In this paper, we present efficient parallel and serial systolic structures for combined multiplication and squaring over GF( ${2^{m}}$ ). The proposed structures have the advantage of computing both modular multiplication and squaring simultaneously for fast execution of modular exponentiation. They share the same hardware components for both operations and this gives them the advantage of saving more space than the case of using a separate structure for each operation. In addition, they have a regular structure and local communication between the processing elements and this makes them more efficient for VLSI implementation. Complexity analysis of the proposed designs and the existing most recent and efficient designs indicates that the proposed serial design has 31.7% lower area-time (AT) complexity than the previously reported most recent serial one. In addition, the proposed parallel design has a significant lower AT complexity over the recent efficient parallel designs by at least 96.9%.

中文翻译：

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GF(${2^{m}}$) 乘法和平方的高效并行和串行收缩结构

在本文中，我们提出了用于 GF(${2^{m}}$) 上组合乘法和平方的高效并行和串行收缩结构。所提出的结构具有同时计算模乘法和平方的优点，以便快速执行模幂运算。它们为两个操作共享相同的硬件组件，这使它们具有比为每个操作使用单独结构的情况节省更多空间的优势。此外，它们在处理元件之间具有规则结构和本地通信，这使得它们对于 VLSI 实施更加有效。对提议的设计和现有的最新有效设计的复杂性分析表明，提议的串行设计有 31 个。与之前报道的最新系列相比，区域时间 (AT) 复杂性降低了 7%。此外，与最近的高效并行设计相比，所提出的并行设计的 AT 复杂性显着降低了至少 96.9%。