Shuffling cards so all orders are equally likely

Given a generator of uniform random numbers, how do you shuffle a deck of $n$ cards so that each of the $n!$ orderings has the same probability?

The Fisher-Yates shuffle, $O(n)$ time. For $i=n$ down to $2$, pick $j$ uniformly from $1,\dots,i$ and swap cards $i$ and $j$. The number of equally likely choice sequences is $n\cdot(n-1)\cdots2=n!$, one per permutation, so the result is uniform. Swapping each card with a position drawn from all $n$ is a common bug that biases the result.