Efficient FHE Radix-2 Arithmetic Operations Based on Redundant Encoding

Fully homomorphic encryption (FHE) is a novel encryption method that can perform operations on encrypted data. The performance of applications based on FHE is still low due to the high computational complexity of operations on the ciphertext. This article combines the characteristics of FHE and the redundant encoding method to achieve faster radix-2 arithmetic operations in BGV-like schemes. First, a ciphertext integer addition with multiplicative depth two, namely, redundant carry-free addition (RCFA), is proposed by applying the carry-free feature of the redundant encoding method. This addition is
$7.207\times $
faster and uses 15.0% of the occupied memory at 512 bits compared with the nonredundant method. After utilizing the single-instruction–multiple-data (SIMD) technique, RCFA with SIMD further improves the efficiency by
$3903\times $
at 512 bits compared with the nonredundant SIMD method. Its ciphertext size and occupied memory are only 34.59% and 1.6% those of the nonredundant SIMD method. Second, to achieve efficient ciphertext multiplications, redundant multiplication (RM) of dual SIMD data (RMDS) and RM of SIMD and non-SIMD data (RMSNS) methods are proposed; they obtain speedups of
$65.6\times $
and
$631.0\times $
, respectively, at 32 bits compared with the nonredundant implementations. Finally, SHA-256 is implemented to validate the efficiency of the proposed arithmetic operations. Compared with the nonredundant SIMD method, this article obtains a speedup of
$4.34\times $
.