I have an LFSR, essentially $x^k \mod p(x)$ for some characteristic polynomial with coefficients in GF(2), as outlined in Clark and Weng's article: it has a period (= order of the associated finite field) that is a "smooth" integer (factors are small).
I know how to compute discrete logarithms using Silver-Pohlig-Hellmann given a completely determined polynomial, so if you told me, Hey you! I have $$x^{23} + x^{16} + x ^ {14} + 1 \equiv x^k \mod p(x)$$ so what's $k$ ? Then I could tell you by following the algorithm.
But what if I don't know all of the coefficients, e.g. all I know is
$$a_{31}x^{31} + a_{30}x^{30} + a_{29}x^{29} + x^{23} + x^{16} + x ^ {14} + 1 \equiv x^k \mod p(x)$$
and I don't know the three unknown $a_j$'s (so $k$ has $2^3 = 8$ principal solutions). Aside from running through each of the possibilities, is there a way to figure out $k$ from one of them, and then determine the rest?
