Is the affine cipher e(x) = 13x + 9 (mod 63) invertible? Yes or no. If yes, find the inverse (fully reduced with positive terms modulo the modulus) if no, explain why. Is the affine cipher e(x) = 22x + 10 (mod 63) invertible? Yes or no. If yes, find the inverse (fully reduced with positive terms modulo the modulus), if no, explain why.
Is the affine cipher e(x) = 13x + 9 (mod 63) invertible? Yes or no. If yes, find the inverse (fully reduced with positive terms modulo the modulus) if no, explain why. Is the affine cipher e(x) = 22x + 10 (mod 63) invertible? Yes or no. If yes, find the inverse (fully reduced with positive terms modulo the modulus), if no, explain why.
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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Transcribed Image Text:**Learning Target N5**: *I can encode English sentences into numbers and reverse the process. I can perform basic affine ciphers and find decryption functions. I can correctly state why an affine cipher is or is not invertible.*
1. Is the affine cipher \( e(x) = 13x + 9 \pmod{63} \) invertible? Yes or no. If yes, find the inverse (fully reduced with positive terms modulo the modulus); if no, explain why.
2. Is the affine cipher \( e(x) = 22x + 10 \pmod{63} \) invertible? Yes or no. If yes, find the inverse (fully reduced with positive terms modulo the modulus); if no, explain why.
![**Transcription for Educational Website**
**Topic: Modular Arithmetic and Inverses**
1. Since \( \gcd(9, 63) = 9 \neq 1 \), not invertible.
2. \( \gcd(10, 63) = 1 \), so it is invertible.
\[ 10^{-1} \equiv 19 \, (\text{mod} \, 63) \]
So,
\[
x \equiv 10y + 17 \, (\text{mod} \, 63)
\]
\[
x - 17 \equiv 10y \, (\text{mod} \, 63)
\]
\[
\Rightarrow 19x - 323 = y \, (\text{mod} \, 63)
\]
So,
\[
e^{-1}(x) = 19x + 55 \, (\text{mod} \, 63)
\]
**Explanation:**
This text explores the concepts of modular arithmetic and the conditions under which a number is invertible. The key focus is on determining when a modular inverse exists, using examples and calculations for clarity. In this example, the greatest common divisor (GCD) is used to determine invertibility.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd375a7e5-ad67-4b21-afad-3fb7503a0e18%2Fee18457e-9b5b-4124-8e83-13384508ad8b%2Fq7qemg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website**
**Topic: Modular Arithmetic and Inverses**
1. Since \( \gcd(9, 63) = 9 \neq 1 \), not invertible.
2. \( \gcd(10, 63) = 1 \), so it is invertible.
\[ 10^{-1} \equiv 19 \, (\text{mod} \, 63) \]
So,
\[
x \equiv 10y + 17 \, (\text{mod} \, 63)
\]
\[
x - 17 \equiv 10y \, (\text{mod} \, 63)
\]
\[
\Rightarrow 19x - 323 = y \, (\text{mod} \, 63)
\]
So,
\[
e^{-1}(x) = 19x + 55 \, (\text{mod} \, 63)
\]
**Explanation:**
This text explores the concepts of modular arithmetic and the conditions under which a number is invertible. The key focus is on determining when a modular inverse exists, using examples and calculations for clarity. In this example, the greatest common divisor (GCD) is used to determine invertibility.
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