Björklund, Husfeldt, and Kaski gave a randomized $\widetilde{O}(n^{ω+3})$-time algorithm for computing the length of a shortest even cycle in an $n$-vertex directed graph, where $ω$ is the square matrix multiplication exponent. Their algebraic framework evaluates a parity cycle-cover enumerator over a field of characteristic two by lifting the computation to a ring of characteristic four. We give a direct evaluator for nonsingular matrices that stays in characteristic two. If $A$ is nonsingular over a field of characteristic two, the sum of the monomials indexed by odd permutations can be computed deterministically using $O(n^3)$ field operations. The key is an inverse-based closed form for a matrix with two proportional rows, combined with row elimination and dynamic inverse maintenance. For a directed graph, we apply the evaluator to $I+zW$, where $W$ is a randomly weighted adjacency matrix. The determinant $\det(I+zW)$ has constant coefficient one, so among any $2n+1$ distinct field elements at least $n+1$ yield nonsingular matrices. Evaluating at these points and interpolating gives a Monte Carlo algorithm that computes the shortest even-cycle length in $\widetilde{O}(n^4)$ time, with error probability $O(n^{-3})$. The error is one-sided with respect to existence: on a graph without an even cycle the algorithm always reports that fact. The result concerns the length; the standard self-reduction produces a cycle in $\widetilde{O}(n^5)$ time.

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