In today's technology, a sheer number of Internet of Things applications use hardware security modules for secure communications. The widely used algorithms in security modules, for example, digital signatures and key agreement, are based upon elliptic curve cryptography (ECC). A core operation used in ECC is the point multiplication, which is computationally expensive for many Internet of things applications. In many IoT applications, such as intelligent transportation systems and distributed control systems, thousands of safety messages need to be signed and verified within a very short time-frame. Considerable research has been conducted in the design of a fast elliptic curve arithmetic on finite fields using residue number systems (RNS). In this article, we propose an RNS-based ECC core hardware for the two families of elliptic curves that are short Weierstraß and twisted Edwards curves. Specifically, we present RNS implementations for SECP256K1 and ED25519 standard curves. We propose an RNS hardware architecture supporting fast elliptic curve point-addition (ECPA), point-doubling (ECPD), and point-tripling (ECPT). We implemented different ECC point multiplication algorithms on the Xilinx FPGA platform. The test results confirm that the performance of our fully RNS ECC point multiplication is better than the fastest ECC point multiplication cores in the literature.

中文翻译：

---

硬件安全模块的椭圆曲线加密点乘核心

在当今的技术中，大量物联网应用程序使用硬件安全模块来实现安全通信。安全模块中广泛使用的算法，例如数字签名和密钥协商，都是基于椭圆曲线密码术（ECC）。ECC 中使用的一个核心操作是点乘法，这对于许多物联网应用来说计算量很大。在许多物联网应用中，例如智能交通系统和分布式控制系统，需要在很短的时间内签署和验证数以千计的安全消息。已经在使用残数系统 (RNS) 的有限域上设计快速椭圆曲线算法方面进行了大量研究。在本文中，我们为短 Weierstraß 和扭曲 Edwards 曲线这两个椭圆曲线系列提出了基于 RNS 的 ECC 核心硬件。具体来说，我们展示了 SECP256K1 和 ED25519 标准曲线的 RNS 实现。我们提出了一种支持快速椭圆曲线点加法 (ECPA)、点倍增 (ECPD) 和点三倍化 (ECPT) 的 RNS 硬件架构。我们在 Xilinx FPGA 平台上实现了不同的 ECC 点乘法算法。测试结果证实，我们完全 RNS ECC 点乘法的性能优于文献中最快的 ECC 点乘法核心。点倍增 (ECPD) 和点三倍 (ECPT)。我们在 Xilinx FPGA 平台上实现了不同的 ECC 点乘法算法。测试结果证实，我们完全 RNS ECC 点乘法的性能优于文献中最快的 ECC 点乘法核心。点倍增 (ECPD) 和点三倍 (ECPT)。我们在 Xilinx FPGA 平台上实现了不同的 ECC 点乘法算法。测试结果证实，我们完全 RNS ECC 点乘法的性能优于文献中最快的 ECC 点乘法核心。