An experimental study of encrypted polynomial arithmetics for private set operations

Recent studies on the performance of private set operations have examined the use of homomorphic public-key encryption and the technique of representing sets as polynomials in a cryptographic model. These polynomial-based solutions require intensive polynomial arithmetic that exhibit quadratic computational complexity. In an effort to develop practical algorithms for private set operations, various well-known techniques such as Karatsuba's algorithm or the fast Fourier transform (FFT) can be used to reduce the complexity of polynomial computations. The FFT appears to be the obvious best choice; however, the use of FFTs in the implementation of polynomial-based set operations may lead to certain subtle technical problems. These problems become particularly serious when the cardinality of the sets is dynamic. Furthermore, our experiment shows that Karatsuba's algorithm delivers higher performance than the FFT in our application, provided that a reasonable response time is important. In addition, our experimental implementation demonstrates the heuristic bound on cardinality for which Karatsuba's algorithm outperforms the FFT. This value can be used to determine the superior optimization techniques and settings in the deployment of private set operation schemes.