Computer hardware

Algorithm

Scheme (mathematics)

Embedded system

Theoretical computer science

Encryption

Telecommunications

Parallel computing

Computer science

Visual arts

Homomorphic encryption

Architecture

Computer network

Software

Computer architecture

Mathematics

Computer engineering

Mathematical analysis

Programming language

Wireless

Throughput

Field-programmable gate array

Hardware architecture

Residue number system

Arithmetic

Homomorphic encryption (HE) is one of the most promising solutions to secure cloud computing. The number theoretic transform (NTT) that is widely used for convolution operations in HE requires a large amount of computation and has high parallelism, and therefore it has been a good candidate for hardware acceleration. Nevertheless, prior NTT hardware solutions for HE-based applications are impractical in most applications because they do not seriously consider the critical bootstrapping procedure that allows unlimited homomorphic operations on encrypted data. In this paper, we suggest practical bootstrappable parameters, specifically for an established residue number system (RNS)based HE scheme, and apply them to our NTT hardware design. In addition, to limit the size of internal memory for roots of unity increased by the bootstrappable parameters, only a few roots of unity are stored and others are generated on the fly. In our NTT hardware architecture, multiple NTT butterfly units (BUs) are efficiently deployed for high throughput and high resource utilization. In particular, several groups of BUs for respective moduli work in a parallel and pipelined manner, which is effective in an RNS-based HE scheme with a number of moduli. Our implementation on a Xilinx UltraScale FPGA with the bootstrappable parameters achieves a $118 \times$ faster processing speed than a software implementation, and it further provides various trade-off choices such as the number of DSP slices against BRAMs based on available FPGA resources.

Hardware Architecture of a Number Theoretic Transform for a Bootstrappable RNS-based Homomorphic Encryption Scheme