gf(2) sqrt(f)

f = b0000000000000000000000000000000000000000000000000001010000010101
f = 0x0000000000001415
f = x12 + x10 + x4 + x2 + 1

Separate each bit of f. f=
0000000000000000000000000000000000000000000000000000000000000001 0000000000000001 1
0000000000000000000000000000000000000000000000000000000000000100 0000000000000004 x2
0000000000000000000000000000000000000000000000000000000000010000 0000000000000010 x4
0000000000000000000000000000000000000000000000000000010000000000 0000000000000400 x10
0000000000000000000000000000000000000000000000000001000000000000 0000000000001000 x12

Shift each bit right by half the degree. sqrt(f)=
0000000000000000000000000000000000000000000000000000000000000001 0000000000000001 1
0000000000000000000000000000000000000000000000000000000000000010 0000000000000002 x
0000000000000000000000000000000000000000000000000000000000000100 0000000000000004 x2
0000000000000000000000000000000000000000000000000000000000100000 0000000000000020 x5
0000000000000000000000000000000000000000000000000000000001000000 0000000000000040 x6

sqrt(f) = b0000000000000000000000000000000000000000000000000000000001100111
sqrt(f) = 0x0000000000000067
sqrt(f) = x6 + x5 + x2 + x + 1