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 ( )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 mod b
q: =
k: =k+1
end {the base b expansion of n is (ak a1 a0)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 a1 a0)2 and
(bn b1 b0)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 s1 s0)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
procedure multiply(a, b : positive integers)
{the binary expansions of and b are(an a1 a0)2 and
(bn b1 b0)2 respectively}
for j:=0 to n-1
Begin
if bj =1 then cj=a shifted j places
else cj:=0
end
{c0 c cn-1 are the partial products}
p :=0
for j:=0 to n-1
p: = p +cj {p is the value of ab}

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
r := d-r
q := -(q+1)
{q = a div d is the quotient, r = a mod d is the remainder}

12. Modular Exponentiation
In cryptography it is important to be able to find bn mod 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 a1 a0)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=
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 a1 a0)2 ,m: positive integer)
x: = 1
power := b mod m
for i=0 to k-1
begin
if ai =1 then x :=(x*power) mod m
power :=(power*power) mod m
End
{x equals bn mod 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.