The document introduces integer factorization, which involves decomposing composite numbers into their prime factors, and discusses its significance in cryptography, particularly in public-key systems like RSA. It highlights challenges such as the threat of quantum computing and vulnerabilities associated with small key sizes, along with various algorithms used for factorization. Additionally, it mentions mitigations like post-quantum cryptography and the importance of larger RSA key sizes for enhanced security.

1. INTRODUCTION TO CYBER SECURITY
Topic: INTEGER FACTORIZATION
Under the guidance: Dr. PRADEEP KANCHAN
Name: Ishwar Pavan
USN: NU24CYB05
Date: 20/11/2024

2. INTRODUCTION
• Integer factorization is the mathematical process of decomposing a composite
number into a product of smaller integers, specifically its prime factors. A prime
factor is a prime number that divides the given integer exactly (without leaving a
remainder).
• For example:
Let's take the number 36.
we can write 36 as a product of its prime factors:
36 = 2 × 2 × 3 × 3
Or, using exponential notation:
36 = 2² × 3²
So, the prime factorization of 36 is 2² × 3².

3. Types of Numbers
• Prime Numbers:
• Numbers greater than 1 that have no divisors other than 111 and themselves
(e.g., 2,3,5,7,…).
• Composite Numbers:
• Composite Numbers: Numbers that can be expressed as a product of primes
(e.g., 4 = 2^2, 15 = 3 . 5).

4. IMPORTANCE
• Cryptography: Integer factorization plays a key role in public-key
cryptographic systems like RSA, which rely on the difficulty of
factoring large composite numbers.
• Number Theory: It is fundamental in understanding properties of
integers, divisors, and modular arithmetic.
• Algorithm Design: Efficient algorithms for factorization have practical
applications in computer science and mathematics.

5. CHALLENGES
• Difficulty of Factorizing Large Numbers
• Vulnerabilities Due to Small Key Sizes
• Quantum Computing Threat
• Increasing Computational Costs
• Emerging Algorithms

6. Advanced Factorization Threats
• Quantum Computing:
• Shor’s algorithm can factor large numbers in polynomial time.
• Could render RSA insecure.
• Emerging Algorithms:
• Research into faster classical algorithms for factorization.

7. Mitigations and Alternatives
• Post-Quantum Cryptography:
• Lattice-based cryptography.
• Hash-based cryptography.
• Increasing RSA Key Sizes:
• Larger keys provide greater security but increase computational overhead.

8. RELATED ALGORITHMS
• Trial Division
• Pollard’s Rho Algorithm
• Elliptic Curve Factorization
• Quadratic Sieve (QS)
• General Number Field Sieve (GNFS)

9. APPLICATIONS
• RSA Encryption
• Digital Signatures
• Secure Key Exchange
• Factoring-Based Attacks

10. Thank You