Finding non-zero coefficients in 8 bit polynomial LFSR I have a following crypto problem and I have been struggling with for some time

An 8-bit linear feedback shift register with connection polynomial
C(X) = 1+aX +bX^2 +cX^3 +dX^4 +eX^5 + fX^6 +gX^7 +hX^8
is used to generate a pseudo-random binary sequence. This pseudo-random
sequence is used as the enciphering key of a stream cipher. It is known that
when the cipher is applied to the plaintext string
[0,0,0,1,0,0,0,0,1,1,0,0,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,1,0,1]
the corresponding ciphertext string is
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,1,0,0]

What would be the approach to determine all non-zero coefficients in this polynomial? I have no clue how to start...

Thanks in advance
Berlekamp Massey algorithm does this for you.

See Dilip Sarwate's nice answer