Modular Polynomial Arithmetic
Module 03 / Lesson 05
Mathematical Visualization
Polynomials and Finite Fields
Modular polynomial arithmetic is a method where coefficients and polynomial results are reduced within a Finite Field (like $GF(2)$). It is the core logic used in AES (Advanced Encryption Standard) to ensure data blocks remain fixed in size.
Key Components:
- Irreducible Polynomial: Acts like a prime number for reduction.
- Modulo Reduction: Finding the remainder after division.
- GF(2) Coefficients: Coefficients are only 0 or 1.
The Logic:
- Multiplication: Perform standard algebraic multiplication.
- Reduction: Divide the result by an irreducible polynomial.
- Result: The remainder is our final encrypted/transformed state.
Example in GF(2³):
Irreducible: $x^3 + x + 1$
$(x^2) \times (x^2) = x^4$
$x^4 \pmod{x^3 + x + 1} = \mathbf{x^2 + x}$
Python Implementation (GF2 Poly Reduction)
def poly_to_int(poly_str):
return int(poly_str, 2)
def gf2_poly_mod(a, m):
# 'a' is the polynomial to reduce, 'm' is irreducible
# Both are integers representing binary coefficients
a_len = a.bit_length()
m_len = m.bit_length()
while a_len >= m_len:
# Align 'm' with the highest bit of 'a'
shift = a_len - m_len
a ^= (m << shift) # XOR reduction
a_len = a.bit_length()
return a
# Example: x^4 (10000) mod x^3 + x + 1 (1011)
result = gf2_poly_mod(0b10000, 0b1011)
print(f"Remainder (Binary): {bin(result)}") # 0b11 -> x + 1