We give a Las Vegas algorithm which computes the shifted Popov form of a nonsingular polynomial matrix $A \in \mathbb{K}^{ m \times m }$ in expected $O\tilde{~}( m^w \lceil \sigma(A) / m \rceil ) \subseteq O\tilde{~}( m^w \deg(A) )$ operations in $\mathbb{K}$, where $\deg(A)$ is the degree of $A$, $\sigma(A)$ is some quantity such that $\sigma(A) / m$ is bounded from above by both the average row degree and the average column degree of $A$, $w$ is the exponent of matrix multiplication, and $O\tilde{~}( \cdot )$ indicates that logarithmic factors are omitted. This improves upon the cost bound of the fastest known algorithms for row reduction and Hermite form computation, which are deterministic. This is the first algorithm for shifted row reduction with cost bound $O\tilde{~}( m^w \deg(A) )$ for an arbitrary shift. This algorithm uses partial linearization to reduce to the case $\deg(A) \le \sigma(A)$, and builds a system of modular equations whose solution set is the row space of $A$. It remains to find the basis in shifted Popov form of this solution set: we give a deterministic algorithm for this problem in $O\tilde{~}( m^{w-1} \sigma )$ operations, where $m$ is the number of unknowns and $\sigma$ is the sum of the degrees of the moduli. This extends previous results with the same cost bound in the specific cases of order basis computation and M-Pad\'e approximation, in which the moduli are products of known linear factors.