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

v₀ = x⁰ (mod f)
v₀ = b0000000000000000000000000000000000000000000000000000000000000001 v₀ = 0x0000000000000001 v₀ = 1
v₁ = x² (mod f)
v₁ = b0000000000000000000000000000000000000000000000000000000000000100 v₁ = 0x0000000000000004 v₁ = x²
v₂ = x⁴ (mod f)
v₂ = b0000000000000000000000000000000000000000000000000000000000010000 v₂ = 0x0000000000000010 v₂ = x⁴
v₃ = x⁶ (mod f)
v₃ = b0000000000000000000000000000000000000000000000000000000001000000 v₃ = 0x0000000000000040 v₃ = x⁶
v₄ = x⁸ (mod f)
v₄ = b0000000000000000000000000000000000000000000000000000000100000000 v₄ = 0x0000000000000100 v₄ = x⁸
v₅ = x¹⁰ (mod f)
v₅ = b0000000000000000000000000000000000000000000000000000010000000000 v₅ = 0x0000000000000400 v₅ = x¹⁰
v₆ = x¹² (mod f)
v₆ = b0000000000000000000000000000000000000000000000000001000000000000 v₆ = 0x0000000000001000 v₆ = x¹²
v₇ = x¹⁴ (mod f)
v₇ = b0000000000000000000000000000000000000000000000000100000000000000 v₇ = 0x0000000000004000 v₇ = x¹⁴
v₈ = x¹⁶ (mod f)
v₈ = b0000000000000000000000000000000000000000000000010000000000000000 v₈ = 0x0000000000010000 v₈ = x¹⁶
v₉ = x¹⁸ (mod f)
v₉ = b0000000000000000000000000000000000000000000000101010101111101101 v₉ = 0x000000000002abed v₉ = x¹⁷ + x¹⁵ + x¹³ + x¹¹ + x⁹ + x⁸ + x⁷ + x⁶ + x⁵ + x³ + x² + 1
v₁₀ = x²⁰ (mod f)
v₁₀ = b0000000000000000000000000000000000000000000000010101001110000011 v₁₀ = 0x0000000000015383 v₁₀ = x¹⁶ + x¹⁴ + x¹² + x⁹ + x⁸ + x⁷ + x + 1
v₁₁ = x²² (mod f)
v₁₁ = b0000000000000000000000000000000000000000000000111110010111100001 v₁₁ = 0x000000000003e5e1 v₁₁ = x¹⁷ + x¹⁶ + x¹⁵ + x¹⁴ + x¹³ + x¹⁰ + x⁸ + x⁷ + x⁶ + x⁵ + 1
v₁₂ = x²⁴ (mod f)
v₁₂ = b0000000000000000000000000000000000000000000000101100000001011110 v₁₂ = 0x000000000002c05e v₁₂ = x¹⁷ + x¹⁵ + x¹⁴ + x⁶ + x⁴ + x³ + x² + x
v₁₃ = x²⁶ (mod f)
v₁₃ = b0000000000000000000000000000000000000000000000001111110101001111 v₁₃ = 0x000000000000fd4f v₁₃ = x¹⁵ + x¹⁴ + x¹³ + x¹² + x¹¹ + x¹⁰ + x⁸ + x⁶ + x³ + x² + x + 1
v₁₄ = x²⁸ (mod f)
v₁₄ = b0000000000000000000000000000000000000000000000111111010100111100 v₁₄ = 0x000000000003f53c v₁₄ = x¹⁷ + x¹⁶ + x¹⁵ + x¹⁴ + x¹³ + x¹² + x¹⁰ + x⁸ + x⁵ + x⁴ + x³ + x²
v₁₅ = x³⁰ (mod f)
v₁₅ = b0000000000000000000000000000000000000000000000101000001100101010 v₁₅ = 0x000000000002832a v₁₅ = x¹⁷ + x¹⁵ + x⁹ + x⁸ + x⁵ + x³ + x
v₁₆ = x³² (mod f)
v₁₆ = b0000000000000000000000000000000000000000000000011111000010011111 v₁₆ = 0x000000000001f09f v₁₆ = x¹⁶ + x¹⁵ + x¹⁴ + x¹³ + x¹² + x⁷ + x⁴ + x³ + x² + x + 1
v₁₇ = x³⁴ (mod f)
v₁₇ = b0000000000000000000000000000000000000000000000010110100110010001 v₁₇ = 0x0000000000016991 v₁₇ = x¹⁶ + x¹⁴ + x¹³ + x¹¹ + x⁸ + x⁷ + x⁴ + 1

Q = 000000000000000001 000000000000000100 000000000000010000 000000000001000000 000000000100000000 000000010000000000 000001000000000000 000100000000000000 010000000000000000 101010101111101101 010101001110000011 111110010111100001 101100000001011110 001111110101001111 111111010100111100 101000001100101010 011111000010011111 010110100110010001

I = 000000000000000001 000000000000000010 000000000000000100 000000000000001000 000000000000010000 000000000000100000 000000000001000000 000000000010000000 000000000100000000 000000001000000000 000000010000000000 000000100000000000 000001000000000000 000010000000000000 000100000000000000 001000000000000000 010000000000000000 100000000000000000

M = 000000000000000000 000000000000000110 000000000000010100 000000000001001000 000000000100010000 000000010000100000 000001000001000000 000100000010000000 010000000100000000 101010100111101101 010101011110000011 111110110111100001 101101000001011110 001101110101001111 111011010100111100 100000001100101010 001111000010011111 110110100110010001

nv₀ = b0000000000000000000000000000000000000000000000100110101110101010 nv₀ = 0x0000000000026baa nv₀ = x¹⁷ + x¹⁴ + x¹³ + x¹¹ + x⁹ + x⁸ + x⁷ + x⁵ + x³ + x
nv₁ = b0000000000000000000000000000000000000000000000011100010001011110 nv₁ = 0x000000000001c45e nv₁ = x¹⁶ + x¹⁵ + x¹⁴ + x¹⁰ + x⁶ + x⁴ + x³ + x² + x
nv₂ = b0000000000000000000000000000000000000000000000000000000000000001 nv₂ = 0x0000000000000001 nv₂ = 1

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

nv₀ = b0000000000000000000000000000000000000000000000100110101110101010 nv₀ = 0x0000000000026baa nv₀ = x¹⁷ + x¹⁴ + x¹³ + x¹¹ + x⁹ + x⁸ + x⁷ + x⁵ + x³ + x

New factor (nf) = f/gcd(f,nv₀)

nf = b0000000000000000000000000000000000000000000000000000000101110001 nf = 0x0000000000000171 nf = x⁸ + x⁶ + x⁵ + x⁴ + 1
Search though known factors (kf) for which (nf) factors evenly

kf₀ = b0000000000000000000000000000000000000000000001101010101111101101 kf₀ = 0x000000000006abed kf₀ = x¹⁸ + x¹⁷ + x¹⁵ + x¹³ + x¹¹ + x⁹ + x⁸ + x⁷ + x⁶ + x⁵ + 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 = b0000000000000000000000000000000000000000000000000000011111011101 t_quotient = 0x00000000000007dd t_quotient = x¹⁰ + x⁹ + x⁸ + x⁷ + x⁶ + x⁴ + x³ + x² + 1

nv₁ = b0000000000000000000000000000000000000000000000011100010001011110 nv₁ = 0x000000000001c45e nv₁ = x¹⁶ + x¹⁵ + x¹⁴ + x¹⁰ + x⁶ + x⁴ + x³ + x² + x

New factor (nf) = f/gcd(f,nv₁)

nf = b0000000000000000000000000000000000000000000000000011111101100111 nf = 0x0000000000003f67 nf = x¹³ + x¹² + x¹¹ + x¹⁰ + x⁹ + x⁸ + x⁶ + x⁵ + x² + x + 1
Search though known factors (kf) for which (nf) factors evenly

kf₀ = b0000000000000000000000000000000000000000000000000000000101110001 kf₀ = 0x0000000000000171 kf₀ = x⁸ + x⁶ + x⁵ + x⁴ + 1
t = kf₀/nf

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

t_remainder != 0, nf is not a factor.

kf₁ = b0000000000000000000000000000000000000000000000000000011111011101 kf₁ = 0x00000000000007dd kf₁ = x¹⁰ + x⁹ + x⁸ + x⁷ + x⁶ + x⁴ + x³ + x² + 1
t = kf₁/nf

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

t_remainder != 0, nf is not a factor.

kf₀ = b0000000000000000000000000000000000000000000000000000000101110001 kf₀ = 0x0000000000000171 kf₀ = x⁸ + x⁶ + x⁵ + x⁴ + 1
kf₁ = b0000000000000000000000000000000000000000000000000000011111011101 kf₁ = 0x00000000000007dd kf₁ = x¹⁰ + x⁹ + x⁸ + x⁷ + x⁶ + x⁴ + x³ + x² + 1

f = (b101110001)(b11111011101) f = (0x171)(0x7dd) f = (x⁸ + x⁶ + x⁵ + x⁴ + 1)(x¹⁰ + x⁹ + x⁸ + x⁷ + x⁶ + x⁴ + x³ + x² + 1)

gf(2) Berlekamp Algorithm