Important Applications on Linear Cipher

Number Theory

1. Divisibility
- 1.1. Greatest Common Divisor
  - 1.1.1. Important Problems On Greatest Common Divisor
- 1.2. Expressing Cubes As Differences Of Squares
2. Prime Numbers
- 2.1. Important Problems On Prime Numbers
- 2.2. Prime Number Theorem
  - 2.2.1. The Prime Number Theorem Explained With Interpretation
3. Congruence
- 3.1. Linear Congruence
  - 3.1.1. Important Applications On Linear Congruence
4. Number Theoretic Functions
- 4.1. Greatest Integer Functions
  - 4.1.1. Divisibility Of Binomial Coefficients
5. Quadratic Reciprocity
- 5.1. Legendre Symbol
  - 5.1.1. Legendre Symbol And Its Properties
6. Fermats Last Theorem
- 6.1. Proof Of Fermats Last Theorem
- 6.2. Pythagorean Triples
  - 6.2.1. Important Problems On Pythagorean Triples
7. Integers As Sum Of Squares
- 7.1. Sum Of More Than Two Squares
  - 7.1.1. Representation Of Integers As Sums Of Five Nonzero Squares For All Values Greater Than Or Equal To 170
- 7.2. Sum Of Two Squares
  - 7.2.1. Perfect Squares In Pythagorean Triples
8. Encryption And Decryption
- 8.1. Linear Cipher
  - 8.1.1. Important Applications On Linear Cipher
- 8.2. Hill Cipher
  - 8.2.1. Hill Cipher Decryption In Number Theory Step By Step With Example

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Important Applications

The Linear Cipher is a foundational encryption technique with several important applications in the field of cryptography and information security. This topic highlights how the Linear Cipher is used to protect data, enhance privacy, and ensure secure communication in various digital systems. From encoding simple messages to laying the groundwork for more advanced encryption methods, these applications show its relevance in both academic learning and real-world cybersecurity. With clear explanations and practical scenarios, this guide helps learners understand the significance of the Linear Cipher in modern encryption.

Decrypt the message “RXQTGU HOZTKGH FJ KTMMTG” which was encrypted using the linear cipher

C = (3P + 7) \mod 26.

Answer:

To decrypt the given linear cipher without using modular inverses, we proceed by computing all possible plaintext values \( P \in \{0, 1, \dots, 25\} \) and corresponding ciphertext values using the encryption formula:

C = (3P + 7) \mod 26.

We compute each \( C \) value for every \( P \):

\begin{array}{|c|c|c|c|c|} \hline \text{Letter (P)} & P & C = (3P + 7) \pmod{26} & \text{C (number)} & \text{Cipher Letter (C)} \\ \hline A & 0 & 3(0) + 7 = 7\mod 26 & 7 & H \\ B & 1 & 3(1) + 7 = 10\mod 26 & 10 & K \\ C & 2 & 3(2) + 7 = 13\mod 26 & 13 & N \\ D & 3 & 3(3) + 7 = 16\mod 26 & 16 & Q \\ E & 4 & 3(4) + 7 = 19\mod 26 & 19 & T \\ F & 5 & 3(5) + 7 = 22\mod 26 & 22 & W \\ G & 6 & 3(6) + 7 = 25\mod 26 & 25 & Z \\ H & 7 & 3(7) + 7 = 28\mod 26 & 2 & C \\ I & 8 & 3(8) + 7 = 31\mod 26 & 5 & F \\ J & 9 & 3(9) + 7 = 34\mod 26 & 8 & I \\ K & 10 & 3(10) + 7 = 37\mod 26 & 11 & L \\ L & 11 & 3(11) + 7 = 40\mod 26 & 14 & O \\ M & 12 & 3(12) + 7 = 43\mod 26 & 17 & R \\ N & 13 & 3(13) + 7 = 46\mod 26 & 20 & U \\ O & 14 & 3(14) + 7 = 49\mod 26 & 23 & X \\ P & 15 & 3(15) + 7 = 52\mod 26 & 0 & A \\ Q & 16 & 3(16) + 7 = 55\mod 26 & 3 & D \\ R & 17 & 3(17) + 7 = 58\mod 26 & 6 & G \\ S & 18 & 3(18) + 7 = 61\mod 26 & 9 & J \\ T & 19 & 3(19) + 7 = 64\mod 26 & 12 & M \\ U & 20 & 3(20) + 7 = 67\mod 26 & 15 & P \\ V & 21 & 3(21) + 7 = 70\mod 26 & 18 & S \\ W & 22 & 3(22) + 7 = 73\mod 26 & 21 & V \\ X & 23 & 3(23) + 7 = 76\mod 26 & 24 & Y \\ Y & 24 & 3(24) + 7 = 79\mod 26 & 1 & B \\ Z & 25 & 3(25) + 7 = 82\mod 26 & 4 & E \\ \hline \end{array}

Decrypt each letter in the message.

Given ciphertext: “RXQTGU HOZTKGH FJ KTMMTG”

We decode word by word:

RXQTGU → R (12) M, X (14) O, Q (3) D, T (4) E, G (17) R, U (13) N → MODERN

HOZTKGH → H (0) A, O (11) L, Z (6) G, T (4) E, K (1) B, G (17) R, H (0) A → ALGEBRA

FJ → F (8) I, J (18) S → IS

KTMMTG → K (1) B, T (4) E, M (19) T, M (19) T, T (4) E, G (17) R → BETTER

Final Decrypted Message:

\boxed{\text{MODERN ALGEBRA IS BETTER}}