COT5937Lec11 - Cryptography(COT 5937 Lecture 13 Last time...

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6/21/01 Last time we summarized RSA encryption. One of the main reasons that RSA works is Euler’s Theorem, which was simply stated in the last lecture. This time, we’ll prove it: Euler’s Theorem: If gcd(a,n) = 1, then a φ (n) 1 (mod n). Definition of a reduced residue system modulo n: A set of φ (n) numbers r 1 , r 2 , r 3 , . .. r φ (n) such that r i r j , for all 1 i < j φ (n) with gcd(r i , n) = 1 for all 1 i φ (n). Theorem about reduced residue systems: If r 1 , r 2 , r 3 , . .. r φ (n) is a reduced residue system modulo n, and gcd(a,n) = 1, then ar 1 , ar 2 , ar 3 , . .. ar φ (n) is ALSO a reduced residue system modulo n. Proof: We need to prove two things in order to verify the theorem above: 1) ar i ar j , for all 1 i < j φ (n) 2) gcd(ar i , n) = 1 for all 1 i φ (n) Proof of 1: Assume to the contrary that there exist distinct integers i and j such that ar i ar j (mod n). We can deduce the following: ar i ar j (mod n) (ar i - ar j ) 0(mod n). n | (a(r i – r j )) We know that gcd(a,n) = 1. Thus, based on a theorem proved earlier, it follows that n | (r i – r j ). But, this infers that r i r j (mod n). This contradicts our premise that r 1 , r 2 , r 3 , . .. r φ (n) is a reduced residue system modulo n. Thus, we can conclude that ar i ar j , for all 1 i < j φ (n). Proof of 2: Since gcd(a,n)=1 and gcd(r i ,n)=1, it follows that n shares no common factors with a or r i . Thus, it shares no common factors with their product and we can conclude that gcd(ar
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COT5937Lec11 - Cryptography(COT 5937 Lecture 13 Last time...

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