GF(2${}^m$) multiplication is a complex and performance-critical operation in Elliptic curve cryptography algorithms. Many techniques have been proposed in the literature for efficient implementation of GF(2${}^m$) multipliers. Montgomery multiplication is a technique used for fast GF(2${}^m$) multiplications, which is more efficient when there is a need for computation of many consecutive multiplications. In this paper, we present two modified bit-serial algorithms namely most significant bit (MSB) first algorithm and least significant bit (LSB) first algorithm for Montgomery multiplication where the modification involves employing more efficient logical relations in the formulation of the algorithms. Furthermore, the hardware structures developed for the proposed modified algorithms using bit-serial sequential architectures are also presented in this paper. Comparison of the analytical as well as implementation results of the proposed multipliers with the existing multipliers shows that the proposed multipliers require low area and time complexities. The proposed MSB-first multiplier and the proposed LSB-first multiplier achieve around 17% and 13% reduction in area-delay-product (ADP) complexities for $m=409,$ respectively, when compared with the respective best multipliers available in the literature. The proposed bit-serial sequential multipliers can be used in low-hardware and low-cost applications such as IoT edge devices.

GF（2${}^{米}$）乘法是椭圆曲线密码算法中一项复杂且对性能至关重要的操作。文献中已经提出了许多有效实施GF（2）的技术。${}^{米}$）乘数。蒙哥马利乘法是用于快速GF（2${}^{米}$）乘法，当需要计算许多连续的乘法时，效率更高。在本文中，我们提出了两种改进的位串行算法，即蒙哥马利乘法的最高有效位（MSB）第一算法和最低有效位（LSB）第一算法，其中修改涉及在算法的制定中采用更有效的逻辑关系。此外，本文还介绍了使用位串行顺序体系结构为所提出的改进算法开发的硬件结构。所提出的乘法器与现有的乘法器的分析结果和实施结果的比较表明，所提出的乘法器需要较低的面积和时间复杂度。$米=409，$分别与文献中可用的最佳乘数相比较。所建议的比特串行顺序乘法器可用于物联网边缘设备等低硬件和低成本应用。