嵌入式的RSA非对称加密算法

汽车电子技术 2023-01-20 2008

汽车电子技术

1333 文章 434.2w阅读 187粉丝

描述

1、对称加密算法

对称加密算法是应用较早的加密算法，数据发送方将明文和密钥经加密算法处理，使其变成密文发送出去；接收方收到密文后，使用和加密算法相同的密钥进行逆算法解密，还原出明文。在对称加密算法中，使用的密钥只有一个，收发双方使用相同的密钥对数据进行加密或解密。

数据

双方都必须保管好密钥，任一方的密钥泄露，都会导致加密信息不安全；尤其是双方协商更换密钥过程中，密钥会出现在传输过程中，严重影响数据的安全性。

对称加密算法常用的AES可以参考[ 嵌入式算法6---AES加密/解密算法 ]

2、非对称加密算法

和对称加密算法最大的区别是，非对称加密算法需要两个密钥，公开密钥（public key 简称公钥）和私有密钥（private key 简称私钥），且公钥与私钥是互相关联的一对。使用公钥对数据进行加密，只有用对应的私钥才能解密，私钥加密签名也只有公钥能解密验签。

数据

非对称加密算法实现机密信息交换的基本过程：

1、甲方生成一对密钥并将公钥公开，私钥保密

2、乙方使用甲方的提供的公钥，对机密信息加密后再发送给甲方；甲方使用自己私钥对加密后的信息进行解密

3、甲方也可以使用自己的私钥对机密信息进行签名后再发送给乙方，乙方用甲方的提供公钥对甲方发来的密文进行验签

非对称加密算法的特点：

1、公钥公开，私钥私藏，无需双方传输密钥协商，所以安全性比对称加密算法更高

2、非对称加密的算法复杂，运算速度比对称加密解密的速度慢很多

3、一般情况下使用非对称加密保护对称加密的密钥，密钥协商后使用对称加密进行通信

4、最佳实现是双方各自保存自己的私钥，使用对方的公钥加密数据传输

3、RSA算法与密钥

非对称加密算法中最常用的当属 RSA ，其算法本身基于一个简单的数论知识，给出两个素数，很容易将它们相乘，然而给出它们的乘积，想得到这两个素数就显得尤为困难。具体的私钥与公钥生成原理和加密、解密过程，不是本文关注的重点。

私钥和公钥的生成，可以借助mbedtls源码或openSSL工具生成，举例如下：

1、安装openSSL，下载地址

https://www.openssl.org/

2、安装后进入openSSL命令行界面，使用命令生成RSA2048的私钥，存入private.key文件

OpenSSL>genrsa -out private.key 2048

3、基于公钥生成私钥，存入文件public.key

OpenSSL> rsa -in private.key -pubout -out public.key

4、有些算法库采用传入指数、模数方式进行加解密，而前面生成的公私钥是PEM格式，需要变成Exponent、Modulus形式，就可以使用以下工具在线转换。

https://www.oren.net.cn/rsa/info.html

5、关于mbedtls应用，可以参考[ mbedtls 基础及其应用 ]，该开源库针对嵌入式系统，且囊括了很多常用的算法。

4、源码

以下是RSA2048的C源码和验证范例，基于Qt测试，也可以结合硬件性能改为RSA1024，移植时注意适配形如 portable_***的三个API。

/************************************/
//关注微信公众号  嵌入式系统
/************************************/
//rsa.h
#include "stdlib.h"

#define RSA_ENCODE_LEN  (2048/8)   //RSA2048即256字节，可以视硬件情况改为1024

typedef unsigned char uint8_t;
typedef unsigned short int uint16_t;
typedef unsigned int  uint32_t;

#define BI_MAXLEN 130
#define DEC 10
#define HEX 16

#define CARRYOVER  0x10000
#define CARRYLAST   0xFFFF

typedef struct
{
    uint32_t m_nLength;    //大数在0x1 00 00 00 00进制下的长度
    uint16_t m_ulValue[BI_MAXLEN];    //用数组记录大数在0x100000000进制下每一位的值
} CBigInt;


//rsa.c
#include "rsa.h"
#include "time.h"

/******************* 适配API *******************/
#define portable_malloc  malloc
#define portable_free    free

//随机数种子源
uint32_t portable_rand_seed(void)
{
    time_t timestamp;
    time(×tamp);
    return timestamp;
}
/******************* 适配API *******************/

/*****************************************************************
基本操作与运算
Init, 构造大数对象并初始化为零
Mov，赋值运算，可赋值为大数或普通整数，可重载为运算符“=”
Cmp，比较运算，可重载为运算符“==”、“!=”、“>=”、“<=”等
Add，加，求大数与大数或大数与普通整数的和，可重载为运算符“+”
Sub，减，求大数与大数或大数与普通整数的差，可重载为运算符“-”
Mul，乘，求大数与大数或大数与普通整数的积，可重载为运算符“*”
Div，除，求大数与大数或大数与普通整数的商，可重载为运算符“/”
Mod，模，求大数与大数或大数与普通整数的模，可重载为运算符“%”
*****************************************************************/
static CBigInt *Mov_Big_Long(CBigInt *X, uint32_t A);
static CBigInt *Mov_Big_Big(CBigInt *X, CBigInt *A);
static CBigInt *Add_Big_Big(CBigInt *X, CBigInt *A);
static CBigInt *Sub_Big_Big(CBigInt *X, CBigInt *A);
static CBigInt *Mul_Big_Big(CBigInt *X, CBigInt *A);
static CBigInt *Div_Big_Big(CBigInt *X, CBigInt *A);
static CBigInt *Mod_Big_Big(CBigInt *X, CBigInt *A);
static CBigInt *Add_Big_Long(CBigInt *X, uint32_t A);
static CBigInt *Sub_Big_Long(CBigInt *X, uint32_t A);
static CBigInt *Mul_Big_Long(CBigInt *X, uint32_t A);
static CBigInt *Div_Big_Long(CBigInt *X, uint32_t A);
static uint32_t Mod_Big_Long(CBigInt *N, uint32_t A);
static int Cmp(CBigInt *N, CBigInt *A);

/*****************************************************************
输入输出
Get，从字符串按10进制或16进制格式输入到大数
Put，将大数按10进制或16进制格式输出到字符串
*****************************************************************/
static CBigInt *Get(CBigInt *N, char *str, uint32_t system);
static char *Put(CBigInt *N, uint32_t system);

/*****************************************************************
RSA相关运算
Rab，拉宾米勒算法进行素数测试
Euc，欧几里德算法求解同余方程
RsaTrans，反复平方算法进行幂模运算
GetPrime，产生指定长度的随机大素数
*****************************************************************/
static int Rab(CBigInt *N);
static CBigInt *Euc(CBigInt *X, CBigInt *A);
static CBigInt *RsaTrans(CBigInt *X, CBigInt *A, CBigInt *B);
static CBigInt *GetPrime(CBigInt *X, int bits);


/*****************************************************************
大数运算库源文件：BigInt.c
说明：适用于C，linux系统 1024位RSA运算
*****************************************************************/
//小素数表
const static int PrimeTable[550] =
{
    3,     5,     7,     11,   13,   17,   19,   23,   29,   31,    37,   41,   43,   47,   53,   59,   61,   67,   71,   73,
    79,   83,   89,   97,   101,   103,   107,   109,   113,   127,    131,   137,   139,   149,   151,   157,   163,   167,   173,   179,
    181,   191,   193,   197,   199,   211,   223,   227,   229,   233,    239,   241,   251,   257,   263,   269,   271,   277,   281,   283,
    293,   307,   311,   313,   317,   331,   337,   347,   349,   353,    359,   367,   373,   379,   383,   389,   397,   401,   409,   419,
    421,   431,   433,   439,   443,   449,   457,   461,   463,   467,    479,   487,   491,   499,   503,   509,   521,   523,   541,   547,
    557,   563,   569,   571,   577,   587,   593,   599,   601,   607,    613,   617,   619,   631,   641,   643,   647,   653,   659,   661,
    673,   677,   683,   691,   701,   709,   719,   727,   733,   739,    743,   751,   757,   761,   769,   773,   787,   797,   809,   811,
    821,   823,   827,   829,   839,   853,   857,   859,   863,   877,    881,   883,   887,   907,   911,   919,   929,   937,   941,   947,
    953,   967,   971,   977,   983,   991,   997,   1009, 1013, 1019,    1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087,
    1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153,    1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229,
    1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297,    1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381,
    1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453,    1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523,
    1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597,    1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663,
    1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741,    1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823,
    1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901,    1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993,
    1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063,    2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131,
    2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221,    2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293,
    2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371,    2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437,
    2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539,    2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621,
    2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689,    2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749,
    2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833,    2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909,
    2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001,    3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083,
    3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187,    3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259,
    3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343,    3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433,
    3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517,    3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581,
    3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659,    3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, 3733,
    3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823,    3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911,
    3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, 4001
};

/****************************************************************************************
大数比较
调用方式：Cmp(N,A)
返回值：若NA返回1
****************************************************************************************/
static int Cmp(CBigInt *N, CBigInt *A)
{
    int i;
    if(N->m_nLength > A->m_nLength)
    {
        return 1;
    }
    if(N->m_nLength < A->m_nLength)
    {
        return -1;
    }
    for(i = N->m_nLength - 1; i >= 0; i--)
    {
        if(N->m_ulValue[i] > A->m_ulValue[i])
        {
            return 1;
        }
        if(N->m_ulValue[i] < A->m_ulValue[i])
        {
            return -1;
        }
    }
    return 0;
}

/****************************************************************************************
大数赋值
调用方式：__Mov_Big_Big(A)
返回值：N,被赋值为A
****************************************************************************************/
static CBigInt *Mov_Big_Big(CBigInt *X, CBigInt *A)
{
    memcpy(X, A, sizeof(CBigInt));
    return X;
}

static CBigInt *Mov_Big_Long(CBigInt *N, uint32_t A)
{
    int i;
    if(A > CARRYLAST)
    {
        N->m_nLength = 2;
        N->m_ulValue[1] = (uint16_t)(A >> 16);
        N->m_ulValue[0] = (uint16_t)A;
    }
    else
    {
        N->m_nLength = 1;
        N->m_ulValue[0] = (uint16_t)A;
    }
    memset((unsigned char*)&N->m_ulValue[N->m_nLength], 0, sizeof(uint16_t) * (BI_MAXLEN - N->m_nLength));
    return N;
}

/****************************************************************************************
大数相加
调用形式：Add_Big_Big(X,A)
返回值：X=X+A
****************************************************************************************/
static CBigInt *Add_Big_Big(CBigInt *X, CBigInt *A)
{
    uint32_t i;
    uint16_t carry = 0;
    uint32_t sum = 0;
    if(X->m_nLength < A->m_nLength)
    {
        X->m_nLength = A->m_nLength;
    }
    for(i = 0; i < X->m_nLength; i++)
    {
        sum = A->m_ulValue[i];
        sum = sum + X->m_ulValue[i] + carry;
        X->m_ulValue[i] = (uint16_t)sum;
        carry = (uint16_t)(sum >> 16);
    }
    X->m_ulValue[X->m_nLength] = carry;
    X->m_nLength += carry;
    return X;
}

static CBigInt *Add_Big_Long(CBigInt *X, uint32_t A)
{
    uint32_t sum;
    sum = X->m_ulValue[0];
    sum += A;
    X->m_ulValue[0] = (uint16_t)sum;
    if(sum > CARRYLAST)
    {
        uint32_t i = 1;
        while(X->m_ulValue[i] == CARRYLAST)
        {
            X->m_ulValue[i] = 0;
            i++;
        }
        X->m_ulValue[i]++;
        if(X->m_nLength == i)
        {
            X->m_nLength++;
        }
    }
    return X;
}

/****************************************************************************************
大数相减
调用形式：Sub_Big_Big(X,A)
返回值：X=X-A
****************************************************************************************/
static CBigInt *Sub_Big_Big(CBigInt *X, CBigInt *A)
{
    if(Cmp(X, A) <= 0)
    {
        memset(X, 0, sizeof(CBigInt));
        return X;
    }
    else
    {
        uint16_t carry = 0;
        uint32_t num;
        uint32_t i;
        for(i = 0; i < X->m_nLength; i++)
        {
            if((X->m_ulValue[i] > A->m_ulValue[i]) || ((X->m_ulValue[i] == A->m_ulValue[i]) && (carry == 0)))
            {
                X->m_ulValue[i] = X->m_ulValue[i] - carry - A->m_ulValue[i];
                carry = 0;
            }
            else
            {
                num = CARRYOVER + X->m_ulValue[i];
                X->m_ulValue[i] = (uint32_t)(num - carry - A->m_ulValue[i]);
                carry = 1;
            }
        }
        while(X->m_ulValue[X->m_nLength - 1] == 0)
        {
            X->m_nLength--;
        }
        return X;
    }
}

static CBigInt *Sub_Big_Long(CBigInt *X, uint32_t A)
{
    if(X->m_ulValue[0] >= A)
    {
        X->m_ulValue[0] -= A;
        return X;
    }
    if(X->m_nLength == 1)
    {
        memset(X, 0, sizeof(CBigInt));
        return X;
    }
    else
    {
        uint32_t num = CARRYOVER + X->m_ulValue[0];
        int i = 1;
        X->m_ulValue[0] = (uint16_t)(num - A);
        while(X->m_ulValue[i] == 0)
        {
            X->m_ulValue[i] = CARRYLAST;
            i++;
        }
        X->m_ulValue[i]--;
        if(X->m_ulValue[i] == 0)
        {
            X->m_nLength--;
        }
        return X;
    }
}

/****************************************************************************************
大数相乘
调用形式：Mul_Big_Big(N,A)
返回值：X=N*A
    A a 0
    N c d
        0     d*0
        1   c*0
                    d*a
        2 c*a

****************************************************************************************/
static CBigInt *Mul_Big_Big(CBigInt *X, CBigInt *A)
{
    if(A->m_nLength == 1)
    {
        return Mul_Big_Long(X, A->m_ulValue[0]);
    }
    else
    {
        uint32_t sum, mul = 0, carry = 0;
        uint32_t i, j;
        CBigInt N = {0};
        memcpy(&N, X, sizeof(CBigInt));
        memset(X, 0, sizeof(CBigInt));
        X->m_nLength = N.m_nLength + A->m_nLength - 1;
        for(i = 0; i < X->m_nLength; i++)
        {
            sum = carry;
            carry = 0;
            for(j = 0; j < A->m_nLength; j++)
            {
                if(((i - j) >= 0) && ((i - j) < N.m_nLength))
                {
                    mul = N.m_ulValue[i - j];
                    mul *= A->m_ulValue[j];
                    carry += mul >> 16;
                    mul = mul & CARRYLAST;
                    sum += mul;
                }
            }
            carry += sum >> 16;
            X->m_ulValue[i] = (uint16_t)sum;
        }
        if(carry)
        {
            X->m_nLength++;
            X->m_ulValue[X->m_nLength - 1] = (uint16_t)carry;
        }
        return X;
    }
}

static CBigInt *Mul_Big_Long(CBigInt *X, uint32_t A)
{
    uint32_t mul;
    uint32_t carry = 0;
    uint32_t i;
    for(i = 0; i < X->m_nLength; i++)
    {
        mul = X->m_ulValue[i];
        mul = mul * A + carry;
        X->m_ulValue[i] = (uint16_t)mul;
        carry = (uint16_t)(mul >> 16);
    }
    if(carry)
    {
        X->m_nLength++;
        X->m_ulValue[X->m_nLength - 1] = carry;
    }
    return X;
}

/****************************************************************************************
大数相除
调用形式：Div_Big_Big(N,A)
返回值：X=N/A
****************************************************************************************/
static CBigInt *Div_Big_Big(CBigInt *X, CBigInt *A)
{
    CBigInt Y = {0}, Z = {0}, T;
    if(A->m_nLength == 1)
    {
        return Div_Big_Long(X, A->m_ulValue[0]);
    }
    else
    {
        uint32_t i, len;
        uint32_t num, div;
        memcpy(&Y, X, sizeof(CBigInt));
        while(Cmp(&Y, A) >= 0)
        {
            div = Y.m_ulValue[Y.m_nLength - 1];
            num = A->m_ulValue[A->m_nLength - 1];
            len = Y.m_nLength - A->m_nLength;
            if((div == num) && (len == 0))
            {
                Add_Big_Long(X, 1);
                break;
            }
            if((div <= num) && len)
            {
                len--;
                div = (div << 16) + Y.m_ulValue[Y.m_nLength - 2];
            }
            div = div / (num + 1);
            Mov_Big_Long(&Z, div);
            if(len)
            {
                Z.m_nLength += len;
                for(i = Z.m_nLength - 1; i >= len; i--)
                {
                    Z.m_ulValue[i] = Z.m_ulValue[i - len];
                }
                for(i = 0; i < len; i++)
                {
                    Z.m_ulValue[i] = 0;
                }
            }
            Add_Big_Big(X, &Z);
            memcpy(&T, A, sizeof(CBigInt));
            Mul_Big_Big(&T, &Z);
            Sub_Big_Big(&Y, &T);
        }
        return X;
    }
}
static CBigInt *Div_Big_Long(CBigInt *X, uint32_t A)
{
    if(X->m_nLength == 1)
    {
        X->m_ulValue[0] = X->m_ulValue[0] / A;
        return X;
    }
    else
    {
        uint32_t div, mul;
        uint32_t carry = 0;
        int i;
        for(i = X->m_nLength - 1; i >= 0; i--)
        {
            div = carry;
            div = (div << 16) + X->m_ulValue[i];
            X->m_ulValue[i] = (uint16_t)(div / A);
            mul = (div / A) * A;
            carry = (uint16_t)(div - mul);
        }
        if(X->m_ulValue[X->m_nLength - 1] == 0)
        {
            X->m_nLength--;
        }
        return X;
    }
}

/****************************************************************************************
大数求模
调用形式：Mod_Big_Big(N,A)
返回值：X=N%A
****************************************************************************************/
static CBigInt *Mod_Big_Big(CBigInt *X, CBigInt *A)
{
    CBigInt Y = {0}, Z;
    uint32_t div, num;
    uint32_t carry = 0;
    uint32_t i, len;
    while(Cmp(X, A) >= 0)
    {
        div = X->m_ulValue[X->m_nLength - 1];
        num = A->m_ulValue[A->m_nLength - 1];
        len = X->m_nLength - A->m_nLength;
        if((div == num) && (len == 0))
        {
            Sub_Big_Big(X, A);
            break;
        }
        if((div <= num) && len)
        {
            len--;
            div = (div << 16) + X->m_ulValue[X->m_nLength - 2];
        }
        div = div / (num + 1);
        Mov_Big_Long(&Y, div);
        memcpy(&Z, A, sizeof(CBigInt));
        Mul_Big_Big(&Z, &Y);
        memcpy(&Y, &Z, sizeof(CBigInt));
        if(len)
        {
            Y.m_nLength += len;
            for(i = Y.m_nLength - 1; i >= len; i--)
            {
                Y.m_ulValue[i] = Y.m_ulValue[i - len];
            }
            for(i = 0; i < len; i++)
            {
                Y.m_ulValue[i] = 0;
            }
        }
        Sub_Big_Big(X, &Y);
    }
    return X;
}

static uint32_t Mod_Big_Long(CBigInt *N, uint32_t A)
{
    if(N->m_nLength == 1)
    {
        return(N->m_ulValue[0] % A);
    }
    else
    {
        uint32_t div;
        uint32_t carry = 0;
        int i;
        for(i = N->m_nLength - 1; i >= 0; i--)
        {
            div = N->m_ulValue[i];
            div += carry * CARRYOVER;
            carry = (uint16_t)(div % A);
        }
        return carry;
    }
}

/****************************************************************************************
从字符串按10进制或16进制格式输入到大数
调用格式：Get(N,str,sys)
返回值：N被赋值为相应大数
sys暂时只能为10或16
****************************************************************************************/
static CBigInt *Get(CBigInt *N, char *s, uint32_t system)
{
    int i;
    int len = strlen(s), k;
    memset(N, 0, sizeof(CBigInt));
    N->m_nLength = 1;
    for(i = 0; i < len; i++)
    {
        Mul_Big_Long(N, system);
        if((s[i] >= '0') && (s[i] <= '9'))
        {
            k = s[i] - 48;
        }
        else if((s[i] >= 'A') && (s[i] <= 'F'))
        {
            k = s[i] - 55;
        }
        else if((s[i] >= 'a') && (s[i] <= 'f'))
        {
            k = s[i] - 87;
        }
        else
        {
            k = 0;
        }
        Add_Big_Long(N, k);
    }
    return N;
}
static CBigInt *GetHex(CBigInt *N, unsigned char *s, unsigned short len, uint32_t system)
{
    int i, j;
    unsigned char *p = (unsigned char*)N->m_ulValue;
    memset(N, 0, sizeof(CBigInt));
    N->m_nLength = 1;
    for(i = len - 1, j = 0; i >= 0; i--, j++)
    {
        p[j] = s[i];
    }
    i = len % 2;
    if(i > 0)
    {
        N->m_nLength = len / 2 + 1;
    }
    else
    {
        N->m_nLength = len / 2;
    }
    return N;
}
/****************************************************************************************
将大数按10进制或16进制格式输出为字符串
调用格式：Put(N,str,sys)
返回值：无，参数str被赋值为N的sys进制字符串
sys暂时只能为10或16
****************************************************************************************/
static char *Put(CBigInt *N, uint32_t system)
{
    char t[17] = "0123456789ABCDEF";
    int i, a;
    static char s[2048];

    if((N->m_nLength == 1) && (N->m_ulValue[0] == 0))
    {
        return NULL;
    }
    else
    {
        CBigInt X = {0};
        memcpy(&X, N, sizeof(CBigInt));
        memset(s, 0, 2048);
        for(i = 2046; X.m_ulValue[X.m_nLength - 1] > 0 && i > 0; i--)
        {
            a = Mod_Big_Long(&X, system);
            s[i] = t[a];
            Div_Big_Long(&X, system);
        }
        if(i % 2 == 0)
        {
            return &s[i + 1];
        }
        else
        {
            s[i] = '0';
            return &s[i];
        }
    }
}

static void PutHex(CBigInt *N, uint8_t *out, uint16_t *len)
{
    int i, j, size;
    if((N->m_nLength == 1) && (N->m_ulValue[0] == 0))
    {
        return;
    }
    size = N->m_nLength * sizeof(N->m_ulValue[0]);
    for(i = size - 1, j = 0; i >= 0; i--, j++)
    {
        out[j] = ((uint8_t*)N->m_ulValue)[i];
    }
    *len = size;
}

/****************************************************************************************
求不定方程ax-by=1的最小整数解
调用方式：Euc(N,A)
返回值：X,满足：NX mod A=1
****************************************************************************************/
static CBigInt *Euc(CBigInt *X, CBigInt *A)
{
    CBigInt M = {0}, E = {0}, N = {0}, Y = {0}, I = {0}, J = {0};
    int x, y;
    memcpy(&E, X, sizeof(CBigInt));
    memcpy(&M, A, sizeof(CBigInt));
    Mov_Big_Long(X, 0);
    Mov_Big_Long(&Y, 1);
    x = y = 1;
    while((E.m_nLength != 1) || (E.m_ulValue[0] != 0))
    {
        memcpy(&I, &M, sizeof(CBigInt));
        Div_Big_Big(&I, &E);
        memcpy(&J, &M, sizeof(CBigInt));
        Mod_Big_Big(&J, &E);
        memcpy(&M, &E, sizeof(CBigInt));
        memcpy(&E, &J, sizeof(CBigInt));
        memcpy(&J, &Y, sizeof(CBigInt));
        Mul_Big_Big(&Y, &I);
        if(x == y)
        {
            if(Cmp(X, &Y) >= 0)
            {
                Sub_Big_Big(&Y, X);
            }
            else
            {
                Sub_Big_Big(&Y, X);
                y = 0;
            }
        }
        else
        {
            Add_Big_Big(&Y, X);
            x = 1 - x;
            y = 1 - y;
        }
        memcpy(X, &J, sizeof(CBigInt));
    }
    if(x == 0)
    {
        Sub_Big_Big(X, A);
    }
    return X;
}

/****************************************************************************************
求乘方的模
调用方式：RsaTrans(N,A,B)
返回值：X=N^A MOD B
****************************************************************************************/
static CBigInt *RsaTrans(CBigInt *X, CBigInt *A, CBigInt *B)
{
    CBigInt N = {0}, Y = {0}, Z;
    int i, j, k;
    uint32_t n;
    uint32_t num;
    k = A->m_nLength * 16 - 16;
    num = A->m_ulValue[A->m_nLength - 1];
    while(num)
    {
        num = num >> 1;
        k++;
    }
    memcpy(&N, X, sizeof(CBigInt));
    for(i = k - 2; i >= 0; i--)
    {
        memcpy(&Y, X, sizeof(CBigInt));
        Mul_Big_Long(&Y, X->m_ulValue[X->m_nLength - 1]);
        Mod_Big_Big(&Y, B);
        for(n = 1; n < X->m_nLength; n++)
        {
            for(j = Y.m_nLength; j > 0; j--)
            {
                Y.m_ulValue[j] = Y.m_ulValue[j - 1];
            }
            Y.m_ulValue[0] = 0;
            Y.m_nLength++;
            memcpy(&Z, X, sizeof(CBigInt));
            Mul_Big_Long(&Z, X->m_ulValue[X->m_nLength - n - 1]);
            Add_Big_Big(&Y, &Z);
            Mod_Big_Big(&Y, B);
        }
        memcpy(X, &Y, sizeof(CBigInt));
        if((A->m_ulValue[i >> 4] >> (i & 15)) & 1)
        {
            memcpy(&Y, &N, sizeof(CBigInt));
            Mul_Big_Long(&Y, X->m_ulValue[X->m_nLength - 1]);
            Mod_Big_Big(&Y, B);
            for(n = 1; n < X->m_nLength; n++)
            {
                for(j = Y.m_nLength; j > 0; j--)
                {
                    Y.m_ulValue[j] = Y.m_ulValue[j - 1];
                }
                Y.m_ulValue[0] = 0;
                Y.m_nLength++;
                memcpy(&Z, &N, sizeof(CBigInt));
                Mul_Big_Long(&Z, X->m_ulValue[X->m_nLength - n - 1]);
                Add_Big_Big(&Y, &Z);
                Mod_Big_Big(&Y, B);
            }
            memcpy(X, &Y, sizeof(CBigInt));
        }
    }
    return X;
}

/****************************************************************************************
拉宾米勒算法测试素数
调用方式：Rab(N)
返回值：若N为素数，返回1，否则返回0
****************************************************************************************/
static int Rab(CBigInt *N)
{
    CBigInt S = {0}, A = {0}, I = {0}, K = {0};
    uint32_t i, j, pass;
    for(i = 0; i < 550; i++)
    {
        if(Mod_Big_Long(N, PrimeTable[i]) == 0)
        {
            return 0;
        }
    }
    memcpy(&K, N, sizeof(CBigInt));
    K.m_ulValue[0]--;
    for(i = 0; i < 5; i++)
    {
        pass = 0;
        Mov_Big_Long(&A, rand()*rand());
        memcpy(&S, &K, sizeof(CBigInt));
        while((S.m_ulValue[0] & 1) == 0)
        {
            for(j = 0; j < S.m_nLength; j++)
            {
                S.m_ulValue[j] = S.m_ulValue[j] >> 1;
                if(S.m_ulValue[j + 1] & 1)
                {
                    S.m_ulValue[j] = S.m_ulValue[j] | 0x8000;
                }
            }
            if(S.m_ulValue[S.m_nLength - 1] == 0)
            {
                S.m_nLength--;
            }
            memcpy(&I, &A, sizeof(CBigInt));
            RsaTrans(&I, &S, N);
            if(Cmp(&I, &K) == 0)
            {
                pass = 1;
                break;
            }
        }
        if((I.m_nLength == 1) && (I.m_ulValue[0] == 1))
        {
            pass = 1;
        }
        if(pass == 0)
        {
            return 0;
        }
    }
    return 1;
}

/****************************************************************************************
产生随机素数
调用方法：GetPrime(N,bits)
返回值：N,被赋值为一个bits位（0x100000000进制长度）的素数
****************************************************************************************/
static CBigInt *GetPrime(CBigInt *N, int bits)
{
    uint32_t i;
    CBigInt S = {0}, A = {0}, I = {0}, K = {0};

    memset(N, 0, sizeof(CBigInt));
    N->m_nLength = bits;
begin:
    srand(portable_rand_seed());
    for(i = 0; i < N->m_nLength; i++)
    {
        N->m_ulValue[i] = rand() * 0x100 + rand();
    }
    N->m_ulValue[0] = N->m_ulValue[0] | 1;
    for(i = N->m_nLength - 1; i > 0; i--)
    {
        N->m_ulValue[i] = N->m_ulValue[i] << 1;
        if(N->m_ulValue[i - 1] & 0x8000)
        {
            N->m_ulValue[i]++;
        }
    }
    N->m_ulValue[0] = N->m_ulValue[0] << 1;
    N->m_ulValue[0]++;
    for(i = 0; i < 550; i++)
    {
        if(Mod_Big_Long(N, PrimeTable[i]) == 0)
        {
            goto begin;
        }
    }
    memcpy(&K, N, sizeof(CBigInt));
    K.m_ulValue[0]--;
    for(i = 0; i < 5; i++)
    {
        Mov_Big_Long(&A, rand()*rand());
        memcpy(&S, &K, sizeof(CBigInt));
        Div_Big_Long(&S, 2);
        memcpy(&I, &A, sizeof(CBigInt));
        RsaTrans(&I, &S, N);
        if(((I.m_nLength != 1) || (I.m_ulValue[0] != 1)) && (Cmp(&I, &K) != 0))
        {
            goto begin;
        }
    }
    return N;
}

/***********************************************************************/

static void entropy_poll(unsigned char *output, unsigned int len)
{
    if(len > 0)
    {
        int i;
        srand(portable_rand_seed);
        for(i = 0; i < len; i++)
        {
            output[i] = rand() % 0xff + 1;
        }
    }
}

static char *del_PKCS1Padding(char *src)
{
    int len = strlen(src);
    if(len % 2 == 1)
    {
        src++;
    }
    while(*src != 0 && *(src + 1) != 0)
    {
        if(*src == '0' && *(src + 1) == '0')
        {
            src += 2;
            break;
        }
        src += 2;
    }
    return src;
}

static int add_PKCS1Padding(unsigned char *src, unsigned int len, unsigned char *out)
{
    if(len > RSA_ENCODE_LEN - 11)
    {
        return -1;
    }
    else
    {
        /*要加密的msg*/
        memcpy(&out[RSA_ENCODE_LEN - len], src, len);
        out[0] = 0;
        out[1] = 2;
        /*至少8字节的随机数*/
        entropy_poll(&out[2], RSA_ENCODE_LEN - 3 - len);
        out[RSA_ENCODE_LEN - len - 1] = 0;
        return 0;
    }
}

static int PKCS1PKCS1PaddingHexRemove(unsigned char *input, unsigned short *len, unsigned char *output)
{
    if(input[0] == 0 && (input[1] == 1 || input[1] == 2))
    {
        int i;
        for(i = 2; i < *len; i++)
        {
            if(input[i] == 0)
            {
                *len -= (i + 1);
                memcpy(output, &input[i + 1], *len);
                return *len;
            }
        }
    }
    return -1;
}


int RSA2048_pri_PKCS1Padding_Encode(unsigned char *data, unsigned short len, unsigned char *out, char *publicKey, char *ModulusHex)
{
    unsigned char buf[RSA_ENCODE_LEN];
    CBigInt N, E;
    CBigInt mw, mi, jm;
    uint16_t outlen = RSA_ENCODE_LEN;

    //Get(&N, Modulus, 16);//string
    GetHex(&N, ModulusHex, RSA_ENCODE_LEN, 16);//hex array
    Get(&E, publicKey, 16);

    add_PKCS1Padding(data, len, buf);

    GetHex(&mw, buf, RSA_ENCODE_LEN, 16);

    RsaTrans(&mw, &E, &N);

    PutHex(&mw, out, &outlen);
    return outlen;
}

int RSA2048_pub_PKCS1Padding_Encode(unsigned char *data, unsigned short len, unsigned char *out, char *publicKey, unsigned char *ModulusHex)
{
    unsigned char buf[RSA_ENCODE_LEN];
    CBigInt N, E;
    CBigInt mw, mi, jm;
    uint16_t outlen = RSA_ENCODE_LEN;

    GetHex(&N, ModulusHex, RSA_ENCODE_LEN, 16);//hex array
    Get(&E, publicKey, 16);

    add_PKCS1Padding(data, len, buf);

    GetHex(&mw, buf, RSA_ENCODE_LEN, 16);

    RsaTrans(&mw, &E, &N);

    PutHex(&mw, out, &outlen);
    return outlen;
}

int RSA2048_pri_PKCS1Padding_Decode(unsigned char *data, unsigned short *len, unsigned char *out, char *privateKey, char *ModulusHex)
{
    unsigned char buf[RSA_ENCODE_LEN];
    CBigInt N, D;
    CBigInt mw, jm;

    //Get(&N, Modulus, 16);//string
    GetHex(&N, ModulusHex, RSA_ENCODE_LEN, 16);//hex array
    Get(&D, privateKey, 16);

    GetHex(&mw, data, *len, 16);

    RsaTrans(&mw, &D, &N);
    PutHex(&mw, buf, len);
    PKCS1PKCS1PaddingHexRemove(buf, len, out);
    return 0;
}
int RSA2048_pub_PKCS1Padding_Decode(unsigned char *data, unsigned short *len, unsigned char *out, char *privateKey, unsigned char *ModulusHex)
{
    unsigned char buf[RSA_ENCODE_LEN];
    CBigInt N, D;
    CBigInt mw, jm;
    int t_len = 0;

    GetHex(&N, ModulusHex, RSA_ENCODE_LEN, 16);
    Get(&D, privateKey, 16);

    GetHex(&mw, data, *len, 16);

    RsaTrans(&mw, &D, &N);
    PutHex(&mw, buf, len);
    t_len = PKCS1PKCS1PaddingHexRemove(buf, len, out);
    return t_len;
}


//test
static const unsigned char base64_table[65] = "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/";

//需要释放内存
unsigned char * base64_encode(const unsigned char *src, size_t len, size_t *out_len)
{
    unsigned char *out, *pos;
    const unsigned char *end, *in;
    size_t olen;
    int line_len;

    olen = len * 4 / 3 + 4; /* 3-byte blocks to 4-byte */
    olen += olen / 72; /* line feeds */
    olen++; /* nul termination */
    if(olen < len)
    {
        return NULL;    /* integer overflow */
    }
    out = portable_malloc(olen);
    if(out == NULL)
    {
        return NULL;
    }

    end = src + len;
    in = src;
    pos = out;
    line_len = 0;
    while(end - in >= 3)
    {
        *pos++ = base64_table[in[0] >> 2];
        *pos++ = base64_table[((in[0] & 0x03) << 4) | (in[1] >> 4)];
        *pos++ = base64_table[((in[1] & 0x0f) << 2) | (in[2] >> 6)];
        *pos++ = base64_table[in[2] & 0x3f];
        in += 3;
        line_len += 4;
        if(line_len >= 72)
        {
            *pos++ = '\\n';
            line_len = 0;
        }
    }

    if(end - in)
    {
        *pos++ = base64_table[in[0] >> 2];
        if(end - in == 1)
        {
            *pos++ = base64_table[(in[0] & 0x03) << 4];
            *pos++ = '=';
        }
        else
        {
            *pos++ = base64_table[((in[0] & 0x03) << 4) |
                                                  (in[1] >> 4)];
            *pos++ = base64_table[(in[1] & 0x0f) << 2];
        }
        *pos++ = '=';
        line_len += 4;
    }

    if(line_len)
    {
        *pos++ = '\\n';
    }

    *pos = '\\0';
    if(out_len)
    {
        *out_len = pos - out;
    }
    return out;
}

//需要释放内存
unsigned char * base64_decode(const unsigned char *src, size_t len,
                              size_t *out_len)
{
    unsigned char dtable[256], *out, *pos, block[4], tmp;
    size_t i, count, olen;
    int pad = 0;

    memset(dtable, 0x80, 256);
    for(i = 0; i < sizeof(base64_table) - 1; i++)
    {
        dtable[base64_table[i]] = (unsigned char) i;
    }
    dtable['='] = 0;

    count = 0;
    for(i = 0; i < len; i++)
    {
        if(dtable[src[i]] != 0x80)
        {
            count++;
        }
    }

    if(count == 0 || count % 4)
    {
        return NULL;
    }

    olen = count / 4 * 3;
    pos = out = portable_malloc(olen);
    if(out == NULL)
    {
        return NULL;
    }

    count = 0;
    for(i = 0; i < len; i++)
    {
        tmp = dtable[src[i]];
        if(tmp == 0x80)
        {
            continue;
        }

        if(src[i] == '=')
        {
            pad++;
        }
        block[count] = tmp;
        count++;
        if(count == 4)
        {
            *pos++ = (block[0] << 2) | (block[1] >> 4);
            *pos++ = (block[1] << 4) | (block[2] >> 2);
            *pos++ = (block[2] << 6) | block[3];
            count = 0;
            if(pad)
            {
                if(pad == 1)
                {
                    pos--;
                }
                else if(pad == 2)
                {
                    pos -= 2;
                }
                else
                {
                    /* Invalid padding */
                    portable_free(out);
                    return NULL;
                }
                break;
            }
        }
    }

    *out_len = pos - out;
    return out;
}

int main(int argc, char *argv[])
{
 //原始公钥-私钥
    /*
    -----BEGIN PUBLIC KEY-----
    MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEAwmejRhw/SB2xB3rgJYhg
    OWBDX/DponDVVPzTWhn3J4INv6jUa9HDkeHhys4OOZTNajr8kRy4TIemotnIYONJ
    noW7VyIQEAkEyxcMett5mqRPBLuyc8Fygn4ho/rd9JId4+PgKLmr6NVcuZCpVXPe
    gyqNx0nR/UojISbq/Bu+NlcStmicZUuAeVbkGGUOlvtMzFehkBPwE31EdpYUq+/z
    LuJ8OaxC+zm5PFo2AZJfI5Gz5lgb1g5ud0TG1JUrm9Dl5/JSNSL3SXBEC77mdfd0
    BA5VFl8lV7IfTfSTUE9IKoMevqZxoaGpyN+ZcBby5NgsoqJJ6vmcJRFjI92UrFHV
    1wIDAQAB
    -----END PUBLIC KEY-----

    -----BEGIN PRIVATE KEY-----
    MIIEvQIBADANBgkqhkiG9w0BAQEFAASCBKcwggSjAgEAAoIBAQDCZ6NGHD9IHbEH
    euAliGA5YENf8OmicNVU/NNaGfcngg2/qNRr0cOR4eHKzg45lM1qOvyRHLhMh6ai
    2chg40mehbtXIhAQCQTLFwx623mapE8Eu7JzwXKCfiGj+t30kh3j4+Aouavo1Vy5
    kKlVc96DKo3HSdH9SiMhJur8G742VxK2aJxlS4B5VuQYZQ6W+0zMV6GQE/ATfUR2
    lhSr7/Mu4nw5rEL7Obk8WjYBkl8jkbPmWBvWDm53RMbUlSub0OXn8lI1IvdJcEQL
    vuZ193QEDlUWXyVXsh9N9JNQT0gqgx6+pnGhoanI35lwFvLk2Cyioknq+ZwlEWMj
    3ZSsUdXXAgMBAAECggEAEnEUaw0475VpesUsSEM0pZy9J3fKIg/EHQjS3+RAru3G
    ch0I8aV3gPpFmiCL9uhnyCEKXpWz4gaoRyCTwqUtEa2sBOsFTRAd9Uodc/YoBgR6
    Pn+zwQlj3H8sn8qnjZDi5wBx/ksGxNKgtjXD6ohQXm8F/hbBpd6HkJiJiBr1o1/a
    4Kzh9L1OpYVS5QF0EN3idNkt911vfjA+bqBXXjAf9IHvirQ2GeCNzmHygxiRhAWA
    ki2fm0pGbKFUAfo+sgtDPT2jrA4rhSHG4mMr8IWBpr082aKNns7ujxT9ndBCl0Pp
    7efPNCT4+zqfyVAB6x2rYhtXTqR7wsASMIfSLzyoIQKBgQDi2sb9z4OEl0mTV0Ey
    1Lia4F06QhD7Kl1mhBWYPA1RO1ieAHHINFvwPJQLjHR2yMRMDmTOc3S9Hy3H3QMG
    XZlrIl5sgeFzsdqGyd2kscGFUGkH9RY2V/XB0mxQFIGI2PBztqd6e6BJQz4oSyMz
    cBZq/NH5u95QRlyI0I1u5b5lMQKBgQDbYZdhqHcDYmc61aHRNEhnxbK4ph1bdTiM
    QbIF0wUssXQKwEzvYzswRlI27rXeSHpJA8vc+XAoD/+Ut30I8Td3yUnnsHZctU2D
    b/8OTqwFqyrwXM+SvyWZejuvX6IVJuRwPMFG88L88lryEs0ntlQKC/x6wxtcs67X
    6ZGsLSfJhwKBgQDAo09/kIv6OA4+lEXFSGZK/mOsaRXKcztFJry/vZ8BcAfchDwa
    6nt4EbkV5XuwsuQeQcrQlbJ4NtXFdqRu72SsWU8djV1Jxanv89PHWzseXh4Sp8jo
    9OC4aluX1RH6h14IpP6rP/fovrU1ujh2IaSnzXDxRNuQB2/krlSr62Q2wQKBgGNL
    jbgvBwcqH+06So6lKmyFx/nZfgoqSVj6VzhZpcrv2sUO+wOTF3QnMAkbDIg6p9aq
    eDhhUklfzF+kmVxVybRXEDNk5H3bteTa6Uexhhzet4WpjG4wRDVuZNtg3rzSKK1A
    Yn7Z0BSrIUzWA7OIzArsF+/8pULVNTsWxc93dL27AoGAT3eaPv07rhzyztxFYTsr
    dHcgNb+vwroN3Ic7IJnxoa2BjyqczaAG3vMrOcxFg2MLJ7fPmI1W4pPOJyxmUhO0
    mZhk+F7h+dg7XB7h0lqR1usp3Ak3qyu1f4XZj4imzQWWyETpgE4uA9tBniPQIRK6
    miXhZ2YoXfdsSZOatWojGLI=
    -----END PRIVATE KEY-----
    */

    //https://www.oren.net.cn/rsa/info.html
    //在线转换为模数和指数

    //PEM->模数+指数
    char *publicKey_exponent = "10001";
    char *privateKey_exponent = "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";

    //hex array ,faster than string
    //直接使用16进制数据可略微提高速度，软件也支持传入字符串
    // 具体看代码中 GetHex 和 Get
    static unsigned char ModulusHex[RSA_ENCODE_LEN] =
    {
        0xC2, 0x67, 0xA3, 0x46, 0x1C, 0x3F, 0x48, 0x1D, 0xB1, 0x07, 0x7A, 0xE0, 0x25, 0x88, 0x60, 0x39,
        0x60, 0x43, 0x5F, 0xF0, 0xE9, 0xA2, 0x70, 0xD5, 0x54, 0xFC, 0xD3, 0x5A, 0x19, 0xF7, 0x27, 0x82,
        0x0D, 0xBF, 0xA8, 0xD4, 0x6B, 0xD1, 0xC3, 0x91, 0xE1, 0xE1, 0xCA, 0xCE, 0x0E, 0x39, 0x94, 0xCD,
        0x6A, 0x3A, 0xFC, 0x91, 0x1C, 0xB8, 0x4C, 0x87, 0xA6, 0xA2, 0xD9, 0xC8, 0x60, 0xE3, 0x49, 0x9E,
        0x85, 0xBB, 0x57, 0x22, 0x10, 0x10, 0x09, 0x04, 0xCB, 0x17, 0x0C, 0x7A, 0xDB, 0x79, 0x9A, 0xA4,
        0x4F, 0x04, 0xBB, 0xB2, 0x73, 0xC1, 0x72, 0x82, 0x7E, 0x21, 0xA3, 0xFA, 0xDD, 0xF4, 0x92, 0x1D,
        0xE3, 0xE3, 0xE0, 0x28, 0xB9, 0xAB, 0xE8, 0xD5, 0x5C, 0xB9, 0x90, 0xA9, 0x55, 0x73, 0xDE, 0x83,
        0x2A, 0x8D, 0xC7, 0x49, 0xD1, 0xFD, 0x4A, 0x23, 0x21, 0x26, 0xEA, 0xFC, 0x1B, 0xBE, 0x36, 0x57,
        0x12, 0xB6, 0x68, 0x9C, 0x65, 0x4B, 0x80, 0x79, 0x56, 0xE4, 0x18, 0x65, 0x0E, 0x96, 0xFB, 0x4C,
        0xCC, 0x57, 0xA1, 0x90, 0x13, 0xF0, 0x13, 0x7D, 0x44, 0x76, 0x96, 0x14, 0xAB, 0xEF, 0xF3, 0x2E,
        0xE2, 0x7C, 0x39, 0xAC, 0x42, 0xFB, 0x39, 0xB9, 0x3C, 0x5A, 0x36, 0x01, 0x92, 0x5F, 0x23, 0x91,
        0xB3, 0xE6, 0x58, 0x1B, 0xD6, 0x0E, 0x6E, 0x77, 0x44, 0xC6, 0xD4, 0x95, 0x2B, 0x9B, 0xD0, 0xE5,
        0xE7, 0xF2, 0x52, 0x35, 0x22, 0xF7, 0x49, 0x70, 0x44, 0x0B, 0xBE, 0xE6, 0x75, 0xF7, 0x74, 0x04,
        0x0E, 0x55, 0x16, 0x5F, 0x25, 0x57, 0xB2, 0x1F, 0x4D, 0xF4, 0x93, 0x50, 0x4F, 0x48, 0x2A, 0x83,
        0x1E, 0xBE, 0xA6, 0x71, 0xA1, 0xA1, 0xA9, 0xC8, 0xDF, 0x99, 0x70, 0x16, 0xF2, 0xE4, 0xD8, 0x2C,
        0xA2, 0xA2, 0x49, 0xEA, 0xF9, 0x9C, 0x25, 0x11, 0x63, 0x23, 0xDD, 0x94, 0xAC, 0x51, 0xD5, 0xD7
    };

    unsigned char encode_str[RSA_ENCODE_LEN] = {0};
    unsigned char decode_str[RSA_ENCODE_LEN] = {0};

    int outlen = 0;

    unsigned char * base64_str = NULL;
    unsigned int base64_str_len = NULL;

    uint8_t* decode_base64 = NULL;
    char pub_source_string[] = "embedded-system > Public key to RSA encode,hehe";
    char pri_source_string[] = "embedded-system > Private key to RSA encode,haha";

    printf("RSA demo");
    //---------------------------------------------------
    memset(encode_str, 0, sizeof(encode_str));
    memset(decode_str, 0, sizeof(decode_str));
    //公钥加密
    outlen = RSA2048_pub_PKCS1Padding_Encode(pub_source_string, strlen(pub_source_string), encode_str, publicKey_exponent, ModulusHex);

    //密文转base64方便显示，和在线工具对比结果
    // https://the-x.cn/cryptography/Rsa.aspx
    //base64_str = base64_encode(encode_str, outlen, &base64_str_len);
    //printf("Public encode %d\\r\\n%s\\r\\n", base64_str_len, base64_str);
    //portable_free(base64_str);

    //私钥解密
    RSA2048_pri_PKCS1Padding_Decode(encode_str, &outlen, decode_str, privateKey_exponent, ModulusHex);
    printf("Private decode %d\\r\\n%s\\r\\n", outlen, decode_str);

    //---------------------------------------------------
    memset(encode_str, 0, sizeof(encode_str));
    memset(decode_str, 0, sizeof(decode_str));

    //私钥签名
    outlen = RSA2048_pri_PKCS1Padding_Encode(pri_source_string, strlen(pri_source_string), encode_str, privateKey_exponent, ModulusHex);

    //公钥验签
    RSA2048_pub_PKCS1Padding_Decode(encode_str, &outlen, decode_str, publicKey_exponent, ModulusHex);
    printf("Public decode %d\\r\\n%s\\r\\n", outlen, decode_str);

    //---------------------------------------------------
    printf("RSA demo done");
    return 0;
}

5、应用

和AES一样，RSA也是块加密算法（ block cipher algorithm），只针对固定长度数据块，如RSA2048其加密的数据长度需要填充后是2048位即256字节，如果明文长度大于244字节则需要拆分。当然最简单的办法是应用层分配多个244字节缓存，有效数据以外以0x00填充。

RSA算法虽然安全，但其计算量非常大，效率较低，尤其在嵌入式系统中，硬件资源有限的情况下加密、解密时间以秒为单位。而对称加密算法AES算法效率高，但其在密钥协商时，在网络传输中有被拦截的风险，或者任一方保存不当导致密钥泄露，其密钥存在很大的安全隐患。

所以，考虑到安全性和高效性，一般采用多种算法组合加密的方式。使用RSA来加密AES的密钥，密钥协商后，使用AES来对后续数据进行加密。

更多信息，请关注微信公众号 嵌入式系统

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