  We consider the problem of binary string reconstruction from the multiset of
its substring compositions, i.e., referred to as the substring composition
multiset, first introduced and studied by Acharya et al. We introduce a new
algorithm for the problem of string reconstruction from its substring
composition multiset which relies on the algebraic properties of the equivalent
bivariate polynomial formulation of the problem. We then characterize specific
algebraic conditions for the binary string to be reconstructed that guarantee
the algorithm does not require any backtracking through the reconstruction,
and, consequently, the time complexity is bounded polynomially. More
specifically, in the case of no backtracking, our algorithm has a time
complexity of $O(n^2)$ compared to the algorithm by Acharya et al., which has a
time complexity of $O(n^2\log(n))$, where $n$ is the length of the binary
string. Furthermore, it is shown that larger sets of binary strings are
uniquely reconstructable by the new algorithm and without the need for
backtracking leading to codebooks of reconstruction codes that are larger, by a
linear factor in size, compared to the previously known construction by
Pattabiraman et al., while having $O(n^2)$ reconstruction complexity.
