7.4. Modular Arithmetic

Try these first

1. 1.

   What is 10 mod 3?

   Hint

   Think about how many times `3` fits into `10`.

2. 2.

   Is 9 ≡ 3 mod 6 true or false?

   Hint

   What is the remainder of both `9` and `3` when divided by `6`?

3. 3.

   What is -11 mod 3?

   • -1
   • 1
   • 2

   Hint

   Convert negative to positive equivalent.

4. 4.

   True or False: Division is always well-defined in modular arithmetic.

   • True
   • False

   Hint

   Consider the conditions under which division holds.

5. 5.

   Given a = 12, b = 15, and N = 10, calculate (a * b) mod N using modular properties. Then explore how changing N affects the result.

   Hint

   Perform multiplication first, then apply mod.

6. 6.

   Implement and analyze the efficiency of the square-and-multiply algorithm for calculating a^b mod N for large b. Discuss the time complexity.

   Hint

   Focus on breaking down the exponent into binary for efficiency.