Give a modular exponentiation algorithm that examines the bits of b from right to
left instead of left to right.
Why did the computer stand on its head? Because it wanted to do a bottoms-up approach!
Now, let me entertain you with a modular exponentiation algorithm that examines the bits of b from right to left:
1. Start with three variables: base (a), exponent (b), and modular value (m).
2. Initialize a variable named result to 1.
3. While the exponent b is greater than 0:
- Check the least significant bit of b. If it is 1, multiply the result by a (result = (result * a) % m).
- Right-shift the exponent b by 1 (b = b >> 1).
- Square the base a (a = (a * a) % m).
4. The final result would be the desired exponential value (result).
Remember, this algorithm not only computes the modular exponentiation in a different direction but also does it efficiently by considering only the relevant bits of the exponent.
The modular exponentiation algorithm that examines the bits of b from right to left instead of left to right is typically called the "Right-to-Left Binary Method." Here is a step-by-step explanation of the algorithm:
1. Take the base number (a), the exponent (b), and the modulus (m).
2. Initialize a variable called "result" to 1.
3. Start iterating over the bits of the exponent (b), starting from the rightmost bit (least significant bit).
4. For each bit, do the following:
a. Multiply the current value of "result" by itself modulo m. (result = (result * result) % m)
b. If the current bit of the exponent (b) is a 1, multiply the current value of "result" by the base number (a) modulo m. (result = (result * a) % m)
5. After iterating over all the bits of the exponent, the final value of "result" will be the modular exponentiation result.
Here's a Python code snippet that implements the Right-to-Left Binary Method:
```python
def modular_exponentiation(a, b, m):
result = 1
while b > 0:
if b % 2 == 1:
result = (result * a) % m
a = (a * a) % m
b = b // 2
return result
```
Example usage of the algorithm:
```python
base = 5
exponent = 13
modulus = 11
result = modular_exponentiation(base, exponent, modulus)
print(result) # Output: 9
```
In this example, the result of computing 5^13 (mod 11) using the Right-to-Left Binary Method is 9.
To perform modular exponentiation by examining the bits of b from right to left, you can use the Right-to-Left Binary Exponentiation algorithm. This algorithm calculates the exponentiation faster by avoiding unnecessary multiplications.
Here's the step-by-step explanation of the algorithm:
1. Initialize three variables:
- result = 1 (to store the result of exponentiation)
- base = a (the base value)
- exponent = b (the exponent value)
2. Convert the exponent b to its binary representation.
3. Iterate through the binary representation of b from right to left. In each iteration:
- Square the base value: base = (base * base) % modulus
- If the current bit of the exponent b is 1, multiply the result with the current base value: result = (result * base) % modulus
4. After iterating through all the bits of the exponent, the value of result will be the modular exponentiation of a and b.
Here's an example to demonstrate the algorithm:
Let's say we want to calculate (3^13) % 7 using the right-to-left binary exponentiation algorithm.
1. Initialize: result = 1, base = 3, exponent = 13.
2. Convert exponent (13) to binary representation: 13 (base 10) = 1101 (base 2).
3. Iterate through the binary representation from right to left:
- Iteration 1: Current bit = 1, so multiply result by base: result = (1 * 3) % 7 = 3.
- Iteration 2: Current bit = 0, so square the base: base = (3 * 3) % 7 = 2.
- Iteration 3: Current bit = 1, so multiply result by base: result = (3 * 2) % 7 = 6.
- Iteration 4: Current bit = 1, so multiply result by base: result = (6 * 2) % 7 = 5.
4. After iterating through all the bits, the final result is 5. So, (3^13) % 7 = 5.
That's how you can perform modular exponentiation by examining the bits of b from right to left using the right-to-left binary exponentiation algorithm.