Playfair Cipher: Encryption of "ALGORITHM" using key "PSEUDOCODE"

Step 1 - EXTRACT: Given Data

• Plaintext: ALGORITHM
• Key: PSEUDOCODE
• Convention: I/J share one cell (standard Playfair)

Step 2 - SOLVE

Construct the 5x5 Matrix

Key: P S E U D O C O D E

Remove duplicates (keep first occurrence): P, S, E, U, D, O, C
(The second O, D, E are dropped.)

Fill remaining letters A, B, F, G, H, I/J, K, L, M, N, Q, R, T, V, W, X, Y, Z:

Prepare Digrams

Plaintext: A L G O R I T H M

No double letters within pairs. M is left alone, pad with X:

$$AL \mid GO \mid RI \mid TH \mid MX$$

Apply Rules

AL: A(2,3), L(3,5) → rectangle → A→(2,5)=F, L→(3,3)=I → FI

GO: G(3,1), O(2,1) → same column → G→(4,1)=M, O→(3,1)=G → MG

RI: R(4,4), I(3,3) → rectangle → R→(4,3)=Q, I→(3,4)=K → QK

TH: T(4,5), H(3,2) → rectangle → T→(4,2)=N, H→(3,5)=L → NL

MX: M(4,1), X(5,3) → rectangle → M→(4,3)=Q, X→(5,1)=V → QV

Final Ciphertext

$$\boxed{\text{Ciphertext} = \text{FIMGQKNLQV}}$$

Intrusion and Intrusion Detection Systems

What is Intrusion?

Intrusion detection is the process of identifying and responding to malicious activity targeted at a resource. An intrusion refers to any unauthorized attempt to access, manipulate, or compromise a system's resources, data, or services.

An Intrusion Detection System (IDS) is a system designed to test/analyze network system traffic/events against a given set of parameters and alert/capture data when these thresholds are met. IDS uses collected information and pre-defined knowledge-based systems to reason about the possibility of an intrusion. It also provides services to cope with intrusion such as giving alarms and activating programs to deal with intrusion.

Two Types (Approaches) of Intrusion Detection System

1. Statistical Anomaly Detection

Statistical anomaly detection involves the collection of data relating to the behaviours of legitimate users over a period of time. Then statistical tests are applied to observed behaviour to determine with a high level of confidence whether the behaviour is that of a legitimate user or not.

It falls into two broad categories:

• Threshold Detection:

  • Involves counting the number of occurrences of a specified event type over an interval of time.
  • If the count exceeds a reasonable number, an intrusion is assumed.
  • Example: Too many failed login attempts within a short time.

• Profile-Based Anomaly Detection:

  • Focuses on characterizing the past behaviour of individual users or related groups of users.
  • Significant deviations from the established profile are flagged as potential intrusions.

2. Rule-Based Detection

Rule-based detection involves an attempt to define a set of rules that can be used to decide whether a given behaviour is that of an intruder.

It falls into two broad categories:

• Rule-Based Anomaly Detection:

  • Historical audit records are analyzed to identify usage patterns.
  • Rules are generated automatically to describe those patterns.
  • Any behaviour that deviates from these rules is flagged as suspicious.

• Rule-Based Penetration Identification:

  • Uses rules for identifying known penetrations or penetrations that would exploit known weaknesses.
  • Rules are typically defined by security experts based on known attack signatures.
  • Example: A rule that detects a specific sequence of commands known to exploit a vulnerability.

Summary Table:

Multiplicative Inverse in GF(2⁸) using the Extended Euclidean Algorithm

Step 1 - EXTRACT (Given data)

• Element to invert: 95 (hexadecimal), interpreted as an element of $GF(2^8)$.
• $95_{16} = 1001,0101_2$, giving polynomial

$$b(x) = x^7 + x^4 + x^2 + 1$$

• AES irreducible (reduction) polynomial:

$$m(x) = x^8 + x^4 + x^3 + x + 1$$

Note: The question only states "95" and "extended Euclidean algorithm". The irreducible polynomial is not explicitly given; the AES standard modulus is assumed since this is the standard BIT/cryptography context.

Goal: find $b(x)^{-1} \bmod m(x)$.

Step 2 - SOLVE

All arithmetic is in $GF(2)$ (addition = XOR).

Initialize:

• $r_0 = m(x)=x^8+x^4+x^3+x+1$, $; t_0=0$
• $r_1 = b(x)=x^7+x^4+x^2+1$, $; t_1=1$

Recurrence: $r_{i+1}=r_{i-1}-q_i r_i$, $;t_{i+1}=t_{i-1}-q_i t_i$.

Division 1: $r_0 \div r_1$

$$x^8+x^4+x^3+x+1 ;=; x\cdot(x^7+x^4+x^2+1) + r_2$$

$x\cdot b = x^8+x^5+x^3+x$. XOR:

$$r_2 = (x^8+x^4+x^3+x+1)\oplus(x^8+x^5+x^3+x)=x^5+x^4+1$$

• $q_1=x$
• $t_2=t_0-q_1t_1 = 0 - x\cdot1 = x$

Division 2: $r_1 \div r_2$

Divide $x^7+x^4+x^2+1$ by $x^5+x^4+1$:

• $x^2$: $x^2(x^5+x^4+1)=x^7+x^6+x^2$; XOR → $x^6+x^4+1$
• $x$: $x(x^5+x^4+1)=x^6+x^5+x$; XOR → $x^5+x^4+x+1$
• $1$: $x^5+x^4+1$; XOR → $x$

So:

• $q_2=x^2+x+1$, $; r_3=x$
• $t_3=t_1-q_2t_2 = 1 - (x^2+x+1)(x) = 1+(x^3+x^2+x)=x^3+x^2+x+1$

Division 3: $r_2 \div r_3$

Divide $x^5+x^4+1$ by $x$:

$$x^5+x^4+1 = x\cdot(x^4+x^3)+1$$

So:

• $q_3=x^4+x^3$, $; r_4=1$
• $t_4=t_2-q_3t_3 = x - (x^4+x^3)(x^3+x^2+x+1)$

Compute $(x^4+x^3)(x^3+x^2+x+1)$:

$x^4\cdot(x^3+x^2+x+1)=x^7+x^6+x^5+x^4$ $x^3\cdot(x^3+x^2+x+1)=x^6+x^5+x^4+x^3$

XOR (add): $x^6\oplus x^6=0$, $x^5\oplus x^5=0$, $x^4\oplus x^4=0$:

$$=x^7+x^3$$

Then:

$$t_4 = x \oplus (x^7+x^3) = x^7+x^3+x$$

Since $r_4 = 1 = \gcd$, the inverse of $b(x)$ modulo $m(x)$ is $t_4$.

Verification

$$b(x)^{-1} = x^7 + x^3 + x$$

Check $b(x)\cdot t_4 \bmod m(x)$:

$b = x^7+x^4+x^2+1$, $t_4=x^7+x^3+x$.

Product:

• $x^7\cdot t_4 = x^{14}+x^{10}+x^8$
• $x^4\cdot t_4 = x^{11}+x^7+x^5$
• $x^2\cdot t_4 = x^9+x^5+x^3$
• $1\cdot t_4 = x^7+x^3+x$

XOR all: $x^5\oplus x^5=0$, $x^7\oplus x^7=0$, $x^3\oplus x^3=0$:

$$= x^{14}+x^{11}+x^{10}+x^9+x^8+x$$

Reduce mod $m(x)=x^8+x^4+x^3+x+1$ (i.e. $x^8\equiv x^4+x^3+x+1$):

• $x^8\equiv x^4+x^3+x+1$

• $x^9\equiv x^5+x^4+x^2+x$

• $x^{10}\equiv x^6+x^5+x^3+x^2$

• $x^{11}\equiv x^7+x^6+x^4+x^3$

• $x^{14}=x^6\cdot x^8\equiv x^6(x^4+x^3+x+1)=x^{10}+x^9+x^7+x^6$

  reduce: $x^{10}\to x^6+x^5+x^3+x^2$, $x^9\to x^5+x^4+x^2+x$ $x^{14}\equiv (x^6+x^5+x^3+x^2)+(x^5+x^4+x^2+x)+x^7+x^6$ $= x^7+x^4+x^3+x$

Sum all reduced terms (XOR):

XOR column by column:

• $x^7$: $1+1=0$
• $x^6$: $1+1=0$
• $x^5$: $1+1=0$
• $x^4$: $1+1+1+1=0$
• $x^3$: $1+1+1+1=0$
• $x^2$: $1+1=0$
• $x^1$: $1+1+1+1=0$
• $x^0$: $1$

$$= 1 \checkmark$$

Final Answer

$$\boxed{b(x)^{-1} = x^7 + x^3 + x}$$

In binary: $1000,1010$, i.e. hexadecimal 8A.

Completing the computation through $t_4$ gives the inverse $x^7+x^3+x$, which multiplies back to $1$.

DoS Attack and PKI Trust Model

Denial of Service (DoS) Attack:

"Denial of service prevents the normal use or management of communication facilities."

A Denial of Service (DoS) attack is a type of active attack in which an attacker attempts to make a system, network, or service unavailable to its intended users by overwhelming or disabling it.

Key Characteristics:

• It is classified under active attacks in network security.
• The attacker does not steal or alter data but rather disrupts availability.
• It targets one of the core security goals: Availability.

Forms of DoS Attack:

Example:

An attacker floods a web server with thousands of fake requests per second, causing it to become unresponsive to legitimate users. This is known as a flood-based DoS attack.

PKI Trust Model

Public Key Infrastructure (PKI) is a framework that manages digital certificates and public-key encryption to enable secure communication. A PKI Trust Model defines how trust is established and distributed among entities in a PKI system.

Core Components of PKI:

• Certificate Authority (CA): Issues and signs digital certificates.
• Registration Authority (RA): Verifies identity before certificate issuance.
• Digital Certificate: Binds a public key to an entity's identity.
• Certificate Revocation List (CRL): Lists revoked certificates.

Types of PKI Trust Models:

1. Single CA (Monopoly) Model

• One single root CA is trusted by all entities.
• Simple but single point of failure.
• If the root CA is compromised, the entire system fails.

        [Root CA]
       /    |    \
    User1  User2  User3

2. Hierarchical Trust Model

• A root CA sits at the top and delegates trust to subordinate CAs.
• Subordinate CAs issue certificates to end users.
• Most widely used model (e.g., SSL/TLS on the internet).

         [Root CA]
        /          \
  [Sub CA1]      [Sub CA2]
   /     \         /    \
User1  User2   User3   User4

• Trust flows top-down.
• If you trust the Root CA, you trust all certificates issued under it.

3. Web of Trust Model

• Used in PGP (Pretty Good Privacy).
• There is no central CA; users sign each other's public keys.
• Trust is distributed and peer-based.
• Example: If Alice trusts Bob, and Bob trusts Carol, Alice may transitively trust Carol.

4. Cross-Certification Model

• Two separate CA hierarchies mutually certify each other.
• Allows users from different organizations to trust each other.

[CA1] <----cross certify----> [CA2]
  |                              |
Users of Org A              Users of Org B

Summary Table:

Conclusion:

A DoS attack disrupts the availability of services, while PKI trust models provide a structured way to establish and manage trust in public key cryptography. Both concepts are fundamental to understanding network security.

Short Notes (Any Two)

a. Totient Value of Any Positive Integer

Definition: Euler's Totient Function, denoted by φ(n), is defined as the number of positive integers less than n that are relatively prime (coprime) to n. Two numbers are relatively prime if their greatest common divisor (GCD) is 1.

Special Property: If n is a prime number, then:

φ(n) = n - 1

This is because all integers from 1 to (n-1) are relatively prime to a prime number.

Example: Find φ(10):

• Numbers less than 10: {1, 2, 3, 4, 5, 6, 7, 8, 9}
• Numbers relatively prime to 10 (GCD with 10 = 1): {1, 3, 7, 9}
• Therefore, φ(10) = 4

Example with prime: Find φ(7):

• Since 7 is prime: φ(7) = 7 - 1 = 6

Euler's Totient Function is widely used in RSA encryption and number theory in cryptography.

b. Properties of Hash Function

Definition: A hash function is a function that maps a message of any length into a fixed-length hash value, which serves as the authenticator in message authentication and digital signatures.

Key Properties:

Example:

• MD5 produces a 128-bit (16-byte) hash value expressed as a 32-digit hexadecimal number.
• MD4 also produces a 128-bit message digest from an input of arbitrary length.

Hash functions play a fundamental role in efficient and secure information processing in cryptography.