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# Class Note for MATH 455 at UMass(1)

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Date Created: 02/06/15

Math 455 0 April 4 2009 Fermat7s Little Theorem For the RSA encryption system we shall need the following result Theorem 1 Fermat7s Little Theorem Letp be a prime Then for each integer a not divisible by p 01771 E1 mod p Proof Let a be an integer for which pia For eachj12p7 1 let rj ja modp sothat Ogr ltp that is Ogr 1971 We are going to prove that r1 r2 rp1 is a permutation of 1 2 p7 1 For each j 123 p 71 we have ja if 0 mod p why that is 77 7 0 Thus r1r2 rp1 all belong to the set 1 23 p 71 Next if1 j k p71 withj 7 k then 77 7 77 Why Thus T1 72 rp1 is a set ofp 7 1 numbers that is a subset of the set 1 2 3 p 7 1 Hence these two sets are the same 71rgrp1123p71 Since both sets have p 7 1 elements then r1r2rp1 is a permutation of 12p71 In other words each of the p 7 1 numbers a 2a 3a p 7 1a is congruent modulo pto exactly one of thep7 1 numbers 1 23p7 1 Hence a2a3ap71a El23p71 modp In other words 29 DW H 30971 mod p Now each of the factors 1 2 3 p71 of p7 1 is relatively prime to p and so by the Congruence Cancellation Law may be cancelled from both sides of After the cancellations what remains is 01771 E 1 mod p as desired 1 The following corollary is in fact equivalent to Fermat s Little Theorem Corollary 1 Letp be a prime The for every integer a a E a mod p Fermat s Little Theorem may be used to calculate ef ciently modulo a prime powers of an integer not divisible by the prime Example 1 Calculate 2345 mod 11 ef ciently using Fermat s Little Theorem Solution The number 2 is not divisible by the prime 11 so 210 E 1 mod 11 by Fermat s Little Theorem By the division algorithm 34534105 Since 2345 234105 21034 25 then 2345 E 134 25 E 1 32 E 10 mod 11 Thus 2345 mod 11 10 The result actually needed for RSA encryption is the following corollary to Fer mat s Little Theorem Corollary 2 Euler7s Corollary Letp and q be distinct primes Then for each integer a not divisible by eitherp or q awiwq l E 1 mod pq Proof This is an exercise D Both Fermat s Little Theorem and Euler s Corollary are special cases of a more general result To formulate the generalization we need the following de nition De nition 1 Euler7s phi function 1 N a N is de ned by the rule that for each positive integer n gtn k 1 g k lt n and k is relatively prime to n For example lt2gt 1 1 lt3gt m 2 lt4gt 173 2 gt6 15 2 and gt12 15711 4 Then the generalization is as follows Theorem 2 Euler7s Theorem Let in be an integer with m gt 1 Then for each integer a that is relatively prime to m a ltmgt E 1 mod m We will not prove Euler s Theorem here because we do not need it Fermat s Little Theorem is a special case of Euler s Theorem because for a prime p Euler s phi function takes the value gtp p7 1 Note that for a prime p saying that an integer a is relatively prime to p is equivalent to saying that p does not divide a Euler s Corollary is also a special case of Euler s Theorem because for distinct primes p and q Euler s phi function takes the value gtp q p 7 1q 7 1 Copyright 2009 by Murray Eisenherg All rights reserved

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