gcd (2 , 21) = 1 . So we can! 1
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We now calculate φ (21) . Instead of linearly determining(see definitions) φ (21) , remember that 21 = 3 * 7 and that φ ( pq ) = ( p - 1)( q - 1) . Hence, φ (21) = 12 . Putting it altogether: 2 999 2 999 mod 12 mod 21 2 3 mod 21 8 mod 21 That’s how the corollary helps - it allows us to reduce the exponent dramatically. Even using that trick, it may not reduce the exponent to a particularly small number. We may be faced with an irreducible mod calculation of 2 100 . We still don’t want to have to multiply 2(2(2(2( . . . )))). But we can re-examine the order with which we multiply, like 2 8 = 2 4 * 2 4 . So if we already had 2 4 , we could just square it to get 2 8 . In an inductive manner of thought, the same logic can be applied to 2 4 = 2 2 * 2 2 .
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