Question 3 **PROBLEMS 10.1**1. Encrypt the message "RETURN HOME" using the Caesar cipher.2. If the Caesar cipher produced "KOSSVE BOVKB," what is the plaintext message?3. (a) Linear ciphers are defined by the congruence $ C \equiv aP + b \pmod{26} $, where $ a $ and $ b $ are integers with $ \gcd(a, 26) = 1 $. Show that the corresponding decrypting congruence is $ P \equiv (C - b) \cdot a^{-1} \pmod{26} $, where the integer $ a^{-1} $ satisfies $ aa^{-1} \equiv 1 \pmod{26} $. (b) Using the linear cipher $ C \equiv 5P + 11 \pmod{26} $, encrypt the message "NUMBER THEORY IS EASY." (c) Decrypt the message "RXQJGUHOXKGH" which was produced using the linear cipher $ C \equiv 3P + 7 \pmod{26} $.---*Note: This text appears on an educational website discussing cryptography problems, specifically focusing on Caesar and linear ciphers. The problems are designed to practice encryption and decryption techniques using mathematical congruence methods.*

Answered: Image /qna-images/answer/d5055574-ea89-4c86-bbc2-79cbff62e956.jpg

2. If the Caesar cipher produced RDSSB ELUWKGDB, WIlal Is uic plumno 3. (a) A linear cipher is defined by the congruence C = aP +b (mod 26), where a and b are integers with gcd(a, 26) = 1. Show that the corresponding decrypting congruence is P = a'(C – b) (mod 26), where the integer a' satisfies aa' = 1 (mod 26). (b) Using the linear cipher C = 5P+ 11 (mod 26), encrypt the message NUMBER %3D II THEORY IS EASY. (c) Decrypt the message RXQTGU HOZTKGH FJ KTMMTG, which was produced using the linear cipher C = 3P +7 (mod 26). II

2. If the Caesar cipher produced RDSSB ELUWKGDB, WIlal Is uic plumno 3. (a) A linear cipher is defined by the congruence C = aP +b (mod 26), where a and b are integers with gcd(a, 26) = 1. Show that the corresponding decrypting congruence is P = a'(C – b) (mod 26), where the integer a' satisfies aa' = 1 (mod 26). (b) Using the linear cipher C = 5P+ 11 (mod 26), encrypt the message NUMBER %3D II THEORY IS EASY. (c) Decrypt the message RXQTGU HOZTKGH FJ KTMMTG, which was produced using the linear cipher C = 3P +7 (mod 26). II

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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Question

Question 3

**PROBLEMS 10.1**

1. Encrypt the message "RETURN HOME" using the Caesar cipher.

2. If the Caesar cipher produced "KOSSVE BOVKB," what is the plaintext message?

3.
(a) Linear ciphers are defined by the congruence $ C \equiv aP + b \pmod{26} $, where $ a $ and $ b $ are integers with $ \gcd(a, 26) = 1 $. Show that the corresponding decrypting congruence is $ P \equiv (C - b) \cdot a^{-1} \pmod{26} $, where the integer $ a^{-1} $ satisfies $ aa^{-1} \equiv 1 \pmod{26} $.

(b) Using the linear cipher $ C \equiv 5P + 11 \pmod{26} $, encrypt the message "NUMBER THEORY IS EASY."

(c) Decrypt the message "RXQJGUHOXKGH" which was produced using the linear cipher $ C \equiv 3P + 7 \pmod{26} $.

---

*Note: This text appears on an educational website discussing cryptography problems, specifically focusing on Caesar and linear ciphers. The problems are designed to practice encryption and decryption techniques using mathematical congruence methods.*

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