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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Transcribed Image Text:### 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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