AES 多项式求逆

在进行AES求解时，涉及到对多项式的求逆的过程。但是好像没有人写过这方面的博客，所以我想试着写一下。

AES中，不可约多项式选择为：m(x) = .

算法的思想是穷举，对于给定的多项式 f(x)，穷举多项式g(x)，使得 f(x)*g(x) = 1 mod m(x).

（扩展欧几里得算法要比穷举好，但是我不会写....

/*
***  Author: Lizhiwei
***  Date:2021.12.25
*/
#include<bits/stdc++.h>
using namespace std;
int Mod = 283; //模数 
stack<int>Mystack;
void Init(){
	int k = Mod;
	while(k>0){
		int t = (k&1);
		Mystack.push(t);
		k = k >> 1;
	}
	while(!Mystack.empty()){
		int t = Mystack.top();
		cout<<t;
		Mystack.pop(); 
	}
}
string Str(int t){
	stringstream ss;
    string str;
    ss<<t;
    ss>>str;
    return str;
}
string GenString(int f){
	int k = f;
	while(k>0){
		int t = (k&1);
		Mystack.push(t);
		k = k >> 1;
	}
	string s = "";
	while(!Mystack.empty()){
		int t = Mystack.top();
		string str = Str(t);
	    s += str;
		Mystack.pop(); 
	}
	if(s.length()<8){
		int len = 8-s.length();
		string s1 = "";
		string s2 = "";
		for(int i=0;i<len;i++){
			s1 += "0";
		}
		s1 += s;
		for(int i=0;i<s1.length();i++){
			if(s1[i] != '0'){
				s2 += Str(s1.length()-1-i);
			}
		}
		return s2;
	}
	string s2;
	for(int i=0;i<s.length();i++){
		if(s[i] != '0'){
			s2 += Str(s.length()-1-i);
		}
	}
	return s2;
}
void Cheng(string s1, string s2, string s3){
	int len1 = s1.length();
	int len2 = s2.length();
	int t = 0;
	int mem[len1*len2]; //
	int mem2[15]; // 下标存储系数的个数 
	for(int i=0;i<15;i++) mem2[i] = 0;
	//cout<<"s1 = "<<s1<<"  s2 = "<<s2<<"  sm = "<<s3<<endl;
	for(int i=0;i<len1;i++){
		for(int j=0;j<len2;j++){
			mem[t++] = int(s1[i]-'0') + int(s2[j]-'0');
		}
	}
	for(int i=0;i<len1*len2;i++){
		mem2[mem[i]] += 1;
	}
	for(int i=0;i<15;i++){
		mem2[i] %= 2;
		//cout<<mem2[i]<<" ";
	} 
	//cout<<endl;
	
	
	for(int i=8;i<15;i++){
		if(mem2[i] != 0){
			mem2[i] = 0;
			mem2[i-int(s3[0]-'0')+int(s3[1]-'0')] += 1;
			mem2[i-int(s3[0]-'0')+int(s3[2]-'0')] += 1;
			mem2[i-int(s3[0]-'0')+int(s3[3]-'0')] += 1;
			mem2[i-int(s3[0]-'0')+int(s3[4]-'0')] += 1;
		}
	}
	
	for(int j=0;j<15;j++){
		mem2[j] %= 2;
	} 
	for(int i=8;i<15;i++){
		if(mem2[i] != 0){
			mem2[i] = 0;
			mem2[i-int(s3[0]-'0')+int(s3[1]-'0')] += 1;
			mem2[i-int(s3[0]-'0')+int(s3[2]-'0')] += 1;
			mem2[i-int(s3[0]-'0')+int(s3[3]-'0')] += 1;
			mem2[i-int(s3[0]-'0')+int(s3[4]-'0')] += 1;
		}
	}
	for(int j=0;j<15;j++){
		mem2[j] %= 2;
	} 
	for(int i=8;i<15;i++){
		if(mem2[i] != 0){
			mem2[i] = 0;
			mem2[i-int(s3[0]-'0')+int(s3[1]-'0')] += 1;
			mem2[i-int(s3[0]-'0')+int(s3[2]-'0')] += 1;
			mem2[i-int(s3[0]-'0')+int(s3[3]-'0')] += 1;
			mem2[i-int(s3[0]-'0')+int(s3[4]-'0')] += 1;
		}
	}
	
	for(int j=0;j<15;j++){
		mem2[j] %= 2;
	} 
	
	for(int i=8;i<15;i++){
		if(mem2[i] != 0){
			mem2[i] = 0;
			mem2[i-int(s3[0]-'0')+int(s3[1]-'0')] += 1;
			mem2[i-int(s3[0]-'0')+int(s3[2]-'0')] += 1;
			mem2[i-int(s3[0]-'0')+int(s3[3]-'0')] += 1;
			mem2[i-int(s3[0]-'0')+int(s3[4]-'0')] += 1;
		}
	}
	for(int j=0;j<15;j++){
		mem2[j] %= 2;
	} 
	
	for(int i=8;i<15;i++){
		if(mem2[i] != 0){
			mem2[i] = 0;
			mem2[i-int(s3[0]-'0')+int(s3[1]-'0')] += 1;
			mem2[i-int(s3[0]-'0')+int(s3[2]-'0')] += 1;
			mem2[i-int(s3[0]-'0')+int(s3[3]-'0')] += 1;
			mem2[i-int(s3[0]-'0')+int(s3[4]-'0')] += 1;
		}
	}
	
	for(int j=0;j<15;j++){
		mem2[j] %= 2;
	} 
	for(int i=8;i<15;i++){
		if(mem2[i] != 0){
			mem2[i] = 0;
			mem2[i-int(s3[0]-'0')+int(s3[1]-'0')] += 1;
			mem2[i-int(s3[0]-'0')+int(s3[2]-'0')] += 1;
			mem2[i-int(s3[0]-'0')+int(s3[3]-'0')] += 1;
			mem2[i-int(s3[0]-'0')+int(s3[4]-'0')] += 1;
		}
	}

	for(int j=0;j<15;j++){
		mem2[j] %= 2;
	} 
	for(int i=8;i<15;i++){
		if(mem2[i] != 0){
			mem2[i] = 0;
			mem2[i-int(s3[0]-'0')+int(s3[1]-'0')] += 1;
			mem2[i-int(s3[0]-'0')+int(s3[2]-'0')] += 1;
			mem2[i-int(s3[0]-'0')+int(s3[3]-'0')] += 1;
			mem2[i-int(s3[0]-'0')+int(s3[4]-'0')] += 1;
		}
	}
	
	for(int j=0;j<15;j++){
		mem2[j] %= 2;
	} 
	for(int i=8;i<15;i++){
		if(mem2[i] != 0){
			mem2[i] = 0;
			mem2[i-int(s3[0]-'0')+int(s3[1]-'0')] += 1;
			mem2[i-int(s3[0]-'0')+int(s3[2]-'0')] += 1;
			mem2[i-int(s3[0]-'0')+int(s3[3]-'0')] += 1;
			mem2[i-int(s3[0]-'0')+int(s3[4]-'0')] += 1;
		}
	}
	
	for(int i=0;i<15;i++){
		mem2[i] %= 2;
	} 
	int flag1 = mem2[0];
	int flag2 = 0;
	
	for(int i=1;i<15;i++){
		if(mem2[i]!=0) flag2 = 1;
	}
	if(flag1 == 1 && flag2 ==0){
		cout<<"("; 
		for(int i=0;i<s2.length()-1;i++){
			cout<<"x^"<<s2[i]<<"+";
		}
		cout<<"x^"<<s2[s2.length()-1]<<")^{-1} = ";
		
		for(int i=0;i<s1.length()-1;i++){
			cout<<"x^"<<s1[i]<<"+";
		}
		cout<<"x^"<<s1[s1.length()-1]<<endl;
		//cout<<" ans = "<<s1 <<"  "<<s2<<endl;
	}
	
}
void Solve(int f,int X,int Mod){
	string s_f = GenString(f);
	string s_x = GenString(X);
	string s_m = GenString(Mod); 
	//cout<<"s_f = "<<s_f<<"  s_x = "<<s_x<<"  s_m = "<<s_m<<endl;
	Cheng(s_f, s_x, s_m);
}
int main(){
	Init(); 
	cout<<endl;
	for(int X=1;X<255;X++){
		for(int i=1;i<255;i++){
			Solve(i, X, Mod);
		}
	}
	
	return 0;
}