## 5 Analysis on Time and Area Overheads  
This section mainly presents the time and area overheads of our architecture (Style-II) with three different check factors (\(p=3\), \(7\), and \(31\)) under each MFM structure. In order to get time and area overheads, we first modeled the proposed architecture using Verilog, then verified using Modelsim, and finally synthesized the circuit by Synopsys Design Vision with TSMC 90 nm CMOS standard cell library. For comparison, Style-I and Parity methods were also modeled and synthesized under the same conditions. The synthesized results are given in Table 3, where time (area) overhead ratio refers to the quotient between extra time (area) overhead and MFM time (area). Firstly, as shown in Table 3, the time and area overhead ratios of Style-II always surpassed those of Style-I, whichever the check factor, but they were lower than those of Parity at \(p=3\) or \(7\). For example, when the MFM structure belonged to Type-8, the mean time and area overhead ratios of Style-I were 0.51% and 41.31%, respectively; those of Style-II with \(p=3\) were 11.86% and 45.67%, respectively; those of Style-II with \(p=7\) were 20.70% and 77.84%, respectively; those of Parity were 30.07% and 72.23%, respectively; those of Style-II with \(p=31\) were 49.64% and 123.22%. Secondly, we noticed that, when the \(p\) value was fixed, the time overhead ratio increased with the operand size \(n\), the area overhead ratio decreased with the growth in \(n\), and the product of time and area overhead ratios decreased with \(n\). For example, for Style-II with \(p=7\), the mean time and area overhead ratios were 18.50% and 104.80%, respectively, when \(n=128\), and 23.64% and 46.80%, respectively, when \(n=1024\). Judging by only time and area overheads, Style-II's performance is negatively correlated with \(p\)-value. However, considering overall performance including time overheads, area overheads, and error detection capability, Style-II with \(p=7\) should be a better choice compared with Style-II with \(p=3\) or \(31\). Thirdly, under the fixed \(p\)-value, Style-II's performance varied with the MFM structures. The larger the word size of the PE in the MFM, the greater the time and area overhead ratios. For example, when Style-II with \(p=3\) was applied to Type-8 MFM, the mean time and area overhead ratios were 11.86% and 45.67%, respectively; when it was applied to Type-16 MFM, the two ratios were 33.98% and 51.02%, respectively. Hence, Style-II works better in error detection of short-word systolic implementation of the MFM. On the contrary, Style-I is more suitable for error detection of long-word systolic implementation of the MFM. In both methods, the time and area overheads are negatively correlated with the operand size \(n\). In other words, the two methods are more efficient for large integer MD problems. Finally, Style-II showed a much shorter delay in error reporting than Style-I and Parity. The delay of Style-II increased with \(w\) and \(p\), but remained basically constant when the MFM structure was stable and the operand size varied. Overall, Style-II with \(p=7\) and Type-8 MFM strikes a good balance between time overheads, area overheads, and error detection capability.