Time-Area Efficient RNS Base-conversion Architecture for HPS-BFV Homomorphic Encryption using Generalized Solinas Primes

초록

The most expensive operation in the BFV Homomorphic Operation scheme is the tensor multiplication part of the homomorphic multiplication, as it requires polynomial multiplication of two ciphertexts without modular reduction four times. The requirement to do it without modular reduction, making the usual technique to accelerate polynomial multiplication, the Number Theoretic Transform and its inverse (NTT/INTT), cannot be used. It means that the complexity cannot be efficiently reduced from O(n2) to O(nlogn), which is very inefficient in hardware settings. The residue number system (RNS) scheme, which allows base conversion, can be used to overcome this problem, which Halevi, Polyakov, and Shoup (HPS) formalize as one of the variants of the BFV scheme. To make NTT/INTT available for polynomial operations in the homomorphic multiplication, the input ciphertexts' base RNS must be extended from base q = q0,q1,q2,...qk-1 to base Q, where Q = q ∪ p and p is an additional set of prime moduli p = p0,p1,p2,...pl-1. In this paper, we propose an efficient architecture for converting an RNS number from base q to Q, by using a specialized set of prime numbers called the Generalized Solinas Prime applied in sets of q and p. They have a small absolute Hamming weight in the Canonical Signed Digit (CSD) representation, making modular reduction on them easy to do by using add, subtract, and shift operations. © 2026 IEEE.

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