Fast Exponentiation, Prime Factorization & Modular Arithmetic

Modular Arithmetic & Exponentiation

Computes $(A^B) \pmod M$ in $O(\log B)$ time using binary bit representation of $B$.

Production Code Example:

public class FastExpo {
    public long power(long base, long exp, long mod) {
        long res = 1;
        base %= mod;
        while (exp > 0) {
            if ((exp & 1) == 1) res = (res * base) % mod;
            base = (base * base) % mod;
            exp >>= 1;
        }
        return res;
    }
}

Key Complexity & Algorithmic Takeaways:

When implementing Fast Exponentiation, Prime Factorization & Modular Arithmetic in coding interviews and production applications, keep these core guidelines in mind:

• Time Complexity Analysis: Always evaluate best-case, average-case, and worst-case time complexities ($O(1)$, $O(\log n)$, $O(n)$, $O(n \log n)$, $O(n^2)$).
• Space Complexity & Memory Bounds: Account for auxiliary memory usage, call stack frame recursion overhead, and heap allocations.
• Edge Cases & Validation: Test empty inputs, null pointers, single-element collections, duplicate values, and integer overflow bounds.
• Optimal vs Naive Solutions: Start with a clear brute-force solution, then optimize using techniques like Hashing, Two Pointers, Windowing, or Dynamic Programming.

Summary Takeaway:

Mastering Fast Exponentiation, Prime Factorization & Modular Arithmetic provides the foundational problem-solving skills needed to pass technical coding interviews at top tech companies and write ultra-performant software systems.