### Problem 22: Efficient Computation of Large Powers ModuloYou are tasked to compute $123456789^{65537} \mod 581859289607$. A friend offers their assistance with a pricing model of 1 cent per multiplication mod $581859289607$. Your friend aims to earn more than $650. However, your challenge is to strategize so the computation costs below 25 cents.#### Solution Strategy:To minimize the number of multiplications, you can utilize the method of **"Exponentiation by Squaring"**. This is an efficient algorithm that reduces the number of multiplication operations typically from $n$ to $O(\log n)$, significantly lowering your costs.- **Exponentiation by Squaring**: This method takes advantage of binary decomposition of the exponent to repeatedly square and multiply: - Break down the exponent (65537) into powers of 2. - Compute successive squares starting from the base number $123456789$. - Combine the computed powers using the properties of exponents.#### Key Observation:- **RSA Exponent**: The exponent $65537$ mentioned is commonly used in RSA encryption due to its balance of security and computational efficiency, being $2^{16} + 1$.By following this approach, you can achieve the required computation with significantly fewer multiplications, ensuring you remain well within the budget of 25 cents.

Given :- In the above question, the statement is mention below as, Need to compute the according…

Answered: RSA encryption

Engineering

Computer Science

RSA encryption

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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### Problem 22: Efficient Computation of Large Powers Modulo

You are tasked to compute $123456789^{65537} \mod 581859289607$. A friend offers their assistance with a pricing model of 1 cent per multiplication mod $581859289607$. Your friend aims to earn more than $650. However, your challenge is to strategize so the computation costs below 25 cents.

#### Solution Strategy:

To minimize the number of multiplications, you can utilize the method of **"Exponentiation by Squaring"**. This is an efficient algorithm that reduces the number of multiplication operations typically from $n$ to $O(\log n)$, significantly lowering your costs.

- **Exponentiation by Squaring**: This method takes advantage of binary decomposition of the exponent to repeatedly square and multiply:
- Break down the exponent (65537) into powers of 2.
- Compute successive squares starting from the base number $123456789$.
- Combine the computed powers using the properties of exponents.

#### Key Observation:

- **RSA Exponent**: The exponent $65537$ mentioned is commonly used in RSA encryption due to its balance of security and computational efficiency, being $2^{16} + 1$.

By following this approach, you can achieve the required computation with significantly fewer multiplications, ensuring you remain well within the budget of 25 cents.

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