Fermat’s Little Theorem The RSA Cryptosystem will require exponentiation to decrypt messages. Exponentiation Notation Example 1: Compute Exponentials Example 1 Example 2: compute Exponential Mod Example 2 In Example 2 it is seen that modulus arithmetic on small exponentials is easy. However, Modular arithmetic on large numbers can be quite difficult and is subject to computer errors due to computer round off. – Example: Large exponential… Example

Exponential Notation All laws of exponents in the real number system carry over to MOD arithmetic, except division. Laws of Exponents Method of Successive Squaring for Arithmetic Modulo m – Step 1: Break the exponent down into the sum of powers of 2. – Step 2: Write the base as a succession of the same base with the exponents from step 1. – Step 3: Write the base to each power of 2, up to the highest one used from step 1 (1, 2, 4, 8, …, highest), and perform the modular arithmetic on these. – Step 4: Multiply the numbers obtained in 3 modulo m. – Example 3: Compute Modulo 23 Example 3 – Example 4: Compute Modulo 41… Example 4

Fermat’s Little Theorem Example 5: Compute powers modulo m Example 5 Example 6: Solve Equation…! Example 6