GF(P)
What is GF(P)?
GF(P) or Galois Field of prime order is a finite mathematical field containing exactly p elements (where p is a prime number). It's a fundamental concept in abstract algebra and cryptography, providing a structured way to perform arithmetic operations within a finite, prime-sized set.
Key Components:
1. Prime number p
2. Elements {0, 1, 2, ..., p-1}
3. Modular arithmetic
4. Closed algebraic operations
Core Rules:
1. All arithmetic done modulo p
2. Elements range from 0 to p-1
3. Supports addition, subtraction, multiplication, division
3. Division uses modular multiplicative inverse
Basic Operations:
1. Addition: (a + b) mod p
2. Subtraction: (a - b) mod p
3. Multiplication: (a * b) mod p
3. Division: Multiplicative inverse
Example with GF(5):
Elements: {0, 1, 2, 3, 4}
Addition:
• 3 + 4 = 2 (mod 5)
• 2 + 3 = 0 (mod 5)
Multiplication:
• 3 * 4 = 2 (mod 5)
• 2 * 3 = 1 (mod 5)
Multiplicative Inverse:
• In GF(5), 2 * 3 = 1 (mod 5))
• So 3 is multiplicative inverse of 2
Significant Properties:
• Commutative operations
• No element has zero divisors
• Every non-zero element has inverse
• Closed under all operations