  The proof of Toda's celebrated theorem that the polynomial hierarchy is
contained in $\P^{# P}$ relies on the fact that, under mild technical
conditions on the complexity class $C$, we have $\exists C \subset BP \cdot
\oplus C$. More concretely, there is a randomized reduction which transforms
nonempty sets and the empty set, respectively, into sets of odd or even size.
The customary method is to invoke Valiant's and Vazirani's randomized reduction
from NP to UP, followed by amplification of the resulting success probability
from $1/\poly(n)$ to a constant by combining the parities of $\poly(n)$ trials.
Here we give a direct algebraic reduction which achieves constant success
probability without the need for amplification. Our reduction is very simple,
and its analysis relies on well-known properties of the Legendre symbol in
finite fields.
