ZUC算法

由于在看ZUC算法的时候会有很多的字符等符号不太知道于是！

运算符

+ 算术加法运算
ab 整数a和b的乘积
= 赋值操作符
mod 整数模运算
⊕ 按比特位逐位异或运算
田模2加法运算
‖ 字符串或字节串连接符
下标H 取字的最高16比特
下标L 取字的最低16比特
<<<k 32比特字循环左移k位
>>k 32比特字右移K位
a->b 向量a赋值给向量b，即按分量逐分量赋值

符号

S1，S2，S3，……S15 线性反馈移位寄存器的16个31比特寄存器单元变量
X1，X2，X3，X4 比特重组输出的4个32比特字
R1，R2 非线性函数F的2个32比特记忆单元变量
W 非线性函数F输出的32比特字
W1 R1与X1进行模2^32加法运算输出的32比特字
W2 R2与X2按比特位逐位异或运算输出的32比特字
Z 算法每个节拍输出的32比特密钥字
K 初始种子密钥
IV 初始向量
D1，D2，D3……D15 15比特的字符串常量
F 非线性函数
L 输出密钥字长度

线性反馈移位寄存器(LFSR)

由于这个算法涉及到了LFSR，于是翻起了自己深入浅出密码学，复习一下知识点
定义:一个LFSR由若干时钟存储元件(触发器)和一个反馈路径组成，存储元件的数目给出了LFSR的度，一个拥有m个触发器的LFSR可以称为“度为m”，反馈网络计算移位寄存器中某些触发器的XOR的和，并将其作为上一个触发器的输入

简单LFSR

书上的例子挺好的,可以帮助理解LFSR，给了初始状态(S2=1,S1=0,S0=0)
看图：用自己话理解了一下，将S0与S1进行XOR，出来的结果给S3，这时候S0输出出去，S1，S2进行左平移，S3加入S2的位置，依次类推，因为是异或且每次是一位0和1那么可以写成公式就是:S3=S1+S0 mod 2
那我们看一下LFSR的状态序列

推导：S(i+3)=S(i)+S(i+1) mod 2
我们可以看到在第6个时钟周期后开始重复，0010111，意味着LFSR输出的周期长度为7

LFSR的数学描述

继续调用了书上的图

我们可以看到，LFSR拥有了m个触发器和m个可能的反馈位置，并且这些触发器和反馈的位置之间通过XOR操作连接，看是否活跃取决于反馈系数P0…Pn
这里有了个定义：
如果Pi=1，此反馈是活跃的(即关闭了开关)
如果Pi=0，对应的触发器的输出将不会被反馈(即打开了开关)
将触发器的i的输出与对应的系数Pi相乘，如果Pi=1那么结果就是输出值(对应关闭开关)，如果Pi=0，则输出值也为0(即打开开关)即公式:

即满足了依次左移的原则，更新Sm，则输出序列描述为:

因为m位状态向量只能得到2^m -1个非零状态(自己理解为是可能性最后遍历回来)，比如最大长度输出序列的LFSR，给定一个度m=4，反馈路径为(P3=0,P2=0,P1=1,P0=1)，周期为2^4-1=15;非最大长度输出序列，给定一个度m=4，反馈路径为(P3=1,P2=1,P1=1,P0=1)周期为5

0 0 1 1/0 0 0 1/1 0 0 0/0 1 0 0/0 0 1 0/1 0 0 1/1 1 0 0/0 1 1 0/1 0 1 1/0 1 0 1/1 0 1 0/1 1 0 1/1 1 1 0/1 1 1 1/0 1 1 1/（0 0 1 1）所以为15
1 1 1 1/0 1 1 1/1 0 1 1/1 1 0 1/1 1 1 0/(1 1 1 1 )所以为5
LFSR通常用：反馈系数向量(Pm-1,…,P1,P0)则P(x)=X^m+Pm-1X^m-1+…+P1x+p0
如上面的例子中的系数(P3=0,P2=0,P1=1,P0=1)的LFSR可以表示为多项式x^4+x+1,最大长度LFSR拥有所谓的本源多项式(设f(x)是一个整系数多项式, 若f(x)的系数的公因子只有±1, 则称f(x)是一个本原多项式.)查了很久总结一下：代数式系数对应的0 1 字串,满足,用8进制变为对应的十进制,若为素数,则为本原多项式，所以本原多项式是一种特殊的不可约分多项式，只有1和他的本身.所以例如(0,2,5)为1+x^2+x^5为10011八进制为23为素数

算法构架

上层是16级线性反馈移位寄存器(LFSR),中层是比特重组(BR),下层是非线性函数F

第一层：线性反馈移位寄存器（LFSR）

LFSR包括16个31比特寄存器单元变量S1，S2，S3，……S15,最开始进行了初始化

因为是mod(2^31-1) 2^31-1=2147483647为素数，所以为素域，采用16次本原多项式P(x)=x^16-2^15*x^15-2^17*x^13-2^21*x^10-2^20*x^4-(2^8+1)所以根据线性反馈移位寄存器的公式来说(看上面的介绍)V=2^15*S^15-2^17*S^13-2^21*S^10-2^20*S^4-(2^8+1)S0 mod(2^31-1)先放到一边，准备开始移位即：(s1,s2…s16)变为(s0,s1…s16)所以输出的为32位比特，但是我们要舍去一个最开始的把s16进行放入到s15的位置s0的位置进行保存
所以可以得到公式为是:S16=(V+u)mod(2^31-1)其中v是上面公式得到的与LFSR接受的31比特进行相加之后与2^31-1进行求余就是S16，如果我们的S16=0 那么S16置为2^31-1
以上是初始化的模式，初始化和工作模式的差异为，初始化时需要引入非线性函数F输出W通过舍弃最低为比特得到u，而工作模式不需要
一般分为两种情况:

第二层：比特重组（BR）

比特重组在LFSR和F之间，起到数据组合和关联的作用
在LFSR的寄存器单元中抽取128比特进行4个分组4个32比特为X0,X1,X2,X3传递给F
计算过程：
X0=S15H || S14L
X1=S11L || S19H
X2=S7L || S5H
X3=S2L || S0H
其中||表示两个字符首位进行拼接，H代表高16位，L代表低16位

第三层：非线性函数F

非线性函数F内部包含2个32比特存储单元R1和R2，F的输入为来自比特重组的3个32比特字X0,X1,X2，F的输出为一个32比特的W，计算过程：

F层核心部件，S盒:非线性部件，为ZUC提供混淆，L变换:线性变换，为ZUC提供扩散，F使用了两个8*8的S盒：

F使用了两个L变换：L1和L2

最后结果

代码实现：加密

#pragma once

#include<malloc.h>
#include<stdio.h>


/* ————————————W—————————- */
typedef unsigned char u8;
typedef unsigned int u32;
/* —————————————t————————- */
/* the state registers of LFSR 16 cells*/
u32 LFSR_S0;
u32 LFSR_S1;
u32 LFSR_S2;
u32 LFSR_S3;
u32 LFSR_S4;
u32 LFSR_S5;
u32 LFSR_S6;
u32 LFSR_S7;
u32 LFSR_S8;
u32 LFSR_S9;
u32 LFSR_S10;
u32 LFSR_S11;
u32 LFSR_S12;
u32 LFSR_S13;
u32 LFSR_S14;
u32 LFSR_S15;
/* the registers of f算法中的R1,R2 */
u32 F_R1;
u32 F_R2;
/* the outputs of BitReorganization BR中的X0,X1,X2,X3*/
u32 BRC_X0;
u32 BRC_X1;
u32 BRC_X2;
u32 BRC_X3;
/* the s-boxes 两个S盒*/
u8 S0[256] = {
0x3e,0x72,0x5b,0x47,0xca,0xe0,0x00,0x33,0x04,0xd1,0x54,0x98,0x09,0xb9,0x6d,0xcb,
0x7b,0x1b,0xf9,0x32,0xaf,0x9d,0x6a,0xa5,0xb8,0x2d,0xfc,0x1d,0x08,0x53,0x03,0x90,
0x4d,0x4e,0x84,0x99,0xe4,0xce,0xd9,0x91,0xdd,0xb6,0x85,0x48,0x8b,0x29,0x6e,0xac,
0xcd,0xc1,0xf8,0x1e,0x73,0x43,0x69,0xc6,0xb5,0xbd,0xfd,0x39,0x63,0x20,0xd4,0x38,
0x76,0x7d,0xb2,0xa7,0xcf,0xed,0x57,0xc5,0xf3,0x2c,0xbb,0x14,0x21,0x06,0x55,0x9b,
0xe3,0xef,0x5e,0x31,0x4f,0x7f,0x5a,0xa4,0x0d,0x82,0x51,0x49,0x5f,0xba,0x58,0x1c,
0x4a,0x16,0xd5,0x17,0xa8,0x92,0x24,0x1f,0x8c,0xff,0xd8,0xae,0x2e,0x01,0xd3,0xad,
0x3b,0x4b,0xda,0x46,0xeb,0xc9,0xde,0x9a,0x8f,0x87,0xd7,0x3a,0x80,0x6f,0x2f,0xc8,
0xb1,0xb4,0x37,0xf7,0x0a,0x22,0x13,0x28,0x7c,0xcc,0x3c,0x89,0xc7,0xc3,0x96,0x56,
0x07,0xbf,0x7e,0xf0,0x0b,0x2b,0x97,0x52,0x35,0x41,0x79,0x61,0xa6,0x4c,0x10,0xfe,
0xbc,0x26,0x95,0x88,0x8a,0xb0,0xa3,0xfb,0xc0,0x18,0x94,0xf2,0xe1,0xe5,0xe9,0x5d,
0xd0,0xdc,0x11,0x66,0x64,0x5c,0xec,0x59,0x42,0x75,0x12,0xf5,0x74,0x9c,0xaa,0x23,
0x0e,0x86,0xab,0xbe,0x2a,0x02,0xe7,0x67,0xe6,0x44,0xa2,0x6c,0xc2,0x93,0x9f,0xf1,
0xf6,0xfa,0x36,0xd2,0x50,0x68,0x9e,0x62,0x71,0x15,0x3d,0xd6,0x40,0xc4,0xe2,0x0f,
0x8e,0x83,0x77,0x6b,0x25,0x05,0x3f,0x0c,0x30,0xea,0x70,0xb7,0xa1,0xe8,0xa9,0x65,
0x8d,0x27,0x1a,0xdb,0x81,0xb3,0xa0,0xf4,0x45,0x7a,0x19,0xdf,0xee,0x78,0x34,0x60
};
//s0盒
u8 S1[256] = {
0x55,0xc2,0x63,0x71,0x3b,0xc8,0x47,0x86,0x9f,0x3c,0xda,0x5b,0x29,0xaa,0xfd,0x77,
0x8c,0xc5,0x94,0x0c,0xa6,0x1a,0x13,0x00,0xe3,0xa8,0x16,0x72,0x40,0xf9,0xf8,0x42,
0x44,0x26,0x68,0x96,0x81,0xd9,0x45,0x3e,0x10,0x76,0xc6,0xa7,0x8b,0x39,0x43,0xe1,
0x3a,0xb5,0x56,0x2a,0xc0,0x6d,0xb3,0x05,0x22,0x66,0xbf,0xdc,0x0b,0xfa,0x62,0x48,
0xdd,0x20,0x11,0x06,0x36,0xc9,0xc1,0xcf,0xf6,0x27,0x52,0xbb,0x69,0xf5,0xd4,0x87,
0x7f,0x84,0x4c,0xd2,0x9c,0x57,0xa4,0xbc,0x4f,0x9a,0xdf,0xfe,0xd6,0x8d,0x7a,0xeb,
0x2b,0x53,0xd8,0x5c,0xa1,0x14,0x17,0xfb,0x23,0xd5,0x7d,0x30,0x67,0x73,0x08,0x09,
0xee,0xb7,0x70,0x3f,0x61,0xb2,0x19,0x8e,0x4e,0xe5,0x4b,0x93,0x8f,0x5d,0xdb,0xa9,
0xad,0xf1,0xae,0x2e,0xcb,0x0d,0xfc,0xf4,0x2d,0x46,0x6e,0x1d,0x97,0xe8,0xd1,0xe9,
0x4d,0x37,0xa5,0x75,0x5e,0x83,0x9e,0xab,0x82,0x9d,0xb9,0x1c,0xe0,0xcd,0x49,0x89,
0x01,0xb6,0xbd,0x58,0x24,0xa2,0x5f,0x38,0x78,0x99,0x15,0x90,0x50,0xb8,0x95,0xe4,
0xd0,0x91,0xc7,0xce,0xed,0x0f,0xb4,0x6f,0xa0,0xcc,0xf0,0x02,0x4a,0x79,0xc3,0xde,
0xa3,0xef,0xea,0x51,0xe6,0x6b,0x18,0xec,0x1b,0x2c,0x80,0xf7,0x74,0xe7,0xff,0x21,
0x5a,0x6a,0x54,0x1e,0x41,0x31,0x92,0x35,0xc4,0x33,0x07,0x0a,0xba,0x7e,0x0e,0x34,
0x88,0xb1,0x98,0x7c,0xf3,0x3d,0x60,0x6c,0x7b,0xca,0xd3,0x1f,0x32,0x65,0x04,0x28,
0x64,0xbe,0x85,0x9b,0x2f,0x59,0x8a,0xd7,0xb0,0x25,0xac,0xaf,0x12,0x03,0xe2,0xf2
};
//s1盒
/* the constants D */
u32 EK_d[16] = {
0x44D7, 0x26BC, 0x626B, 0x135E, 0x5789, 0x35E2, 0x7135, 0x09AF,
0x4D78, 0x2F13, 0x6BC4, 0x1AF1, 0x5E26, 0x3C4D, 0x789A, 0x47AC
};
/* ——————————————————————- */
/* c = a + b mod (2^31 – 1)  那个附录里复杂的计算*/
u32 AddM(u32 a, u32 b) {
	u32 c = a + b;
	return (c & 0x7FFFFFFF) + (c >> 31);
}
/* LFSR with initialization mode */
#define MulByPow2(x, k) ((((x) << k) | ((x) >> (31 - k))) & 0x7FFFFFFF) 
void LFSRWithInitialisationMode(u32 u) {
	u32 f, v; f = LFSR_S0;
	v = MulByPow2(LFSR_S0, 8);
	f = AddM(f, v);

	v = MulByPow2(LFSR_S4, 20);
	f = AddM(f, v);

	v = MulByPow2(LFSR_S10, 21);
	f = AddM(f, v);

	v = MulByPow2(LFSR_S13, 17);
	f = AddM(f, v);

	v = MulByPow2(LFSR_S15, 15);
	f = AddM(f, v);

	f = AddM(f, u);
	/* update the state */
	LFSR_S0 = LFSR_S1;
	LFSR_S1 = LFSR_S2;
	LFSR_S2 = LFSR_S3;
	LFSR_S3 = LFSR_S4;
	LFSR_S4 = LFSR_S5;
	LFSR_S5 = LFSR_S6;
	LFSR_S6 = LFSR_S7;
	LFSR_S7 = LFSR_S8;
	LFSR_S8 = LFSR_S9;
	LFSR_S9 = LFSR_S10;
	LFSR_S10 = LFSR_S11;
	LFSR_S11 = LFSR_S12;
	LFSR_S12 = LFSR_S13;
	LFSR_S13 = LFSR_S14;
	LFSR_S14 = LFSR_S15;
	LFSR_S15 = f;
}
/* LFSR with work mode 工作模式*/
void LFSRWithWorkMode() {
	u32 f, v; f = LFSR_S0;
	v = MulByPow2(LFSR_S0, 8);
	f = AddM(f, v);

	v = MulByPow2(LFSR_S4, 20);
	f = AddM(f, v);

	v = MulByPow2(LFSR_S10, 21);
	f = AddM(f, v);

	v = MulByPow2(LFSR_S13, 17);
	f = AddM(f, v);

	v = MulByPow2(LFSR_S15, 15);
	f = AddM(f, v);

	/* update the state */
	LFSR_S0 = LFSR_S1;
	LFSR_S1 = LFSR_S2;
	LFSR_S2 = LFSR_S3;
	LFSR_S3 = LFSR_S4;
	LFSR_S4 = LFSR_S5;
	LFSR_S5 = LFSR_S6;
	LFSR_S6 = LFSR_S7;
	LFSR_S7 = LFSR_S8;
	LFSR_S8 = LFSR_S9;
	LFSR_S9 = LFSR_S10;
	LFSR_S10 = LFSR_S11;
	LFSR_S11 = LFSR_S12;
	LFSR_S12 = LFSR_S13;
	LFSR_S13 = LFSR_S14;
	LFSR_S14 = LFSR_S15;
	LFSR_S15 = f;
}

/* BitReorganization */
void BitReorganization() {
	BRC_X0 = ((LFSR_S15 & 0x7FFF8000) << 1) | (LFSR_S14 & 0xFFFF);
	BRC_X1 = ((LFSR_S11 & 0xFFFF) << 16) | (LFSR_S9 >> 15);
	BRC_X2 = ((LFSR_S7 & 0xFFFF) << 16) | (LFSR_S5 >> 15);
	BRC_X3 = ((LFSR_S2 & 0xFFFF) << 16) | (LFSR_S0 >> 15);
}
#define ROT(a, k) (((a) << k) | ((a) >> (32 - k)))
/* L1 */
u32 L1(u32 X) {
	return (X ^ ROT(X, 2) ^ ROT(X, 10) ^ ROT(X, 18) ^ ROT(X, 24));
}
/* L2 */
u32 L2(u32 X) {
	return (X ^ ROT(X, 8) ^ ROT(X, 14) ^ ROT(X, 22) ^ ROT(X, 30));
}
#define MAKEU32(a, b, c, d) (((u32)(a) << 24) | ((u32)(b) << 16) | ((u32)(c) << 8) | ((u32)(d)))
/* F */
u32 F() {
	u32 W, W1, W2, u, v;
	W = (BRC_X0 ^ F_R1) + F_R2;
	W1 = F_R1 + BRC_X1;
	W2 = F_R2 ^ BRC_X2;
	u = L1((W1 << 16) | (W2 >> 16));
	v = L2((W2 << 16) | (W1 >> 16));
	F_R1 = MAKEU32(S0[u >> 24], S1[(u >> 16) & 0xFF],
		S0[(u >> 8) & 0xFF], S1[u & 0xFF]);
	F_R2 = MAKEU32(S0[v >> 24], S1[(v >> 16) & 0xFF],
		S0[(v >> 8) & 0xFF], S1[v & 0xFF]);
	return W;
}
#define MAKEU31(a, b, c)(((u32)(a) << 23)|((u32)(b) << 8)|(u32)(c))
/* initialize */
//初始化
void Initialization(u8* k, u8* iv) {
	u32 w, nCount;

	/* expand key */
	LFSR_S0 = MAKEU31(k[0], EK_d[0], iv[0]);
	LFSR_S1 = MAKEU31(k[1], EK_d[1], iv[1]);
	LFSR_S2 = MAKEU31(k[2], EK_d[2], iv[2]);
	LFSR_S3 = MAKEU31(k[3], EK_d[3], iv[3]);
	LFSR_S4 = MAKEU31(k[4], EK_d[4], iv[4]);
	LFSR_S5 = MAKEU31(k[5], EK_d[5], iv[5]);
	LFSR_S6 = MAKEU31(k[6], EK_d[6], iv[6]);
	LFSR_S7 = MAKEU31(k[7], EK_d[7], iv[7]);
	LFSR_S8 = MAKEU31(k[8], EK_d[8], iv[8]);
	LFSR_S9 = MAKEU31(k[9], EK_d[9], iv[9]);
	LFSR_S10 = MAKEU31(k[10], EK_d[10], iv[10]);
	LFSR_S11 = MAKEU31(k[11], EK_d[11], iv[11]);
	LFSR_S12 = MAKEU31(k[12], EK_d[12], iv[12]);
	LFSR_S13 = MAKEU31(k[13], EK_d[13], iv[13]);
	LFSR_S14 = MAKEU31(k[14], EK_d[14], iv[14]);
	LFSR_S15 = MAKEU31(k[15], EK_d[15], iv[15]);
	/* set F_R1 and F_R2 to zero */
	F_R1 = 0;
	F_R2 = 0;
	nCount = 32;
	while (nCount > 0)
	{
		BitReorganization();
		w = F();
		LFSRWithInitialisationMode(w >> 1);
		nCount--;
	}
}
//生成密钥流
void GenerateKeystream(u32* pKeystream, int KeystreamLen) {
	int i;
	{
		BitReorganization();
		F();		/* discard the output of F */
		LFSRWithWorkMode();
	}
	for (i = 0; i < KeystreamLen; i++) {
		BitReorganization();
		pKeystream[i] = F() ^ BRC_X3;
		LFSRWithWorkMode();
	}
}
/*Test1 :测试密钥的生成 :
input:
	Key: 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
	IV: 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
output: z1: 27bede74
		z2: 018082da
*/


int main() {
	int i;
	u32 z[10];
	u8 k1[] = {0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00};
	u8 iv1[] = {0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x00};
	Initialization(k1, iv1);
	GenerateKeystream(z, 2);
	for (i = 0; i < 2; i++) {
		printf("Z[%d],%08X\n",i, z[i]);
	}
}

代码实现：解密

流密码进行，一般会给出k,iv以及一个密文序列，那么进行异或就可以出来明文

Author: L0x1c

Link: https://l0x1c.github.io/2020/01/03/2020-1-03/

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