Chapter 3

1. Chapter 3 • 3.6 Integers and Algorithms • Representations of integers • Algorithms for integer operations • Modular Exponentiation • The Euclidean Algorithm

2. Representations of integers • Theorem 1: Let b be a positive integer greater than 1. Then if n is a positive integer, it can be expressed uniquely in the form where k is a nonnegative integer, a0, a1, …,ak are nonnegative integers less than b, and ak ≠0.

3. Example 1: What is the decimal expansion of the integer that has (1 0101 1111)2 as its binary expansion? • Example 2: What is the decimal expansion of the hexadecimal expansion of (2AE0B)16 ?

4. Example 3: Find the base 8, or octal, expansion of (12345)10 • Example 4: Find the hexadecimal expansion of (177130)10?

5. Algorithm 1: Construction Base b Expansions procedure base b expansion(n:positive integer) q: = n k: =0 while q ≠ 0 begin ak : =q modb q: = k: =k+1 end {the base b expansion of n is (ak-1. . . a1a0)b}

7. Algorithms for integer operations • Algorithm 2: Addition of Integers Procedure add(a , b:positive integers) {the binary expansions of a and b are (an-1. . . a1a0)2and (bn-1. . . b1b0)2 respectively} c : =0 for j: =0 to n-1 Begin d : = sj: = aj+bj+c-2d c : =d end sn:=c {the binary expansion of the sum if (sn sn-1. . . s1s0)2 }

8. Algorithms for integer operations Example 7: Add a=(1110)2 and b=(1011)2.

9. Algorithms for integer operations • Algorithm 3 : Multiplying Integers proceduremultiply(a, b : positive integers) {the binary expansions of and b are(an-1. . . a1a0)2 and (bn-1. . . b1b0)2 respectively} for j:=0 to n-1 Begin ifbj =1 thencj=a shifted j places else cj:=0 end {c0c1. . . cn-1are the partial products} p :=0 for j:=0 to n-1 p: = p +cj {p is the value ofab}

10. Algorithms for integer operations Example 9: Find the product of a= (110)2 and b=(101)2

11. Algorithms for integer operations • Algorithm 4 : Computing div and mod procedure division algorithm(a :integers ,d: positive integer) q: =0 r: =|a| while r≧d begin r := r-d q :=q+1 end if a<0 then if r=0 then q:=-q else begin r := d-r q := -(q+1) end {q = a divd is the quotient, r = amodd is the remainder}

12. Modular Exponentiation • In cryptography it is important to be able to find bnmod m efficiently, where b, n and m and large integers. It’s impractical to first compute bn and then find its remainder when divided by m because bn will be a huge number. Instead, we can use an algorithm that employ expansion of the exponent n , say n = (ak-1. . . a1a0)2. • Before we present this algorithm, we illustrate its basic idea. We will explain how to use the binary expansion of n to compute bn .First , note that

13. Modular Exponentiation • To compute bn , we find the values of b, b2,(b2)2=b4, (b4)2=b8, . . . , . • We multiply the terms in this list, where aj=1 . This gives us . • For example, to compute 311 we first note that 11 = (1011)2, so that 311= 383231. • By successively squaring, we find that 32=9, 34=81, 38=6561. • Consequently,311=383231=6561*9*3= 177,147

14. Modular Exponentiation • Algorithm 5: Modular Exponentiation procedure modular exponentiation(b:integer , n=(ak-1. . . a1a0)2 ,m: positive integer) x: = 1 power := b mod m for i=0 to k-1 begin ifai =1 then x :=(x*power) modm power :=(power*power) modm End {x equals bnmod m} Example 11: Use Algorithm 5 to find 3644 mod 645.

15. The Euclidean Algorithm • Lemma 1: Let a=bq+r ,where a, b, q, and r are integers. Then gcd(a,b)=gcd(b,r). • Algorithm 6: The Euclidean Algorithm procedure gcd(a.b:integers) x: = a y: = b while y0 begin r := x mod y x := y y := r end {gcd(a,b) is x}

16. The Euclidean Algorithm • Example 12: Find the GCD of 414 and 662 using the Euclidean Algorithm.