I'm trying to understand what are the techniques to prove an exponential time lower bound.
For some problems, we can prove that the size of the output is exponential is the size of the input, thus it takes an exponential number of steps just to write the output. For example for some generalized board games, if the input is the size of the board and the output is the sequence of winning steps.
But what about decision problems, where $|output|=1$ ? Are there any decision problems in $EXPTIME$ with known exponential lower-bounds ?
I don't see how we can prove such a lower bound. Like if we take again the generalized board game and ask the question if there is a winning strategy, we can prove that evaluating all strategies will take (in the worst case) exponential time, but how can we prove that evaluating all strategies is the only way to answer the question ?
Or if I take another example, if my input is a number $x$ and the output is $1$ iff there exist a permutation $p$ such as shuffling the bits of $x$ makes it prime, i.e. $p(x)$ is prime. Exhaustively exploring all permutations and running a primality test will probably be exponential in $|x|$, but what if there is a smart way to answer just by looking at $x$ (in sub-exponential time) ?
I hope that looking at an example of such a proof will help me grasp the principles involved.