gf(2) Square-free factorization (sff)

Find and remove all squared factors from f.

f = b0110011001110111111000100000000101000110010100001000111110110111 f = 0x6677e20146508fb7 f = x⁶² + x⁶¹ + x⁵⁸ + x⁵⁷ + x⁵⁴ + x⁵³ + x⁵² + x⁵⁰ + x⁴⁹ + x⁴⁸ + x⁴⁷ + x⁴⁶ + x⁴⁵ + x⁴¹ + x³² + x³⁰ + x²⁶ + x²⁵ + x²² + x²⁰ + x¹⁵ + x¹¹ + x¹⁰ + x⁹ + x⁸ + x⁷ + x⁵ + x⁴ + x² + x + 1

Find the derivative of f (f')

f' = b0001000100010001010100010000000000000001000000000100010101010001 f' = 0x1111510001004551 f' = x⁶⁰ + x⁵⁶ + x⁵² + x⁴⁸ + x⁴⁶ + x⁴⁴ + x⁴⁰ + x²⁴ + x¹⁴ + x¹⁰ + x⁸ + x⁶ + x⁴ + 1

Since f' does not equal zero, find g=gcd(f,f')

g = b0000000000000000000000000000000000000000000000000000000000000001 g = 0x0000000000000001 g = 1

Since g equals one, f has no square factors.

Result (f factored into square-free factors)

f = (b0110011001110111111000100000000101000110010100001000111110110111) f = (0x6677e20146508fb7) f = (x⁶² + x⁶¹ + x⁵⁸ + x⁵⁷ + x⁵⁴ + x⁵³ + x⁵² + x⁵⁰ + x⁴⁹ + x⁴⁸ + x⁴⁷ + x⁴⁶ + x⁴⁵ + x⁴¹ + x³² + x³⁰ + x²⁶ + x²⁵ + x²² + x²⁰ + x¹⁵ + x¹¹ + x¹⁰ + x⁹ + x⁸ + x⁷ + x⁵ + x⁴ + x² + x + 1)