Split 32-bit LFSR into two 16-bit LFSR which xor to same sequence Today I was thinking about exactly this. 

Is it possible, for a 32-bit LFSR e.g.

x^32 + x^30 + x^28 + x^26 + x^22 + x^21 + x^20 + x^16 + x^14 + x^12 + x^10 + x^8 + x^7 + x^6 + x^2 + x^1 + 1


to generate two 16-bit LFSR which xor to each other generate the 32-bit sequence?



edit:
my wording wasn't exactly

there is a LFSR sequence with a known plaintext of 64 bit e.g. 

1010111000000011
1111001010111001
0001010011010111
1001001111101101


it is known that this LFSR is built upon two 16bit LFSR which xor together give the same sequence as the LFSR above.

is it possible to generate the two unknown 16bit LFSR?

technically you have many lineary connections,

for example you know either the first bit in the first LFSR or the first bit in the second LFSR is 1
If a given 64-bit sequence is known to be the bit-by-bit XOR sum of
the outputs of two 16-bit LFSRs, then the degree-32 LFSR feedback polynomial
found by the Berlekamp-Massey algorithm is indeed the product of the
(16-bit) feedback polynomials of the two 16-bit LFSRs, and as @poncho said, 
you need to factor the degree-32 polynomial into two 16-degree polynomials.
But, be aware that while this is guaranteed to work when the two 16-degree
polynomials are irreducible, there can be issues if the two 16-degree
polynomials are not irreducible or are not relatively prime. For example,
it might be that the Berlekamp-Massey algorithm (which finds the shortest
LFSR that will generate a given sequence) might give you an LFSR of length
less than 32 when the algorithm is applied to the 64-bit sequence.