How can modular arithmetic help you perform fast exponentiation?

1 What is modular arithmetic?

Modular arithmetic is a system of arithmetic where numbers wrap around after reaching a certain value, called the modulus. For example, if the modulus is 7, then 8 is equivalent to 1, 9 is equivalent to 2, and so on. You can think of modular arithmetic as working with a clock, where the modulus is the number of hours on the clock. Modular arithmetic allows you to perform addition, subtraction, multiplication, and division with the same rules as normal arithmetic, but with a different way of handling remainders.

2 How does modular arithmetic relate to exponentiation?

One of the properties of modular arithmetic is that if you perform the same operation on two numbers that are equivalent modulo a certain modulus, the result will also be equivalent modulo that modulus. For example, if the modulus is 7, then 2 + 5 is equivalent to 0, and 2 + 12 is equivalent to 0 as well. This means that you can reduce large numbers to smaller ones by taking their remainder modulo the modulus, and still get the same result. This is useful for exponentiation, because you can reduce the base and the exponent to smaller numbers before raising them to a power, and then take the remainder of the final result.

3 What is the algorithm for fast exponentiation?

The algorithm for fast exponentiation is based on the idea of binary exponentiation, which is a method of breaking down the exponent into a series of powers of two. For example, to compute 2^13, you can write 13 as 8 + 4 + 1, which are powers of two. Then, you can use the following formula:

2^13 = 2^8 * 2^4 * 2^1

To compute each power of two, you can use the property of exponentiation that says that a^b * a^c = a^(b + c). This means that you can repeatedly square the base and multiply it by itself to get higher powers of two. For example, to compute 2^8, you can do:

2^8 = (2^4)^2 = ((2^2)^2)^2 = (((2^1)^2)^2)^2

This reduces the number of multiplications from 13 to 4. To make the algorithm even faster, you can use modular arithmetic to reduce the base and the result after each multiplication. For example, if the modulus is 7, you can do:

2^8 mod 7 = (2^4 mod 7)^2 mod 7 = ((2^2 mod 7)^2 mod 7)^2 mod 7 = (((2^1 mod 7)^2 mod 7)^2 mod 7)^2 mod 7

= (4^2 mod 7)^2 mod 7 = (2^2 mod 7)^2 mod 7 = (4^2 mod 7) mod 7 = 2 mod 7

Using this algorithm, you can compute any exponentiation modulo a modulus in a logarithmic number of steps, which is much faster than the naive approach of multiplying the base by itself the exponent times.

4 How can you implement the algorithm in code?

To implement the algorithm in code, you can use a loop or a recursion to iterate over the bits of the exponent, and use a variable to store the result. You can use the bitwise operators to check the bits of the exponent, and the modulo operator to reduce the numbers. Here is an example of a Python function that performs fast exponentiation modulo a modulus:

def fast_exp(base, exp, mod):
  result = 1 # initialize result
  base = base % mod # reduce base
  while exp > 0: # loop while exponent is positive
    if exp & 1: # if exponent is odd
      result = (result * base) % mod # multiply result by base and reduce
    exp = exp >> 1 # right shift exponent by one bit
    base = (base * base) % mod # square base and reduce
  return result # return final result        

5 What are some applications of fast exponentiation?

Fast exponentiation is useful for many applications that involve large numbers and modular arithmetic, such as cryptography, primality testing, and modular inverse computation. For example, in the RSA encryption scheme, you need to compute c = m^e mod n, where m is the message, e is the public key, and n is the product of two large prime numbers. Using fast exponentiation, you can compute c in a reasonable amount of time, even if e and n are very large. Similarly, in the Diffie-Hellman key exchange protocol, you need to compute s = g^ab mod p, where g is a generator, a and b are private keys, and p is a large prime number. Using fast exponentiation, you can compute s in a secure and efficient way, without revealing your private keys.

6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?