Affine Cipher Decryption Explained
The problem asks us to decrypt a ciphertext character 'Z' using the Affine cipher method. We are given the ciphertext, the key pair used for encryption, and the modulus.
- Ciphertext: "Z"
- Key Pair: $(a, b) = (7, 2)$
- Modulus: $m = 26$
The Affine cipher encryption works as $E(x) = (ax + b) \pmod{m}$, where $x$ is the numerical representation of the plaintext letter. The decryption process reverses this.
Decryption Formula and Steps
The formula for Affine cipher decryption is:
$ D(y) = a^{-1}(y - b) \pmod{m} $
Where:
- $y$ is the numerical value of the ciphertext letter.
- $a^{-1}$ is the modular multiplicative inverse of $a$ modulo $m$.
- $b$ is the additive key component.
- $m$ is the modulus.
Step 1: Convert Ciphertext to Number
First, we convert the ciphertext character "Z" into its corresponding numerical value. Assuming A=0, B=1, ..., Z=25:
Ciphertext 'Z' corresponds to $y = 25$.
Step 2: Identify Keys and Modulus
From the key pair $(7, 2)$, we have:
- $a = 7$
- $b = 2$
- $m = 26$
Step 3: Calculate the Modular Multiplicative Inverse ($a^{-1}$)
We need to find the modular multiplicative inverse of $a=7$ modulo $m=26$. This means finding a number $a^{-1}$ such that:
$ 7 \times a^{-1} \equiv 1 \pmod{26} $
We can find this by trying multiples of 7:
- $7 \times 1 = 7 \pmod{26}$
- $7 \times 2 = 14 \pmod{26}$
- ...
- $7 \times 15 = 105$. Now, let's find $105 \pmod{26}$: $105 = (4 \times 26) + 1$.
- So, $7 \times 15 \equiv 1 \pmod{26}$.
Therefore, the modular inverse $a^{-1}$ is $15$.
Step 4: Apply the Decryption Formula
Now, substitute the values into the decryption formula:
$ D(25) = 15 \times (25 - 2) \pmod{26} $
Calculate the expression inside the parentheses:
$ D(25) = 15 \times (23) \pmod{26} $
Perform the multiplication:
$ 15 \times 23 = 345 $
Now, find the result modulo 26:
$ D(25) = 345 \pmod{26} $
To find $345 \pmod{26}$, we divide 345 by 26:
$ 345 \div 26 = 13 \text{ with a remainder of } 7 $
So, $ 345 \equiv 7 \pmod{26} $
The numerical result of the decryption is $7$. This corresponds to the letter 'H' in the alphabet (A=0, H=7).
Final Decrypted Value
The decrypted numerical value for the ciphertext "Z" with the key pair (7, 2) in modulus 26 is 7.
Important Questions from Cryptography - Teaching
- In symmetric key cryptography, if every person in a group of 12 people needs to communicate with every other person in the group, how many secret keys are required?
- Match the following:
Match correctly the cryptographic algorithm with their corresponding key-length.1. DES a. 128-256 bits 2. Rj indeal b. 1-448 bits 3. RC4 c. 56 bits 4. Blowfish d. 1-2048 bits In the RSA public key cryptosystem, the private and public keys are $(e, n)$ and $(d, n)$ respectively, where $n = p \times q$ and $p$ and $q$ are large primes. Besides, $n$ is public and $p$ and $q$ are private. Let M be an integer such that $0 <M< n$ and $\phi(n) = (p-1)(q - 1)$. now consider the following equations.
I. M’ =\( M^e\) mod n M =\( (M')^d\) mod nII. ed ≡1 mod n
III. ed ≡1 mod f(n)
IV. M’ = \(M^e \)mod f(n)
M = \( (M')^d\) mod
Which of the above equations correctly represent RSA cryptosystem?- Let M be a message consisting of 50 AES blocks. Alice encrypts using CBC mode and transmits the resulting ciphertext to Bob. Due to a network error, ciphertext block number 45 is corrupted during transmission. All other ciphertext blocks are transmitted and received correctly. Once Bob decrypts the received ciphertext, number of plaintext blocks corrupted is
- Match the type of attack and its prevention mechanism:
1. Brute-Force Attack a. Change of keys 2. Statistical Attack b. Number of possible keys are large 3. Pattern Attack c. Hide the characteristics of the
language4. Known - Plain Text Attack d. Ciphertext is as random as
possible
