From 1aa9dfcce92cc455d37d38a28803edbc166e8af1 Mon Sep 17 00:00:00 2001
From: Sun Yimin <emmansun@users.noreply.github.com>
Date: Wed, 21 Feb 2024 17:00:49 +0800
Subject: [PATCH] Updated MFMM (markdown)

---
 MFMM.md | 22 +++++++++++++++++++---
 1 file changed, 19 insertions(+), 3 deletions(-)

diff --git a/MFMM.md b/MFMM.md
index 6bcb0e8..fdd185f 100644
--- a/MFMM.md
+++ b/MFMM.md
@@ -2,9 +2,6 @@ MFMM=[Montgomery Friendly modules Montgomery Multiplication](https://eprint.iacr
 
 首先NIST P256 / SM2 256 的素数P都是Montgomery Friendly modules。  
 
-SM2 256 的素数P=0xfffffffeffffffffffffffffffffffffffffffff00000000ffffffffffffffff，也可以表示为
-
-$P = 2^{256}-(2^{32} \ast 2^{192} + 0 \ast 2^{128} + (2^{32} - 1) \ast 2^{64} + 1)$  
 
     输入：
     X, Y都是Montgomery数值表示
@@ -285,3 +282,22 @@ acc0, acc1, acc2, acc3, acc4, acc5是64位寄存器
     BenchmarkMoreThan32_P256SM2-2   	    4550	    263296 ns/op
     PASS
     ok  	github.com/emmansun/gmsm/sm2	4.753s
+
+### 续
+SM2 256 的素数P=0xfffffffeffffffffffffffffffffffffffffffff00000000ffffffffffffffff，也可以表示为
+
+$P = 2^{256}-(2^{32} \ast 2^{192} + 0 \ast 2^{128} + (2^{32} - 1) \ast 2^{64} + 1)$  
+
+这样，可以通过移位和加减操作来实现模约减。  
+假设：  
+$T=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 $  
+则共四次约减，第一次约减为：  
+
+$T_1=t_0$  
+
+$T_2=T_1 \ast P=t_0 \ast P= t_0 \ast (2^{256}-(2^{32} \ast 2^{192} + 0 \ast 2^{128} + (2^{32} - 1) \ast 2^{64} + 1))$  
+$T_2=t_0 \ast 2^{256} - t_0 \ast 2^{32} \ast 2^{192} - t_0 \ast (2^{32} - 1) \ast 2^{64} - t_0$  
+
+$T_3=T + T_2=t_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_0 $  
+$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}  $  
+