Fully Homomorphic Encryption (FHE) enables computations on encrypted data, proving itself to be an essential building block for privacy-preserving applications. However, it involves computationally demanding operations such as polynomial multiplication, with the Number Theoretic Transform (NTT) being the state-of-the-art solution to perform it. Considering that most FHE schemes operate over the negacyclic ring of polynomials, we introduce a novel formulation of the hierarchical Four-Step NTT approach for the negacyclic ring, eliminating the need for pre- and post-processing steps found in the existing methods. To accelerate NTT operations, the Field-Programmable Gate Array (FPGA) devices offer flexible and powerful computing platforms. We propose an FPGA-based, high-speed, parametric and fully pipelined architecture that implements the improved Seven-Step NTT algorithm, which builds upon the four-step algorithm. Our design supports a wide range of parameters, including ring sizes up to 216 and modulus sizes up to 64-bit. We focus on achieving configurable throughput, as constrained by the bandwidth of High-Bandwidth Memory (HBM), which is an additional in-package memory common in high-end FGPA devices such as Alveo U280. We aim to maximize throughput through an IO parametric design on the Alveo U280 FPGA. The implementation results demonstrate that the average latency of our design for batch NTT operation is 8.32μs for the ring size 216 and 64-bit width; a speed-up of 7.96 × compared to the current state-of-the-art designs.

Lattice-based cryptographic schemes such as Crystals-Kyber and Dilithium are post-quantum algorithms selected to be standardized by NIST as they are considered to be secure against quantum computing attacks. The multiplication in polynomial rings is the most time-consuming operation in many lattice-based cryptographic schemes, which is also subject to side-channel attacks. While NTT-based polynomial multiplication is almost a norm in a wide range of implementations, a relatively new method, incomplete NTT is preferred to accelerate lattice-based cryptography, especially on some computing platforms that feature special instructions. In this paper, we present a novel, efficient and non-profiled power/EM side-channel attack targeting polynomial multiplication based on the incomplete NTT algorithm. We apply the attack on the Crystals-Dilithium signature algorithm and Crystals-Kyber KEM. We demonstrate that the method accelerates attack run-time when compared to the existing approaches. While a conventional non-profiled side-channel attack tests a much larger hypothesis set because it needs to predict two coefficients of secret polynomials together, we propose a much fasterzero-value filtering attack(ZV-FA), which reduces the size of the hypothesis set by targeting the coefficients individually. We also propose an effective and efficient validation and correction technique employing the inverse NTT to estimate and modify the mispredicted coefficients. Our experimental results show that we can achieve a speed-up of 1915×over brute-force.