The Montgomery kP algorithm i.e. the Montgomery ladder is reported in literature as resistant against simple SCA due to the fact that the processing of each key bit value of the scalar k is done using the same sequence of operations. We implemented the Montgomery kP algorithm using Lopez-Dahab projective coordinates for the NIST elliptic curve B-233. We instantiated the same VHDL code for a wide range of clock frequencies for the same target FPGA and using the same compiler options. We measured electromagnetic traces of the kP executions using the same input data, i.e. scalar k and elliptic curve point P, and measurement setup. Additionally, we synthesized the same VHDL code for two IHP CMOS technologies, for a broad spectrum of frequencies. We simulated the power consumption of each synthesized design during an execution of the kP operation, always using the same scalar k and elliptic curve point P as inputs. Our experiments clearly show that the success of simple electromagnetic analysis attacks against FPGA implementations as well as the one of simple power analysis attacks against synthesized ASIC designs depends on the target frequency for which the design was implemented and at which it is executed significantly. In our experiments the scalar k was successfully revealed via simple visual inspection of the electromagnetic traces of the FPGA for frequencies from 40 to 100 MHz when standard compile options were used as well as from 50 MHz up to 240 MHz when performance optimizing compile options were used. We obtained similar results attacking the power traces simulated for the ASIC. Despite the significant differences of the here investigated technologies the designs’ resistance against the attacks performed is similar: only a few points in the traces represent strong leakage sources allowing to reveal the key at very low and very high frequencies. For the “middle” frequencies the number of points which allow to successfully reveal the key increases when increasing the frequency.

蒙哥马利kP算法，即蒙哥马利阶梯在文献中被报道为抵抗简单 SCA，因为标量k的每个关键位值的处理是使用相同的操作序列完成的。我们使用 NIST 椭圆曲线B-233 的Lopez-Dahab 投影坐标实现了 Montgomery kP算法。我们为同一目标 FPGA 的各种时钟频率实例化了相同的 VHDL 代码，并使用相同的编译器选项。我们使用相同的输入数据测量了kP执行的电磁轨迹，即标量k和椭圆曲线点P, 和测量设置。此外，我们为两种 IHP CMOS 技术合成了相同的 VHDL 代码，适用于广泛的频率。我们在执行kP操作期间模拟了每个综合设计的功耗，始终使用相同的标量k和椭圆曲线点P作为输入。我们的实验清楚地表明，针对 FPGA 实现的简单电磁分析攻击以及针对合成 ASIC 设计的简单功耗分析攻击的成功取决于实现设计的目标频率以及在该频率下显着执行设计。在我们的实验中，标量k当使用标准编译选项时，通过对 40 至 100 MHz 频率以及使用性能优化编译选项时 50 MHz 至 240 MHz 频率的 FPGA 电磁迹线的简单目视检查，成功揭示了 FPGA 的电磁轨迹。我们在攻击为 ASIC 模拟的电源迹线时获得了类似的结果。尽管这里研究的技术存在显着差异，但设计对所执行攻击的抵抗力是相似的：迹线中只有少数点代表强泄漏源，允许在非常低和非常高的频率下显示密钥。对于“中间”频率，当增加频率时，允许成功显示密钥的点数增加。