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(a) Write 23 as a sum of distinct powers of 2. (b) Use the fast factoring algorithm to compute 323 (mod 31).

(a) Write 23 as a sum of distinct powers of 2. (b) Use the fast factoring algorithm to compute 323 (mod 31).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem 7:** **(a)** Write 23 as a sum of distinct powers of 2. **(b)** Use the fast factoring algorithm to compute \( 3^{23} \pmod{31} \). --- **Explanation:** - **Part (a)** requires expressing the number 23 as a sum of different powers of 2 such as \(2^0, 2^1, 2^2, \ldots\). - **Part (b)** involves calculating \( 3^{23} \mod{31} \) using an efficient method known as the fast factoring algorithm. **Educational Website Notes:** For part (a), you will break down the number 23 into a sum where each term is a power of 2 and all terms are distinct. This relates to representing numbers in binary form. For part (b), fast exponentiation methods such as exponentiation by squaring or modular exponentiation can be applied to compute large powers under a modulo operation efficiently. The fast factoring algorithm is commonly used in cryptographic applications where large calculations are needed to be performed securely and efficiently.
Transcribed Image Text:**Problem 7:** **(a)** Write 23 as a sum of distinct powers of 2. **(b)** Use the fast factoring algorithm to compute \( 3^{23} \pmod{31} \). --- **Explanation:** - **Part (a)** requires expressing the number 23 as a sum of different powers of 2 such as \(2^0, 2^1, 2^2, \ldots\). - **Part (b)** involves calculating \( 3^{23} \mod{31} \) using an efficient method known as the fast factoring algorithm. **Educational Website Notes:** For part (a), you will break down the number 23 into a sum where each term is a power of 2 and all terms are distinct. This relates to representing numbers in binary form. For part (b), fast exponentiation methods such as exponentiation by squaring or modular exponentiation can be applied to compute large powers under a modulo operation efficiently. The fast factoring algorithm is commonly used in cryptographic applications where large calculations are needed to be performed securely and efficiently.
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