Number Theory Benchaporn Jantarakongkul 1 The Integers and

Number Theory Benchaporn Jantarakongkul 1

The Integers and Division • ��������������� �������������� hash functions, cryptography, digital signatures 2

Divisors. Examples ����������� ? 1. 77 | 7 2. 7 | 77 3. 24 | 24 4. 0 | 24 5. 24 | 0 4

Facts re: the Divides Relation • Theorem: a, b, c Z: 1. a≠ 0 a|0 ��� a|a 2. (a|b a|c) a | (b + c) 3. a|bc 4. (a|b b|c) a|c 5. [a|(b+c) a|b)] a|c ���� : w w 17|0 17|34 17|170 17|204 17|340 6|12 12|144 6 | 144 6

Division ����� d the divisor a the dividend q the quotient r the remainder 24 + 3· 31 = 117 a = dq + r 18

Greatest Common Divisor Relatively Prime Q: 1. 2. 3. 4. ���� �. ���� gcd ���� : gcd(11, 77) gcd(33, 77) gcd(24, 36) gcd(24, 25) 24

Greatest Common Divisor Relatively Prime A: 1. 2. 3. 4. gcd(11, 77) = 11 gcd(33, 77) = 11 gcd(24, 36) = 12 gcd(24, 25) = 1 ������� 24 ��� 25 ����������� (relatively prime( ����� : ��������� relatively prime ����������� 25

Pairwise relatively prime • �������� {a 1, a 2, …} ����������� (pai rwise relatively prime) ����� (ai, aj), ����� i j, ���� relatively prime Q: ���������� pairwise relatively prime ���������� { 17 , 169 , 15 , 21 , 28 , 44 } 26

Pairwise relatively prime A: ������������ pairwise relatively prime ��� {44, 28, 21, 15, 169, 17} : ������� {17, 169, 28, 15} ���� {17, 169, 44, 15} ���� : 15, 17, ��� 27 ���� pairwise relatively prime ������� ? • ������ , ����� gcd(15, 27) = 3 15, 17, ��� 28 ���� pairwise relatively prime ������� ? • ��� , ����� gcd(15, 17) = 1, gcd(15, 28) = 1 ��� gcd(17, 28) =1 27

������ (Least Common Multiple( Q: 1. 2. 3. 4. 5. ���� �. ���� lcm ���� : lcm(10, 100) lcm(7, 5) lcm(9, 21) lcm(3, 7) lcm(4, 6) 29

������ (Least Common Multiple( A: 1. 2. 3. 4. 5. lcm(10, 100) = 100 lcm(7, 5) = 35 lcm(9, 21) = 63 lcm(3, 7) = 21 lcm(4, 6) =12 ���� : a = 60 =22 31 51 b = 54 = 21 33 50 lcm(a, b) = lcm(60, 54) = 22 33 51 = 4 27 5 = 540 30

GCD ��� LCM a = 60 = 22 31 51 b = 54 = 21 33 50 gcd(a, b) = 21 3 1 5 0 =6 lcm(a, b) = 22 3 3 5 1 = 540 Theorem: a b = gcd(a, b) lcm(a, b) 31

mod function Q: ������� 1. 113 mod 24 2. -29 mod 7 33

mod function A: ������� 1. 113 mod 24: 2. -29 mod 7 34

mod function A: ������� 1. 113 mod 24: 2. -29 mod 7 35

mod function A: ������� 1. 113 mod 24: 2. -29 mod 7 36

mod function A: ������� 1. 113 mod 24: 2. -29 mod 7 37

Modular Congruence Q: 1. 2. 3. 4. ���������� ? 3 3 (mod 17) 3 -3 (mod 17) 172 177 (mod 5) -13 13 (mod 26) 39

Spiral Visualization of mod �������≡ 0 (mod 5) modulo-5 20 15 10 ≡ 4 19 14 9 (mod 5) 4 8 13 18 ≡ 3 (mod 5) ≡ 1 (mod 5) 5 0 3 2 1 6 11 16 21 7 12 17 22 ≡ 2 (mod 5) 41

Modular arithmetic harder examples Q: ���������� 1. 3071001 mod 102 2. (-45 · 77) mod 17 3. 45

Modular arithmetic harder examples 1. �� 3071001 mod 102 ������� ��� a b (mod m) ������� an bn (mod m) : ��� 307 1 (mod 102) ������� 307 n 1 n (mod 102) : 3071001 mod 102 3071001 (mod 102) ������� , 3071001 mod 102 = 1 46

Modular arithmetic harder examples 2. �� (-45 · 77) mod 17 ������� ��� a b (mod m) ��� c d (mod m), ������� ac bd (mod m) ��� -45 6 (mod 17) ��� 77 9 (mod 17), ������� -45· 77 6· 9 (mod 17): (-45· 77) mod 17 (-45· 77) (mod 17) (6· 9) (mod 17) 54 (mod 17) 3 (mod 17) ������� (-45· 77) mod 17 = 3 47

Hashing Functions Example • �������� m=111 ���������� �� 64212848 ��� 37149212 ���������� 14 ��� 65 ����� 5/20/2021 h(64212848) = 64212848 mod 111 = 14 h(37149212) = 37149212 mod 111 = 65 ��������� 24666707 ����������� 65 ����� 50

Letter Number Conversion Table A B C D E 1 2 3 4 5 F 6 G H 7 8 I J K L M 9 10 11 12 13 N O P Q R S T U V W X Y Z 14 15 16 17 18 19 20 21 22 23 24 25 26 5/20/2021 54

Caesar’s Cipher ����������� f (a) = (a+3) mod 26 ������ “YESTERDAY” 1. YESTERDAY 2. 25 1 4 18 5 20 19 5 25 3. 2 4 7 21 8 23 22 8 2 4. “BHVWHUGDB” 5/20/2021 55

Caesar’s Cipher • ����� “WHQ” • A : 23 8 17 f -1(a) = (a-3) mod 26 20 5 14 “TEN” 5/20/2021 56

Euclid’s Algorithm Example • gcd(372, 164) = gcd(372 mod 164, 164). – 372 mod 164 = 372 164 372/164 = 372 164· 2 = 372 328 = 44 • gcd(164, 44) = gcd(164 mod 44, 44). – 164 mod 44 = 164 44 164/44 = 164 44· 3 = 164 132 = 32 • gcd(44, 32) = gcd(44 mod 32, 32) = gcd(12, 32) = gcd(32 mod 12, 12) = gcd(8, 12) = gcd(12 mod 8, 8) = gcd(4, 8) = gcd(8 mod 4, 4) = gcd(0, 4) = 4 58

Euclid’s Algorithm Pseudocode procedure gcd(a, b: positive integers) while b 0 begin r ≔ a mod b; a ≔ b; b ≔ r; end return a 59

Euclidean Algorithm. Example gcd(33, 77): Step r = a mod b a b 0 - 33 77 60

Euclidean Algorithm. Example gcd(33, 77): Step r = a mod b a b 0 - 33 77 1 33 mod 77 = 33 77 33 61

Euclidean Algorithm. Example gcd(33, 77): Step r = a mod b a b 0 - 33 77 77 33 33 11 1 2 33 mod 77 = 33 77 mod 33 = 11 62

Euclidean Algorithm. Example gcd(33, 77): Step r = a mod b a b 0 - 33 77 77 33 33 11 11 0 1 2 3 33 mod 77 = 33 77 mod 33 = 11 33 mod 11 =0 63

Euclidean Algorithm. Example gcd(244, 117): Step r = a mod b a b 0 - 244 117 64

Euclidean Algorithm. Example gcd(244, 117): Step r = a mod b a b 0 - 244 117 1 244 mod 117 = 10 117 10 65

Euclidean Algorithm. Example gcd(244, 117): Step r = a mod b a b 0 - 244 117 1 244 mod 117 = 10 117 10 2 117 mod 10 = 7 10 7 66

Euclidean Algorithm. Example gcd(244, 117): Step r = a mod b a b 0 - 244 117 1 244 mod 117 = 10 117 10 2 3 117 mod 10 = 7 10 mod 7 = 3 10 7 7 3 67

Euclidean Algorithm. Example gcd(244, 117): Step r = a mod b a b 0 - 244 117 1 244 mod 117 = 10 117 10 2 3 4 117 mod 10 = 7 10 mod 7 = 3 7 mod 3 = 1 10 7 3 1 68

Euclidean Algorithm. Example gcd(244, 117): Step r = a mod b a b 0 - 244 117 1 244 mod 117 = 10 117 10 2 3 4 117 mod 10 = 7 10 mod 7 = 3 7 mod 3 = 1 10 7 3 1 5 3 mod 1=0 1 0 ���� 244 ��� 117 ���� relatively prime ����� gcd(244, 117)=1 69

Exercise on Board Q: ���� gcd(414, 662) ������ Euclidean algorithm A: 414 = 662 1 + 248 414 = 248 1 + 166 248 = 166 1 + 82 166 = 82 2 + 2 82 = 2 41 ������� gcd(414, 662) = 2 70