An Efficient Method for Comparing Numbers and Determining the Sign of a Number in RNS for Even Ranges

Abstract

Fully Homomorphic Encryption (FHE) permits processing information in the form of ciphertexts without decryption. It can ensure the security of information in common technologies used today, such as cloud computing, the Internet of Things, and machine learning, among others. A primary disadvantage for its practical application is the low efficiency of sign and comparison operations. Several FHE schemes use the Residue Number System (RNS) to decrease the time complexity of these operations. Converting from the RNS to the positional number system and calculating the positional characteristic of a number are standard approaches for both operations in the RNS domain. In this paper, we propose a new method for comparing numbers and determining the sign of a number in RNS. We focus on the even ranges that are computationally simple due to their peculiarities. We compare the performance of several state-of-art algorithms based on an implementation in C++ and relatively simple moduli with a bit depth from 24 to 64 bits. The experimental analysis shows a better performance of our approach for all the test cases; it improves the sign detection between 1.93 and 15.3 times and the number comparison within 1.55–11.35 times with respect to all the methods and configurations.

Privacy Preserving Epigenetic PaceMaker Stronger Privacy and Improved Efficiency

“…Formally, for a given set of primes 𝒫 , there exists two distinct integers, x and y , such that − N < y < 0 < x < N , yet x mod 𝒫 = y mod 𝒫 . To tackle this sign issue, we follow [6,31], and use an additional bit. This extra bit can be employed as a sign bit, ensuring that x mod 𝒫 and y mod 𝒫 possess unique representations.…”

“…Despite this, due to the modulo operations inherent in our system, our value image range is confined to [0, N − 1], where N is the product of all primes used. Formally, for a given set of primes P, there exists two distinct integers, x and y, such that −N < y < 0 < x < N , yet x mod P = y mod P. To tackle this sign issue, we follow [6,31], and use an additional bit. This extra bit can be employed as a sign bit, ensuring that x mod P and y mod P possess unique representations.…”

Performance Analysis of Hardware Implementations of Reverse Conversion from the Residue Number System

“…The architecture of computing systems operating in RNS generally consists of forward conversion of positional numbers into RNS [3], computation of the main problem using modular addition [4], multiplication [5], number comparison and sign determination [6] operations, reconversion from RNS to positional number system [7][8][9][10][11][12][13][14]. Furthermore, if addition, subtraction, multiplication in RNS have a parallel structure with numbers of small dimensions, then the operations of sign determination, comparison, conversion from RNS to the positional number system require the calculation of the positional characteristic of the number, which is computationally complex.…”