4 research outputs found
Efficient Homomorphic Comparison Methods with Optimal Complexity
Comparison of two numbers is one of the most frequently used operations, but it has been a challenging task to efficiently compute the comparison function in homomorphic encryption (HE) which basically support addition and multiplication.
Recently, Cheon et al. (Asiacrypt 2019) introduced a new approximate representation of the comparison function with a rational function, and showed that this rational function can be evaluated by an iterative algorithm. Due to this iterative feature, their method achieves a logarithmic computational complexity compared to previous polynomial approximation methods; however, the computational complexity is still not optimal, and the algorithm is quite slow for large-bit inputs in HE implementation.
In this work, we propose new comparison methods with optimal asymptotic complexity based on composite polynomial approximation. The main idea is to systematically design a constant-degree polynomial by identifying the \emph{core properties} to make a composite polynomial get close to the sign function (equivalent to the comparison function) as the number of compositions increases. We additionally introduce an acceleration method applying a mixed polynomial composition for some other polynomial with different properties instead of . Utilizing the devised polynomials and , our new comparison algorithms only require computational complexity to obtain an approximate comparison result of satisfying within error.
The asymptotic optimality results in substantial performance enhancement: our comparison algorithm on encrypted -bit integers for takes milliseconds in amortized running time, which is times faster than the previous work