This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ....
View
Full
Document
This note was uploaded on 08/31/2011 for the course CSE 207 taught by Professor Daniele during the Winter '08 term at UCSD.
 Winter '08
 daniele

Click to edit the document details