 Solution

A binary linear-feedback shift register of length $L$ keeps $L$ bits and produces a sequence satisfying a recurrenceAfter each output it shifts its stored bits and inserts the feedback bit. Its connection polynomial is $C(z)=1+c_{1}z+\cdots +c_L{z}^L$. This convention reverses the usual characteristic polynomial $z^L+c_1{z}^{L-1}+\cdots +c_L$.

The Berlekamp-Massey algorithm recovers the shortest recurrence compatible with observed bits. Maintain the current $C$, length $L$, an earlier polynomial $B$ that last caused a length increase, and the offset $m$ since that increase. Initially $C=B=1,L=0,m=1$. At position $n$ calculate the discrepancy $d=s_n+{\sum }_{j=1}^L{c}_j{s}_{n-j}$. If $d=0$, advance $m$. If $d=1$, save $T=C$ and replace $C$ by $C+z^{m}B$. If also $2L\le n$, replace $L$ by $n+1-L$, $B$ by $T$ and $m$ by one; otherwise increase $m$. Over a general field the correction is $(d/b)z^{m}B$, where $b$ is the saved nonzero discrepancy. The correction cancels the failed equation while retaining the previous successful ones; when it forces a length increase the smallest possible new length is $n+1-L$.

Here, indexing the first bit by zero, the nonzero discrepancies occur at positions $1,4,6$. The corresponding updates areEvery subsequent supplied discrepancy is zero. ThusEquivalently the characteristic feedback polynomial is $z^4+z+1$. No length-three recurrence works: positions $3,4,5$ force $c_2=1,c_1=0,c_3=1$, but position six then has nonzero discrepancy. The recovered length is therefore minimal. Observing at least twice the true linear complexity ordinarily identifies the recurrence, after which a known consecutive block determines all future key bits. This is why a bare register is unsuitable as a cryptographic keystream generator; a finite prefix alone does not exclude a longer generator that later diverges.