Performance Analysis of Sorting of FHE Data: Integer-Wise Comparison vs Bit-Wise Comparison

In this paper, we present the exact method for an integer-wise comparison technique in FHE domain using polynomial interpolaion and analyze the performance against bit-wise comparison techniques. We observe that even though the integer-wise comparison requires only one ciphertext unit for integer in contrast to l ciphertext units for an l-bit integer for bit-wise comparison, bit-wise comparison schemes have better performance due to less multiplicative depth of the comparison circuit. Our analysis shows that bit-wise comparison based on depth optimized circuits have O(log(l)) multiplicative depth, where as the integer-wise comparison has O(l) multiplicative depth and bit-wise comparison based on two's complement arithmetic techniques has O(l) multiplicative depth. We have evaluated the performance of odd-even merge sort and direct sort by considering all the three above stated comparison techniques on FHE data using HElib library and analyzed their complexities.