6.6 - 156 ≡ log 2 2 2 +log 2 7 (mod 226) Î 156 ≡ 2log...

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6.6 (a) 2 32 mod 227 = ((2 8 ) 2 ) 2 mod 227 = ((29) 2 ) 2 mod 227 = (160) 2 mod 227 = 176 = 2 4 × 11 2 40 mod 227 = (2 32 × 2 8 ) mod 227 = (176 × 256) mod 227 = (176 × 29) mod 227 =110 = 2 × 5 × 11 2 59 mod 227 = (2 40 × (2 8 ) 2 × 2 3 ) mod 227 = (110 × 160 × 8) mod 227 = 60 = 2 2 × 3 × 5 2 156 mod 227 = ((2 40 ) 3 × (2 32 ) × 2 4 ) mod 227 = ((110) 3 × 176 × 16) mod 227 = ( (110) 2 × 110 × 176 × 16) mod 227 = (69 × 110 × 176 × 16) mod 227 = (99 × 176 × 16) mod 227 =(172 × 16)mod 227 =28 =2 2 × 7 (b) log 2 2=1 x=32, 2 32 mod 227 = 2 4 × 11 Î log 2 2 32 log 2 2 4 +log 2 11(mod 226) Î 32 4 log 2 2+log 2 11(mod 226) Î log 2 11 32 4=28(mod 226) x=40, 2 40 mod 227 = 2 × 5 × 11 Î log 2 2 40 log 2 2 +log 2 5+log 2 11(mod 226) Î 40 log 2 2+log 2 5+log 2 11(mod 226) Î log 2 5 40 1 28=11(mod 226) x=59, 2 59 mod 227 = 2 2 × 3 × 5 Î log 2 2 59 log 2 2 2 +log 2 3+log 2 5(mod 226) Î 59 2log 2 2+log 2 3+log 2 5(mod 226) Î log 2 3 59 2 11=46 (mod 226) x=156, 2 156 mod 227 = 2 2 × 7 Î log 2 2
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Unformatted text preview: 156 ≡ log 2 2 2 +log 2 7 (mod 226) Î 156 ≡ 2log 2 2+log 2 7(mod 226) Î log 2 7 ≡ 156 − 2=154(mod 226) (c) (2 177 ) mod 227 = ((2 8 ) 22 × 2) mod 227 = ((29) 22 × 2)mod 227 = (((29) 2 ) 11 × 2)mod 227 = ((160) 11 × 2)mod 227 =(((160) 2 ) 5 × 160 × 2)mod 227 =((176) 5 × 160 × 2) mod 227 =(((176) 2 ) 2 × 176 × 160 × 2) mod 227 =((104) 2 × 176 × 160 × 2) mod 227 =(147 × 176 × 160 × 2) mod 227 =(221 × 160 × 2) mod 227 =123 (173 × 2 177 ) mod 227 = (173 × 123) mod 227 =168 =2 3 × 3 × 7 Î (log 173 + log 2 177 ) mod 226 = (log 2 3 + log 3+ log7) mod 226 Î log 173 mod 226 = (3 log 2 + log 3+ log7 − log 2 177 ) mod 226 = (3+46+154 − 177) mod 226 = 2 6...
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This note was uploaded on 12/25/2010 for the course ALL 0204 taught by Professor 79979 during the Spring '10 term at National Chiao Tung University.

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6.6 - 156 ≡ log 2 2 2 +log 2 7 (mod 226) Î 156 ≡ 2log...

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