The fast powering algorithm

One of the foundational operations in many cryptographic systems involves computing large exponents of numbers modulo a large modulus. This process, known as modular exponentiation, is crucial in many cryptographic algorithms. However, naive methods of exponentiation can be prohibitively slow when dealing with very large numbers. To address this, the Square and Multiply algorithm offers a streamlined approach to perform these calculations efficiently. Let's explore this concept in detail.

The Need for Efficient Modular Exponentiation in Cryptographic Systems

Many cryptographic systems, such as RSA and Diffie-Hellman, rely on the ability to compute large exponents of a number n^k modulo a large modulus m. For instance, in RSA encryption and decryption processes, operations like n^k mod m are fundamental. Given that m is typically a very large number (often hundreds or thousands of bits long), performing these calculations efficiently is critical for the system's performance and security.

The Limitations of Naive Exponentiation

A straightforward approach to compute n^k involves multiplying n by itself k times. While conceptually simple, this naive exponentiation method becomes exceedingly slow as k increases. The time complexity is linear with respect to the exponent k, making it impractical for large-scale cryptographic applications where exponents can be in the order of 10¹⁰⁰ or higher.

Question: How can we speed up this process?

Binary Decomposition: Breaking Down the Exponent

To accelerate modular exponentiation, we can leverage the binary representation of the exponent k. By decomposing k into powers of 2, we can reduce the number of multiplications required. Here's how it works:

Example: Compute 3³¹⁵ mod 100

1. Binary Decomposition of the Exponent:

   Break down 315 into sums of powers of 2:

   • 315 = 2⁸ + 2⁵ + 2⁴ + 2³ + 2¹ + 2⁰
   • Or, ordered from smallest to largest:
   • 315 = 2⁰ + 2¹ + 2³ + 2⁴ + 2⁵ + 2⁸
2. Expressing the Power:

   Using the decomposition:

   3³¹⁵ = 3²⁰ × 3²¹ × 3²³ × 3²⁴ × 3²⁵ × 3²⁸

3. Calculating Powers via Squaring:

   Notice that each subsequent power 3²ⁱ is obtained by squaring the previous one:

   • 3²⁰ = 3¹ = 3
   • 3²¹ = (3²⁰)² = 3² = 9
   • 3²² = (3²¹)² = 9² = 81
   • 3²³ = (3²²)² = 81² = 6561 ≡ 61 mod 100
     and so on

   Importantly, since we are working modulo 100, we only multiply numbers ever exceeding two-digit numbers in this case (in this case)

4. Final Multiplication:

   Multiply the relevant powers together, each already reduced modulo 100:

   3³¹⁵ mod 100 = (3) × (9) × (61) × … mod 100

   The exact multiplication steps would continue until the final result is obtained.