SIMD HSS and aHMAC from Interval Encoding with Application to One-Bit-Per-Gate Garbling

Primitives enabling homomorphic computation over secret-shared values--Homomorphic Secret Sharing (HSS) and algebraic Homomorphic MACs (aHMAC)--have recently emerged as efficient alternatives to ciphertext-based primitives such as fully homomorphic encryption (FHE) and attribute-based encryption (ABE). Leveraging the distributed nature of secret sharing, direct constructions of HSS and aHMAC are simple, lightweight, avoid costly bootstrapping, and have many applications including one-bit-per-gate garbled circuits. Despite encouraging progress, all existing direct schemes still lack one key feature: efficient Single Instruction Multiple Data (SIMD) evaluation, a capability that has been critical to the efficiency of FHE. This gap leaves the potential of substantial efficiency improvements untapped. We present the first SIMD evaluation techniques for HSS and aHMAC, based on variants of the RLWE assumption. Using a new interval coefficient encoding, our approach embeds $\sqrt{n}$ integer-valued slots per ring element and supports $\sqrt{n}$-fold batch addition and multiplication in just $O(\log n)$ ring operations, achieving a multiplicative $\tilde O(\sqrt{n})$ improvement in amortized efficiency over prior direct constructions. Building on top of these improvements, we show a streamlined one-bit-per-gate SIMD garbling scheme with similar efficiency gains in the online phase. Our efficiency gains are concrete. Concrete operation counts and microbenchmark based estimates show $6\times$--$10\times$ improvements in amortized multiplication cost over prior non-SIMD constructions, with up to $25\times$--$50\times$ speedups for aggregation-heavy workloads such as matrix--vector multiplication. These results demonstrate the practical potential of SIMD techniques for secret-sharing-based homomorphic computation.