Double Modular Exponentiation

Double Modular Exponentiation

What is this problem about?

The Double Modular Exponentiation coding problem provides an array of queries, where each query contains four integers [a, b, c, m] and a target value. For each query, you need to calculate ((a^b % 10)^c) % m. You are asked to return the indices of all queries where the result of this double modular exponentiation matches the given target.

Why is this asked in interviews?

Amazon uses the Double Modular Exponentiation interview question to test a candidate's understanding of modular arithmetic properties and efficient exponentiation. It evaluates whether a candidate can implement the "Binary Exponentiation" algorithm to avoid overflows and slow O(N) multiplication. It's a test of simulation combined with mathematical optimization.

Algorithmic pattern used

The core math interview pattern is Binary Exponentiation (Exponentiation by Squaring).

1. Standard a^b % m can be calculated in O(log b) time.
2. In this problem, the calculation has two stages:
   • Stage 1: temp = power(a, b, 10)
   • Stage 2: final_result = power(temp, c, m)
3. Use a loop to process each query and collect the valid indices.

Example explanation

Query: a=2, b=3, c=2, m=10, Target: 4

1. Step 1: 2^3 % 10 = 8 % 10 = 8.
2. Step 2: 8^2 % 10 = 64 % 10 = 4.
3. Result 4 matches target 4. Add index to result list.

Query: a=3, b=3, c=3, m=8, Target: 1

1. Step 1: 3^3 % 10 = 27 % 10 = 7.
2. Step 2: 7^3 % 8 = 343 % 8 = 7.
3. Result 7 does not match target 1. Skip index.

Common mistakes candidates make

• Integer Overflow: Trying to calculate a^b or temp^c directly before applying the modulo. For large inputs, this will exceed 64-bit integer limits.
• Inefficient Power Function: Using a standard pow() function that returns floating-point numbers or using a loop for multiplication, which is O(N) instead of O(log N).
• Modulo Mix-up: Using the wrong modulo in the first stage (it must be 10 as per the problem description).

Interview preparation tip

Modular exponentiation is a fundamental building block for many cryptographic and number theory problems. Practice writing a robust modular_pow(base, exp, mod) function; it is a very common utility needed in technical interviews.