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

v₀ = x⁰ (mod f)
v₀ = b00000000000000000000000000000001 v₀ = 0x00000001 v₀ = 1
v₁ = x² (mod f)
v₁ = b00000000000000000000000000000100 v₁ = 0x00000004 v₁ = x²
v₂ = x⁴ (mod f)
v₂ = b00000000000000000000000000010000 v₂ = 0x00000010 v₂ = x⁴
v₃ = x⁶ (mod f)
v₃ = b00000000000000000000000001000000 v₃ = 0x00000040 v₃ = x⁶
v₄ = x⁸ (mod f)
v₄ = b00000000000000000000000100000000 v₄ = 0x00000100 v₄ = x⁸
v₅ = x¹⁰ (mod f)
v₅ = b00000000000000000000001111011101 v₅ = 0x000003dd v₅ = x⁹ + x⁸ + x⁷ + x⁶ + x⁴ + x³ + x² + 1
v₆ = x¹² (mod f)
v₆ = b00000000000000000000000011001110 v₆ = 0x000000ce v₆ = x⁷ + x⁶ + x³ + x² + x
v₇ = x¹⁴ (mod f)
v₇ = b00000000000000000000001100111000 v₇ = 0x00000338 v₇ = x⁹ + x⁸ + x⁵ + x⁴ + x³
v₈ = x¹⁶ (mod f)
v₈ = b00000000000000000000001101011010 v₈ = 0x0000035a v₈ = x⁹ + x⁸ + x⁶ + x⁴ + x³ + x
v₉ = x¹⁸ (mod f)
v₉ = b00000000000000000000001011010010 v₉ = 0x000002d2 v₉ = x⁹ + x⁷ + x⁶ + x⁴ + x

Q = 0000000001 0000000100 0000010000 0001000000 0100000000 1111011101 0011001110 1100111000 1101011010 1011010010

I = 0000000001 0000000010 0000000100 0000001000 0000010000 0000100000 0001000000 0010000000 0100000000 1000000000

M = 0000000000 0000000110 0000010100 0001001000 0100010000 1111111101 0010001110 1110111000 1001011010 0011010010

nv₀ = b00000000000000000000001001001100 nv₀ = 0x0000024c nv₀ = x⁹ + x⁶ + x³ + x²
nv₁ = b00000000000000000000000000000001 nv₁ = 0x00000001 nv₁ = 1

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

nv₀ = b0000000000000000000000000000000000000000000000000000001001001100 nv₀ = 0x000000000000024c nv₀ = x⁹ + x⁶ + x³ + x²

New factor (nf) = f/gcd(f,nv₀)

nf = b0000000000000000000000000000000000000000000000000000000000110111 nf = 0x0000000000000037 nf = x⁵ + x⁴ + x² + x + 1
Search though known factors (kf) for which (nf) factors evenly

kf₀ = b0000000000000000000000000000000000000000000000000000011111011101 kf₀ = 0x00000000000007dd kf₀ = x¹⁰ + x⁹ + x⁸ + x⁷ + x⁶ + x⁴ + x³ + x² + 1
t = kf₀/nf

t_remainder = b0000000000000000000000000000000000000000000000000000000000000000 t_remainder = 0x0000000000000000 t_remainder = 0

t_remainder == 0, nf is a factor of kf₀

kf₀ = nf // Replace kf₀ with nf
kf₁ = t_quotient // Append t_quotient to known factors list

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

kf₀ = b0000000000000000000000000000000000000000000000000000000000110111 kf₀ = 0x0000000000000037 kf₀ = x⁵ + x⁴ + x² + x + 1
kf₁ = b0000000000000000000000000000000000000000000000000000000000101111 kf₁ = 0x000000000000002f kf₁ = x⁵ + x³ + x² + x + 1

f = (b110111)(b101111) f = (0x37)(0x2f) f = (x⁵ + x⁴ + x² + x + 1)(x⁵ + x³ + x² + x + 1)

gf(2) Berlekamp Algorithm