s-cnt

# s-cnt - COMPUTATIONAL NUMBER THEORY 1 70 Notation Z = − 2...

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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 } • ϕ ( 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 } • ϕ ( 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 } • ϕ (12) = 3 / 70 Integers mod N For N ∈ Z + , let • Z N = { , 1 ,... , N − 1 } • Z ∗ N = { a ∈ Z N : gcd( a , N ) = 1 } • ϕ ( 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 } • ϕ (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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## This note was uploaded on 08/31/2011 for the course CSE 207 taught by Professor Daniele during the Winter '08 term at UCSD.

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s-cnt - COMPUTATIONAL NUMBER THEORY 1 70 Notation Z = − 2...

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