gf(2) Iterate until all irreducible factors of f₀ are found

f₀ = b0000000000000000111101111101100100100100100111111111001100111111 f₀ = 0x0000f7d9249ff33f f₀ = x⁴⁷ + x⁴⁶ + x⁴⁵ + x⁴⁴ + x⁴² + x⁴¹ + x⁴⁰ + x³⁹ + x³⁸ + x³⁶ + x³⁵ + x³² + x²⁹ + x²⁶ + x²³ + x²⁰ + x¹⁹ + x¹⁸ + x¹⁷ + x¹⁶ + x¹⁵ + x¹⁴ + x¹³ + x¹² + x⁹ + x⁸ + x⁵ + x⁴ + x³ + x² + x + 1

Find at least one irreducible factor of f₀, if it exists.

Remove f₀ from list of resulting factors.
Add:

f₁ = b0000000000000000000000000000000000000000001001111000101010011001 f₁ = 0x0000000000278a99 f₁ = x²¹ + x¹⁸ + x¹⁷ + x¹⁶ + x¹⁵ + x¹¹ + x⁹ + x⁷ + x⁴ + x³ + 1
f₂ = b0000000000000000000000000000000000000000000000000000000011000001 f₂ = 0x00000000000000c1 f₂ = x⁷ + x⁶ + 1
f₃ = b0000000000000000000000000000000000000000000001101010101111101101 f₃ = 0x000000000006abed f₃ = x¹⁸ + x¹⁷ + x¹⁵ + x¹³ + x¹¹ + x⁹ + x⁸ + x⁷ + x⁶ + x⁵ + x³ + x² + 1
f₄ = b0000000000000000000000000000000000000000000000000000000000000011 f₄ = 0x0000000000000003 f₄ = x + 1

f₁ = b0000000000000000000000000000000000000000001001111000101010011001 f₁ = 0x0000000000278a99 f₁ = x²¹ + x¹⁸ + x¹⁷ + x¹⁶ + x¹⁵ + x¹¹ + x⁹ + x⁷ + x⁴ + x³ + 1

Find at least one irreducible factor of f₁, if it exists.

f₁ is irreducible.

f₂ = b0000000000000000000000000000000000000000000000000000000011000001 f₂ = 0x00000000000000c1 f₂ = x⁷ + x⁶ + 1

Find at least one irreducible factor of f₂, if it exists.

f₂ is irreducible.

f₃ = b0000000000000000000000000000000000000000000001101010101111101101 f₃ = 0x000000000006abed f₃ = x¹⁸ + x¹⁷ + x¹⁵ + x¹³ + x¹¹ + x⁹ + x⁸ + x⁷ + x⁶ + x⁵ + x³ + x² + 1

Find at least one irreducible factor of f₃, if it exists.

Remove f₃ from list of resulting factors.
Add:

f₅ = b0000000000000000000000000000000000000000000000000000000101110001 f₅ = 0x0000000000000171 f₅ = x⁸ + x⁶ + x⁵ + x⁴ + 1
f₆ = b0000000000000000000000000000000000000000000000000000011111011101 f₆ = 0x00000000000007dd f₆ = x¹⁰ + x⁹ + x⁸ + x⁷ + x⁶ + x⁴ + x³ + x² + 1

f₄ = b0000000000000000000000000000000000000000000000000000000000000011 f₄ = 0x0000000000000003 f₄ = x + 1

Find at least one irreducible factor of f₄, if it exists.

f₄ is irreducible.

f₅ = b0000000000000000000000000000000000000000000000000000000101110001 f₅ = 0x0000000000000171 f₅ = x⁸ + x⁶ + x⁵ + x⁴ + 1

Find at least one irreducible factor of f₅, if it exists.

f₅ is irreducible.

f₆ = b0000000000000000000000000000000000000000000000000000011111011101 f₆ = 0x00000000000007dd f₆ = x¹⁰ + x⁹ + x⁸ + x⁷ + x⁶ + x⁴ + x³ + x² + 1

Find at least one irreducible factor of f₆, if it exists.

Remove f₆ from list of resulting factors.
Add:

f₇ = b0000000000000000000000000000000000000000000000000000000000110111 f₇ = 0x0000000000000037 f₇ = x⁵ + x⁴ + x² + x + 1
f₈ = b0000000000000000000000000000000000000000000000000000000000101111 f₈ = 0x000000000000002f f₈ = x⁵ + x³ + x² + x + 1

f₇ = b0000000000000000000000000000000000000000000000000000000000110111 f₇ = 0x0000000000000037 f₇ = x⁵ + x⁴ + x² + x + 1

Find at least one irreducible factor of f₇, if it exists.

f₇ is irreducible.

f₈ = b0000000000000000000000000000000000000000000000000000000000101111 f₈ = 0x000000000000002f f₈ = x⁵ + x³ + x² + x + 1

Find at least one irreducible factor of f₈, if it exists.

f₈ is irreducible.

Irreducible factors:

f₀ = (b1001111000101010011001)(b11000001)(b11)(b101110001)(b110111)(b101111) f₀ = (0x278a99)(0xc1)(0x3)(0x171)(0x37)(0x2f) f₀ = (x²¹ + x¹⁸ + x¹⁷ + x¹⁶ + x¹⁵ + x¹¹ + x⁹ + x⁷ + x⁴ + x³ + 1)(x⁷ + x⁶ + 1)(x + 1)(x⁸ + x⁶ + x⁵ + x⁴ + 1)(x⁵ + x⁴ + x² + x + 1)(x⁵ + x³ + x² + x + 1)

gf(2) Iterate until all irreducible factors of f0 are found

gf(2) Iterate until all irreducible factors of f₀ are found