No, you don't have a chance. Even if you did find a more efficient way to complete the algorithm, what's stopping those big farms from using that new technique too?
I think this is the ONLY response in this thread so far that has understood what the OP was even asking.
Note: The OP is not asking to change bitcoin to use something other than SHA256. The OP is asking if it is possible to find a more efficient way to compute SHA256.
The answer to the OP is, YES. It is possible to be more efficient.
As a matter of fact, there are MANY efficiencies that have already been discovered and implemented by most of the mining community. For example, SHA256 is no longer computed from the beginning on every nonce change. It was noticed that several steps at the beginning of the hash process had the exact same result as the nonce changed. Now, this result is only computed once (it's called a "midstate"). Then the miner modifies the nonce and continues the calculation FROM THE MIDSTATE, returning to the MIDSTATE each time he doesn't find a successful hash. The midstate is only recomputed after all the available nonces have been attempted and something else in the block needs to change (such as the timestamp or the extranonce).
If the OP is an exceptional mathematician, then he may discover a new mathematical shortcut that nobody else has noticed yet. This would give him an advantage until others discovered the same shortcut.
OK, would he not need to re-program his ASICs in order to perform this new shortcut?
Or existing ASICs can just follow a modified version of CGminer let's say?
Excuse my ignorance by the way