Systolic finite field multiplier over GF(2m), because of its superior features such as high throughput and regularity, is highly desirable for many demanding cryptosystems. On the other side, however, obtaining high-performance systolic multiplier with relatively low hardware cost is still a challenging task due to the fact that the systolic structure usually involves large area complexity. Based on this consideration, in this paper, we propose to carry out two novel coherent interdependent efforts. First, a new digit-serial multiplication algorithm based on polynomial basis over binary field (GF(2m)) is proposed. Novel Toeplitz matrix-vector product (TMVP)-based decomposition strategy is employed to derive an efficient subquadratic space complexity. Second, The proposed algorithm is then innovatively mapped into a low-complexity systolic multiplier, which involves less area-time complexities than the existing ones. A series of resource optimization techniques also has been applied on the multiplier which optimizes further the proposed design (it is the first report on digit-serial systolic multiplier based on TMVP approach covering all irreducible polynomials, to the best of our knowledge). The following complexity analysis and comparison confirm the efficiency of the proposed multiplier, that is, it has lower area-delay product (ADP) than the existing ones. The extension of the proposed multiplier for bit-parallel implementation is also considered in this paper.

Significant innovation has been made in the development of public-key cryptography that is able to withstand quantum attacks, known as post-quantum cryptography (PQC). This paper focuses on the development of an efficient PQC hardware implementation. Specifically, an implementation of the binary Ring-learning-with-errors (BRLWE)-based encryption scheme, a promising lightweight PQC suitable for resource-constrained applications, is proposed. The paper first develops the mathematical formulation to present the proposed algorithmic process. The corresponding hardware accelerators are then described in detail. Finally, comparisons with previous implementations are provided to demonstrate the superior performance of the proposed design. For instance, the proposed low-complexity accelerator has 34.7% less area-delay product (ADP) than the state-of-the-art design for $n=256$ in the field-programmable gate array (FPGA) platform. Apart from the efficiency of the hardware architectures, the proposed design also has a complete input/output processing setup, and thus is feasible for emerging lightweight applications.

Post-quantum cryptography (PQC) refers to the cryptosystem that can resist the attacks launched from mature quantum computers in the not far future and has recently gained intensive attention from the research community as most of the existing public-key cryptosystems are vulnerable to attacks from quantum computers. Ring-Learning-with-Errors (Ring-LWE)-based scheme is an essential type of the lattice-based PQC due to its strong security proof and ease of implementation. As the latest variant of the Ring-LWE, the binary Ring-LWE (BRLWE)-based scheme possesses even smaller computational complexity and thus is more suitable for resource-constrained applications. However, the existing works have not well covered various aspects related to this new scheme, especially on the low-complexity hardware implementation. In this paper, we aim to present a novel implementation of the BRLWE-based scheme on the hardware platform with very low-complexity with this point of view. To carry out the specified work in a successful manner, we have proposed mainly four layers of coherent interdependent efforts: (i) we have provided the necessary algorithmic derivation process in detail to formulate the desired algorithm for the polynomial multiplication over hybrid fields, which is the major arithmetic component of the BRLWE scheme; (ii) we have presented the corresponding hardware architecture in a thorough format with sufficient description of the internal structures; (iii) we have also provided the complexity analysis and implementation-based comparison to demonstrate the superior performance of the proposed polynomial multiplication over the state-of-the-art design; (iv) finally, we have extended the proposed low-complexity polynomial multiplication to the major operational phase of the BRLWE scheme. We have shown that the proposed BRLWE structure involves significantly lower area-time complexities over the existing design, e.g., the proposed design has at least 66.01% less area-delay product (ADP) than the newly reported (Straix V device). Overall, the proposed design and implementation strategies are highly efficient, and the proposed BRLWE structure is desirable for many emerging applications.

Due to the increasing threats from possible large-scale quantum computers, post-quantum cryptography (PQC) has drawn significant attention from various communities recently. In particular, along with the National Institute of Standards and Technology (NIST) PQC standardization process, more works have gradually switched to the PQC hardware implementations. Following this trend, this tutorial brief, led by a group of experts in the field, aims to deliver a comprehensive tutorial on hardware circuits and systems design for PQC. After introducing primary arithmetic operations and algorithmic features of different PQC, we introduced related PQC hardware circuits and systems design techniques (from component to system levels). Future research and directions are also provided. This tutorial will provide useful information for the TCAS-II community and the broader Circuits and Systems Society.

Ring learning-with-errors (RLWE)-based encryption scheme is a lattice-based cryptographic algorithm that constitutes one of the most promising candidates for Post-Quantum Cryptography (PQC) standardization due to its efficient implementation and low computational complexity. Binary Ring -LWE (BRLWE) is a new optimized variant of RLWE, which achieves smaller computational complexity and higher efficient hardware implementations. In this paper, two efficient architectures based on Linear-Feedback Shift Register (LFSR) for the arithmetic used in Inverted Binary Ring -LWE ( Inv BRLWE)-based encryption scheme are presented, namely the operation of $A\cdot B+C$ over the polynomial ring $\mathbb {Z}_{q}/(x^{n}+1)$ . The first architecture optimizes the resource usage for major computation and has a novel input processing setup to speed up the overall processing latency with minimized input loading cycles. The second architecture deploys an innovative serial-in serial-out processing format to reduce the involved area usage further yet maintains a regular input loading time-complexity. Experimental results show that the architectures presented here improve the complexities obtained by competing schemes found in the literature, e.g., involving 71.23% less area-delay product than recent designs. Both architectures are highly efficient in terms of area-time complexities and can be extended for deploying in different lightweight application environments.

Post-quantum cryptography (PQC) has recently drawn substantial attention from various communities owing to the proven vulnerability of existing public-key cryptosystems against the attacks launched from well-established quantum computers. The Ring-Binary-Learning-with-Errors (RBLWE), a variant of Ring-LWE, has been proposed to build PQC for lightweight applications. As more Field-Programmable Gate Array (FPGA) devices are being deployed in lightweight applications like Internet-of-Things (IoT) devices, it would be interesting if the RBLWE-based PQC can be implemented on the FPGA with ultra-low complexity and flexible processing. However, thus far, limited information is available for such implementations. In this article, we propose novel RBLWE-based PQC accelerators on the FPGA with ultra-low implementation complexity and flexible timing. We first present the process of deriving the key operation of the RBLWE-based scheme into the proposed algorithmic operation. The corresponding hardware accelerator is then efficiently mapped from the proposed algorithm with the help of algorithm-to-architecture implementation techniques and extended to obtain higher-throughput designs. The final complexity analysis and implementation results (on a variety of FPGAs) show that the proposed accelerators have significantly smaller area-time complexities than the state-of-the-art designs. Overall, the proposed accelerators feature low implementation complexity and flexible processing, making them desirable for emerging FPGA-based lightweight applications.