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We prove that a uniformized variant of both the Rosenthal walk\cite{Rosenthal} and the Kac random walk \cite{Kac} on SO(n) mixes in$\cO(n^3)$ steps in total variation distance. The proof also extends easily toRosenthal walk with fixed angle $\theta \neq \pi$. To the best of ourknowledge, this is the first polynomial time bound for both walks. Thetechniques employed are mainly from representation theory of SO(n). But acrucial new ingredient is the interpretation of the Fourier coefficients of thecharacter ratio as counting the number of particle cascade paths arising fromthe classical branching rules.

Mixing time upper bound for the uniformized Rosenthal walk on the special orthogonal groups