You might want to read about fully homomorphic encryption, the PCP theorem and many other mind-blowing results to have come out of computational complexity theory community in the past 15 years. These results use powerful mathematical tools like Fourier analysis on the boolean cube. Here are a few links to get you started:

http://www.scottaaronson.com/papers/philos.pdf

http://www.ccs.neu.edu/home/jmitesh/Mitesh_Jain_files/draft....

http://theoryofcomputing.org/articles/gs001/gs001.pdf