We propose a secure integer-wise homomorphic division algorithm on fully homomorphic encryption schemes (FHE). For integer-wise algorithms, we encrypt plaintexts as integers without encoding them into bit values, while in bit-wise algorithms, plaintexts are encoded into binary and bit values are encrypted one by one. All the publicly available division algorithms are constructed in bit-wise style, and to the best of our knowledge there are no known integer-wise algorithm for secure division. We derive some empirical results on the FHE library HElib and show that our algorithm is 2.45x faster than the fastest bit-wise algorithm. We also show that the multiplicative depth of our algorithm is O(l), where l is the integer bit length, while that of existing division algorithms is O(l2). Furthermore, we generalise our secure division algorithm and propose a method for secure calculation of a general 2-variable function. The order of multiplicative depth of the algorithm, which is a main factor of the complexity of a FHE algorithm, is exactly the same as our secure division algorithm.
CITATION STYLE
Okada, H., Cid, C., Hidano, S., & Kiyomoto, S. (2019). Linear depth integer-wise homomorphic division. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 11469 LNCS, pp. 91–106). Springer Verlag. https://doi.org/10.1007/978-3-030-20074-9_8
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