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Unformatted text preview: COMPUTATIONAL NUMBER THEORY 1 / 70 Notation Z = { ... , 2 , 1 , , 1 , 2 ,... } N = { , 1 , 2 ,... } Z + = { 1 , 2 , 3 ,... } d  a means d divides a Example: 2  4. For a , N Z let gcd( a , N ) be the largest d Z + such that d  a and d  N . Example: gcd(30 , 70) = 2 / 70 Notation Z = { ... , 2 , 1 , , 1 , 2 ,... } N = { , 1 , 2 ,... } Z + = { 1 , 2 , 3 ,... } d  a means d divides a Example: 2  4. For a , N Z let gcd( a , N ) be the largest d Z + such that d  a and d  N . Example: gcd(30 , 70) = 10. 2 / 70 Integers mod N For N Z + , let Z N = { , 1 ,... , N 1 } Z N = { a Z N : gcd( a , N ) = 1 } ( N ) =  Z N  Example: N = 12 Z 12 = { , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 } Z 12 = 3 / 70 Integers mod N For N Z + , let Z N = { , 1 ,... , N 1 } Z N = { a Z N : gcd( a , N ) = 1 } ( N ) =  Z N  Example: N = 12 Z 12 = { , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 } Z 12 = { 1 , 5 , 7 , 11 } (12) = 3 / 70 Integers mod N For N Z + , let Z N = { , 1 ,... , N 1 } Z N = { a Z N : gcd( a , N ) = 1 } ( N ) =  Z N  Example: N = 12 Z 12 = { , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 } Z 12 = { 1 , 5 , 7 , 11 } (12) = 4 3 / 70 Division and mod Fact: For any a , N Z with N > 0 there exist unique q , r N such that a = Nq + r r < N Refer to q as the quotient and r as the remainder. Then a mod N = r Z N is the remainder when a is divided by N . Def: a b (mod N ) iff ( a mod N ) = ( b mod N ). Examples: If a = 17 and N = 3 then the quotient and remainder are q = ? and r = ? 4 / 70 Division and mod Fact: For any a , N Z with N > 0 there exist unique q , r N such that a = Nq + r r < N Refer to q as the quotient and r as the remainder. Then a mod N = r Z N is the remainder when a is divided by N . Def: a b (mod N ) iff ( a mod N ) = ( b mod N ). Examples: If a = 17 and N = 3 then the quotient and remainder are q = 5 and r = 2 4 / 70 Division and mod Fact: For any a , N Z with N > 0 there exist unique q , r N such that a = Nq + r r < N Refer to q as the quotient and r as the remainder. Then a mod N = r Z N is the remainder when a is divided by N . Def: a b (mod N ) iff ( a mod N ) = ( b mod N ). Examples: If a = 17 and N = 3 then the quotient and remainder are q = 5 and r = 2 17 mod 3 = 4 / 70 Division and mod Fact: For any a , N Z with N > 0 there exist unique q , r N such that a = Nq + r r < N Refer to q as the quotient and r as the remainder. Then a mod N = r Z N is the remainder when a is divided by N ....
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 Winter '08
 daniele

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