From 18799b4d2c9687cd192c3e3598b93fccceb654da Mon Sep 17 00:00:00 2001
From: Sun Yimin <emmansun@users.noreply.github.com>
Date: Fri, 1 Mar 2024 09:42:22 +0800
Subject: [PATCH] Updated SM2 WWMM (2) (markdown)

---
 SM2-WWMM-(2).md | 6 ++++--
 1 file changed, 4 insertions(+), 2 deletions(-)

diff --git a/SM2-WWMM-(2).md b/SM2-WWMM-(2).md
index 450a1b8..1db2c45 100644
--- a/SM2-WWMM-(2).md
+++ b/SM2-WWMM-(2).md
@@ -83,6 +83,7 @@ $T_2=t_0 \ast 2^{256} - t_0 \ast 2^{32} \ast 2^{192} - t_0 \ast (2^{32} - 1) \as
 
 $T_3=T + T_2=t_7 \ast 2^{448} + t_6 \ast 2^{384} + t_5 \ast 2^{320} + t_4 \ast 2^{256} + t_3 \ast 2^{192} + t_2 \ast 2^{128} + t_1 \ast 2^{64} + t_0 \ast 2^{256} - t_0 \ast 2^{32} \ast 2^{192} - t_0 \ast (2^{32} - 1) \ast 2^{64}$  
 $T_3=t_7 \ast 2^{448} + t_6 \ast 2^{384} + t_5 \ast 2^{320} + (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}  $  
+$T_3=t_7 \ast 2^{448} + t_6 \ast 2^{384} + t_5 \ast 2^{320} + (t_4+t_0-t0>>32) \ast 2^{256}+(t_3 - t_0<<32) \ast 2^{192} + (t_2 - t0>>32) \ast 2^{128} + (t_1 + t_0 - t_0<<32) \ast 2^{64}  $  
 
 先处理加法，后处理减法，后三个加法是带进位加法   
 $t_1=t_0 + t_1$  
@@ -99,8 +100,6 @@ $t_0=t_0 - a_1$
 
 **减法显然是安全的（因为第四步的结果显然是>=0的，而且为零的情况仅限于 $t_0==0$ 的情况 ），所以调整为先做减法，再做加法，确保第四步加法不会产生进位**。
 
-$T_3=t_7 \ast 2^{448} + t_6 \ast 2^{384} + t_5 \ast 2^{320} + (t_4+t_0-t0>>32) \ast 2^{256}+(t_3 - t_0<<32) \ast 2^{192} + (t_2 - t0>>32) \ast 2^{128} + (t_1 + t_0 - t_0<<32) \ast 2^{64}  $  
-
 伪代码： 
 ```asm
   \ // First reduction step, [p3, p2, p1, p0] = [1, -0x100000000, 0, (1 - 0x100000000), -1]
@@ -212,6 +211,7 @@ $T_2=t_0 \ast 2^{256} - t_0 \ast 2^{32} \ast 2^{192} - t_0 \ast (2^{32} - 1) \as
 
 $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_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}  $  
+$T_3=(t_4+t_0-t_0>>32) \ast 2^{256}+(t_3 - t_0<<32) \ast 2^{192} + (t_2 - t_0>>32) \ast 2^{128} + (t_1 + t_0 - t_0<<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   
 $t_1=t_1 + t_0$  
@@ -409,6 +409,7 @@ $T_2=T_1 \ast O=Y \ast O= Y \ast 2^{256}-(Y \ast 2^{32}) \ast 2^{192} - Y \ast 2
 
 $T_3=T + T_2=t_7 \ast 2^{448} + t_6 \ast 2^{384} + t_5 \ast 2^{320} + t_4 \ast 2^{256} + t_3 \ast 2^{192} + t_2 \ast 2^{128} + t_1 \ast 2^{64} + t_0 + Y \ast 2^{256}-(Y \ast 2^{32}) \ast 2^{192} - Y \ast 2^{128} + (Y \ast O_1) \ast 2^{64} + (Y \ast O_0) $  
 $T_3=t_7 \ast 2^{448} + t_6 \ast 2^{384} + t_5 \ast 2^{320} + (t_4+Y) \ast 2^{256}+(t_3 - Y \ast 2^{32}) \ast 2^{192} + (t_2 - Y) \ast 2^{128} + (t_1 + Y \ast O_1) \ast 2^{64} + (t_0 + Y \ast O_0)  $  
+$T_3=t_7 \ast 2^{448} + t_6 \ast 2^{384} + t_5 \ast 2^{320} + (t_4+Y-Y>>32) \ast 2^{256}+(t_3 - Y<<32) \ast 2^{192} + (t_2 - Y) \ast 2^{128} + (t_1 + Y \ast O_1) \ast 2^{64} + (t_0 + Y \ast O_0)  $  
 
 **依然采用先减后加！**  
 ```asm
@@ -591,6 +592,7 @@ $T_2=T_1 \ast O=Y \ast O= Y \ast 2^{256}-(Y \ast 2^{32}) \ast 2^{192} - Y \ast 2
 
 $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 + Y \ast 2^{256}-(Y \ast 2^{32}) \ast 2^{192} - Y \ast 2^{128} + (Y \ast O_1) \ast 2^{64} + (Y \ast O_0) $  
 $T_3=(t_4+Y) \ast 2^{256}+(t_3 - Y \ast 2^{32}) \ast 2^{192} + (t_2 - Y) \ast 2^{128} + (t_1 + Y \ast O_1) \ast 2^{64} + (t_0 + Y \ast O_0)  $  
+$T_3=(t_4+Y-Y>>32) \ast 2^{256}+(t_3 - Y<<32) \ast 2^{192} + (t_2 - Y) \ast 2^{128} + (t_1 + Y \ast O_1) \ast 2^{64} + (t_0 + Y \ast O_0)  $  
 
 **依然采用先减后加！**