I have been studying Wieschebrink paper "Cryptanalysis of the Niederreiter Public Key Scheme Based on GRS Codes". In the paper a cryptosystem using GRS codes is exhibited with an attack proposed to the cryptosystem, this one being the Sidelnikov-Shestakov attack (well, actually a reformulation from the original one that at least to me is easier to understand).
In the attack you try to recover the multipliers and evaluators that generates a GRS code equal to the original one. In a part of it you end with equations of the form:
$$ \frac{b_{1,j}}{b_{2,j}}(\alpha_j-\alpha_1)=\frac{c_{b_1}}{c_{b_i}}(\alpha_j-\alpha_2) $$
for a certains values of $j$ with $\alpha_j$ and $\frac{c_{b_1}}{c_{b_i}}$ as the unknowns. Your objective is to recover this $\alpha_j$, where $\alpha_j$ are the evaluators of the code. This equation cannot be solved directly (two unknwons and one equation) but in the attack they guess a value of $\frac{c_{b_1}}{c_{b_i}}$ and work through it.
Thing is, this process of guessing troubles me, as in the paper it is not explained why is guaranteed that you end successfully. I understand that if your guess is correct you succeed, but what happens when your guess is wrong? I suppose you could end recovering parameters that are not suitable for a GRS code, in this case, some $\alpha_j$ that is equal to $\alpha_i$ for $i\neq j$, and you would know your guess is wrong, but is there the possibility that you end up recovering parameters of a GRS code that is not "equal" to the original one? Thus, the attack would be a failure.
