From b875c6696e22eb9ba52ebf520c15ec58d0b50046 Mon Sep 17 00:00:00 2001
From: Sun Yimin <emmansun@users.noreply.github.com>
Date: Thu, 22 Feb 2024 14:21:53 +0800
Subject: [PATCH] Updated SM2 MFMM (2) (markdown)

---
 SM2-MFMM-(2).md | 49 +++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 49 insertions(+)

diff --git a/SM2-MFMM-(2).md b/SM2-MFMM-(2).md
index 647d1da..ca6c5c8 100644
--- a/SM2-MFMM-(2).md
+++ b/SM2-MFMM-(2).md
@@ -193,3 +193,52 @@ $t_5=0 + 0$
 ```
 乘法: 3  
 加法：8  
+
+### 方案二：(移位、加法、减法)
+$T_2=T_1 \ast P=t_0 \ast P= t_0 \ast (2^{256}-(2^{32} \ast 2^{192} + 0 \ast 2^{128} + (2^{32} - 1) \ast 2^{64} + 1))$  
+$T_2=t_0 \ast 2^{256} - t_0 \ast 2^{32} \ast 2^{192} - t_0 \ast (2^{32} - 1) \ast 2^{64} - t_0$  
+
+$T_3=T + T_2=t_4 \ast 2^{256} + t_3 \ast 2^{192} + t_2 \ast 2^{128} + t_1 \ast 2^{64} + t_0 \ast 2^{256} - t_0 \ast 2^{32} \ast 2^{192} - t_0 \ast (2^{32} - 1) \ast 2^{64} - t_0 $  
+$T_3=(t_4+t_0) \ast 2^{256}+(t_3 - t_0 \ast 2^{32}) \ast 2^{192} + t_2 \ast 2^{128} + (t_1 + t_0 - t_0 \ast 2^{32}) \ast 2^{64}  $  
+
+先处理加法，后处理减法，后四个加法是带进位加法   
+$t_1=t_1 + t_0$  
+$t_2=t_2 + 0$  
+$t_3=t_3 + 0$  
+$t_4=t_4 + t_0$  
+$t_5=0 + 0$  
+
+显然，这里不像平方那样有溢出风险。
+
+接着处理减法，假定a<sub>0</sub>是  $t_0 \ast 2^{32}$  的低64位，a<sub>1</sub>是  $t_0 \ast 2^{32}$  的高64位。后三个减法是带借位减法：  
+$t_1=t_1 - a_0$  
+$t_2=t_2 - a_1$  
+$t_3=t_3 - a_0$  
+$t_4=t_4 - a_1$  
+$t_5=t_5 - 0$  
+
+伪代码： 
+```asm
+	// First reduction step
+	MOVQ acc0, AX
+	MOVQ acc0, DX
+	SHLQ $32, AX
+	SHRQ $32, DX
+
+	ADDQ acc0, acc1
+	ADCQ $0, acc2
+	ADCQ $0, acc3
+	ADCQ acc0, acc4
+	ADCQ $0, acc5
+	
+	SUBQ AX, acc1
+	SBBQ DX, acc2
+	SBBQ AX, acc3
+	SBBQ DX, acc4
+	SBBQ $0, acc5
+```
+移位: 2  
+加法：5  
+减法：5
+
+