1. Raise a number x to the power y

2. Raise a number x to the power y
Example: 645

3. Raise a number x to the power y
Example: 645
6*6*6*6*6*...* Yikes! 45 multiplies

4. Raise a number x to the power y
Example: 645
6*6*6*6*6*...* Yikes! 45 multiplies
6 = n[0] (61)
n[0]*n[0] = n[1] (62)
n[1]*n[1] = n[2] (64)
n[2]*n[2] = n[3] (68)
n[3]*n[3] = n[4] (616)
n[4]*n[4] = n[5] (632)
So, 645 = n[5]*n[3]*n[2]*n[0]
Only 8 multiplies!

5. Raise a number x to the power y
Example: 645
Notice: = (in binary)
6 = n[0] (61)
n[0]*n[0] = n[1] (62)
n[1]*n[1] = n[2] (64)
n[2]*n[2] = n[3] (68)
n[3]*n[3] = n[4] (616)
n[4]*n[4] = n[5] (632)
So, 645 = n[5]*n[3]*n[2]*n[0]

6. Raise a number x to the power y
Example: 645
Notice: = (in binary)
Also notice: x1000 is the square of
x100
and
x is the square of
x101010

7. Raise a number x to the power y
Example: 645
Notice: = (in binary)
Algorithm:
Find binary equivalent of y.
Define an accumulator variable p and initialize it to x.
Starting with the 2nd most significant bit of y,
repeat the following until all bits of y are considered:
Square p. If the bit is 1, multiply p by x.
Consider the next lesser significant bit of y.

8. Raise a number x to the power y
Example: 645

9. Raise a number x to the power y
Example: 645
So: x = 6, y= 45 = (in binary)

10. Raise a number x to the power y
Example: 645
So: x = 6, y= 45 = (in binary)
Set p = 6

11. Raise a number x to the power y
Example: 645
So: x = 6, y= 45 = (in binary) *
Set p = 6
2nd MSB of y is 0 so square p (p = 36)

12. Raise a number x to the power y
Example: 645
So: x = 6, y= 45 = (in binary) *
Set p = 6
2nd MSB of y is 0 so square p (p = 36)
Next MSB is 1: square and mult (p = 7776)

13. Raise a number x to the power y
Example: 645
So: x = 6, y= 45 = (in binary) *
Set p = 6
2nd MSB of y is 0 so square p (p = 36)
Next MSB is 1: square and mult (p = 7776)
Next MSB is 1: square and mult (p = )

14. Raise a number x to the power y
Example: 645
So: x = 6, y= 45 = (in binary) *
Set p = 6
2nd MSB of y is 0 so square p (p = 36)
Next MSB is 1: square and mult (p = 7776)
Next MSB is 1: square and mult (p = )
Next MSB is 0: square (p = )

15. Raise a number x to the power y
Example: 645
So: x = 6, y= 45 = (in binary) *
Set p = 6
2nd MSB of y is 0 so square p (p = 36)
Next MSB is 1: square and mult (p = 7776)
Next MSB is 1: square and mult (p = )
Next MSB is 0: square (p = )
Last MSB is 1: s&m (p = )

16. Raise a number x to the power y
Example: 645
How to find the last MSB?

17. Raise a number x to the power y
Example: 645
How to find the last MSB? *

18. Raise a number x to the power y
Example: 645
How to find the last MSB? *
(45 % 2)

19. Raise a number x to the power y
Example: 645
How to find the last MSB? *
(45 % 2)
How to strip off the last MSB?

20. Raise a number x to the power y
Example: 645
How to find the last MSB? *
(45 % 2)
How to strip off the last MSB?
(45 / 2)

21. Raise a number x to the power y
Example: 645
How to find the last MSB?
(45 % 2)
How to strip off the last MSB?
(45 / 2)
How to store all the bits?
Work from LSB to MSB and put the bits
on a stack.