Chapter 3

1. Chapter 3 • 3.4 The Integers and Division • Division • The Division Algorithm • Modular Arithmetic • Applications of Congruences • Cryptology

2. Division • Definition 1: if a and b are integers with a≠0, we say that a divides b if there is aninteger c such that b=ac. When a divides b we say that a is a factor of b and that b is a multiple of a. the notation a|b denotes that a divides b. we write a | b when a does not divide b. • Example 1: Determine whether 3|7 and whether 3|12. • Example: Determine whether 3|0. /

3. Theorem 1: let a, b, and c be integers. Then • If a|b and a|c, then a|(b+c) • If a|b and a|bc for all integer c • If a|b and b|c, then a|c • Corollary 1: If a, b, c are integers such that a|b and a|c , then a| mb+ nc whenever m and n are integers.

4. The Division Algorithm • Theorem 2 the division algorithm :let a be an integer and d a positive integer. Then there are unique integers q and r, with 0 ≤ r < d, such that a= dq+r • Definition 2: In the equality give in the division algorithm, d is called thedivisor, a is called the dividend, q is called the quotient, and r is called the remainder. This notation is used to express the quotient and remainder. q = adiv d, r = amodd. • Example 4: What are the quotient and remainder when -11 is divided by 3?

5. Modular Arithmetic • Definition 3: if a and b are integers and m is a positive integer, then a is congruent to b modulo mif m divides a - b. • we use the notationa≡b(mod m) to indicate that a is congruent to b modulo m. • if a and b are not congruent modulo m, we write a ≡b (mod m) . /

6. Modular Arithmetic • Theorem 3: let a and b be integers, and let m be a positive integer. Then a≡b (mod m) if and only if amodm = bmodm . • Example 5: determine whether 17 is congruent to 5 modulo 6 and whether 24 and 14 are congruent modulo 6.

7. Modular Arithmetic • Theorem 4 : let m be positive integer. The integers a and b are congruent modulo m if and only if there is an integer k such that a = b + km . • Theorem 5: let m be a positive integer. If a≡b(mod m ) and c ≡d (mod m), then a+c≡b+d (mod m) , ac ≡ bd(mod m) • Example 6: because 7≡2 (mod 5) and 11≡1 (mod 5) , it follows from theorem 5 that 18=7+11 ≡2+1=3(mod 5) , and that 77=7*11 ≡2*1=2 (mod 5)

8. Corollary 2: let m be a positive integer and let a and b be integers. Then (a+b) modm = ((amod m)+(bmod m)) mod m And abmod m =((amodm)(bmodm)) mod m.

9. Applications of Congruences • Hashing Functions • Pseudorandom Numbers • Cryptology

10. Hashing Functions • How can memory locations be assigned so that customer records can be retrieved quickly? • Hashing function and key • h(k) = k mod m; m is the number of available memory locations. • Collision: one way to re solve a collision is to assign the first free location.

11. Pseudorandom Numbers • The numbers generated by systematic method are not truly random, they are called pseudorandom numbers. • Linear Congruential Method(m, a, c, x0 :integers): • Modulus m • Multiplier a, 2  a < m • Increment c, 0  c < m • Seed x0 , 0  x0 < m • xn+1= (axn+c) mod m • For example: m=9, a=7, c=4, x0 =3, then (x1, x2, x3, x4, x5, x6, x7, x8, x9)=(7, 8, 6, 1, 2, 0, 4, 5, 3) x10=x1

12. Cryptology • Important Application of Congruences • Earliest known uses by Julius Caesar. • Shifting each letter three letters forward in the alphabet. • To express the process mathematically: • Let U={0,.., 25}, V={A, .., Z} and g: V -> U is a bijection function defined as the table below. • Define function f : U -> U, where f(p)=(p+3)mod26. • The Encryption function h:V->V, where h(x)=g-1( f(g(x) ) ) • The decryption function f-1(p)=(p-3) mod 26.

13. Applications of Congruences • Example 9: • What is the secret message produced from the message “MEET YOU IN THE PARK” using the Caesar cipher. • HW: Example 10, p208