AES算法重点详解和实现

可以看到，在原始数据长度为16的整数倍时，假如原始数据长度等于16*n，则使用NoPadding时加密后数据长度等于16*n，其它情况下加密数据长度等于16*(n+1)。在不足16的整数倍的情况下，假如原始数据长度等于16*n+m[其中m小于16]，除了NoPadding填充之外的任何方式，加密数据长度都等于16*(n+1)；NoPadding填充情况下，CBC、ECB和PCBC三种模式是不支持的，CFB、OFB两种模式下则加密数据长度等于原始数据长度。

下面举例演示各个填充方式：

NoPadding: 不填充

PKCS5Padding: 填充字符串由一个字节序列组成，每个字节填充该字节序列的长度，假定数据长度为9，则需要填充的长度为16 - 9 = 7，数据等于 FF FF FF FF FF FF FF FF FF，填充后数据为FF FF FF FF FF FF FF FF FF 07 07 07 07 07 07 07

ISO10126Padding: 填充字符串由一个字节序列组成，此字节序列的最后一个字节填充字节序列的长度，其余字节填充随机数据。假定数据长度为9，则需要填充的长度为16 - 9 = 7，数据等于 FF FF FF FF FF FF FF FF FF，填充后数据为FF FF FF FF FF FF FF FF FF 7D 2A 75 EF F8 EF 07

aes.h

/*************************************************************************** AES declarations **************************************************************************/#define AES_MAXROUNDS           14
#define AES_BLOCKSIZE           16
#define AES_IV_SIZE             16typedef struct aes_key_st
{uint16_t rounds;uint16_t key_size;uint32_t ks[(AES_MAXROUNDS+1)*8];uint8_t iv[AES_IV_SIZE];
} AES_CTX;typedef enum
{AES_MODE_128,AES_MODE_256
} AES_MODE;void AES_set_key(AES_CTX *ctx, const uint8_t *key, const uint8_t *iv, AES_MODE mode);
void AES_cbc_encrypt(AES_CTX *ctx, const uint8_t *msg, uint8_t *out, int length);
void AES_cbc_decrypt(AES_CTX *ks, const uint8_t *in, uint8_t *out, int length);

aes.c

static const uint8_t aes_sbox[256] =
{0x63,0x7C,0x77,0x7B,0xF2,0x6B,0x6F,0xC5,0x30,0x01,0x67,0x2B,0xFE,0xD7,0xAB,0x76,0xCA,0x82,0xC9,0x7D,0xFA,0x59,0x47,0xF0,0xAD,0xD4,0xA2,0xAF,0x9C,0xA4,0x72,0xC0,0xB7,0xFD,0x93,0x26,0x36,0x3F,0xF7,0xCC,0x34,0xA5,0xE5,0xF1,0x71,0xD8,0x31,0x15,0x04,0xC7,0x23,0xC3,0x18,0x96,0x05,0x9A,0x07,0x12,0x80,0xE2,0xEB,0x27,0xB2,0x75,0x09,0x83,0x2C,0x1A,0x1B,0x6E,0x5A,0xA0,0x52,0x3B,0xD6,0xB3,0x29,0xE3,0x2F,0x84,0x53,0xD1,0x00,0xED,0x20,0xFC,0xB1,0x5B,0x6A,0xCB,0xBE,0x39,0x4A,0x4C,0x58,0xCF,0xD0,0xEF,0xAA,0xFB,0x43,0x4D,0x33,0x85,0x45,0xF9,0x02,0x7F,0x50,0x3C,0x9F,0xA8,0x51,0xA3,0x40,0x8F,0x92,0x9D,0x38,0xF5,0xBC,0xB6,0xDA,0x21,0x10,0xFF,0xF3,0xD2,0xCD,0x0C,0x13,0xEC,0x5F,0x97,0x44,0x17,0xC4,0xA7,0x7E,0x3D,0x64,0x5D,0x19,0x73,0x60,0x81,0x4F,0xDC,0x22,0x2A,0x90,0x88,0x46,0xEE,0xB8,0x14,0xDE,0x5E,0x0B,0xDB,0xE0,0x32,0x3A,0x0A,0x49,0x06,0x24,0x5C,0xC2,0xD3,0xAC,0x62,0x91,0x95,0xE4,0x79,0xE7,0xC8,0x37,0x6D,0x8D,0xD5,0x4E,0xA9,0x6C,0x56,0xF4,0xEA,0x65,0x7A,0xAE,0x08,0xBA,0x78,0x25,0x2E,0x1C,0xA6,0xB4,0xC6,0xE8,0xDD,0x74,0x1F,0x4B,0xBD,0x8B,0x8A,0x70,0x3E,0xB5,0x66,0x48,0x03,0xF6,0x0E,0x61,0x35,0x57,0xB9,0x86,0xC1,0x1D,0x9E,0xE1,0xF8,0x98,0x11,0x69,0xD9,0x8E,0x94,0x9B,0x1E,0x87,0xE9,0xCE,0x55,0x28,0xDF,0x8C,0xA1,0x89,0x0D,0xBF,0xE6,0x42,0x68,0x41,0x99,0x2D,0x0F,0xB0,0x54,0xBB,0x16,
};/** AES is-box*/
static const uint8_t aes_isbox[256] =
{0x52,0x09,0x6a,0xd5,0x30,0x36,0xa5,0x38,0xbf,0x40,0xa3,0x9e,0x81,0xf3,0xd7,0xfb,0x7c,0xe3,0x39,0x82,0x9b,0x2f,0xff,0x87,0x34,0x8e,0x43,0x44,0xc4,0xde,0xe9,0xcb,0x54,0x7b,0x94,0x32,0xa6,0xc2,0x23,0x3d,0xee,0x4c,0x95,0x0b,0x42,0xfa,0xc3,0x4e,0x08,0x2e,0xa1,0x66,0x28,0xd9,0x24,0xb2,0x76,0x5b,0xa2,0x49,0x6d,0x8b,0xd1,0x25,0x72,0xf8,0xf6,0x64,0x86,0x68,0x98,0x16,0xd4,0xa4,0x5c,0xcc,0x5d,0x65,0xb6,0x92,0x6c,0x70,0x48,0x50,0xfd,0xed,0xb9,0xda,0x5e,0x15,0x46,0x57,0xa7,0x8d,0x9d,0x84,0x90,0xd8,0xab,0x00,0x8c,0xbc,0xd3,0x0a,0xf7,0xe4,0x58,0x05,0xb8,0xb3,0x45,0x06,0xd0,0x2c,0x1e,0x8f,0xca,0x3f,0x0f,0x02,0xc1,0xaf,0xbd,0x03,0x01,0x13,0x8a,0x6b,0x3a,0x91,0x11,0x41,0x4f,0x67,0xdc,0xea,0x97,0xf2,0xcf,0xce,0xf0,0xb4,0xe6,0x73,0x96,0xac,0x74,0x22,0xe7,0xad,0x35,0x85,0xe2,0xf9,0x37,0xe8,0x1c,0x75,0xdf,0x6e,0x47,0xf1,0x1a,0x71,0x1d,0x29,0xc5,0x89,0x6f,0xb7,0x62,0x0e,0xaa,0x18,0xbe,0x1b,0xfc,0x56,0x3e,0x4b,0xc6,0xd2,0x79,0x20,0x9a,0xdb,0xc0,0xfe,0x78,0xcd,0x5a,0xf4,0x1f,0xdd,0xa8,0x33,0x88,0x07,0xc7,0x31,0xb1,0x12,0x10,0x59,0x27,0x80,0xec,0x5f,0x60,0x51,0x7f,0xa9,0x19,0xb5,0x4a,0x0d,0x2d,0xe5,0x7a,0x9f,0x93,0xc9,0x9c,0xef,0xa0,0xe0,0x3b,0x4d,0xae,0x2a,0xf5,0xb0,0xc8,0xeb,0xbb,0x3c,0x83,0x53,0x99,0x61,0x17,0x2b,0x04,0x7e,0xba,0x77,0xd6,0x26,0xe1,0x69,0x14,0x63,0x55,0x21,0x0c,0x7d
};static const unsigned char Rcon[30]=
{0x01,0x02,0x04,0x08,0x10,0x20,0x40,0x80,0x1b,0x36,0x6c,0xd8,0xab,0x4d,0x9a,0x2f,0x5e,0xbc,0x63,0xc6,0x97,0x35,0x6a,0xd4,0xb3,0x7d,0xfa,0xef,0xc5,0x91,
};#define rot1(x) (((x) << 24) | ((x) >> 8))
#define rot2(x) (((x) << 16) | ((x) >> 16))
#define rot3(x) (((x) <<  8) | ((x) >> 24))/* * This cute trick does 4 'mul by two' at once.  Stolen from* Dr B. R. Gladman <brg@gladman.uk.net> but I'm sure the u-(u>>7) is* a standard graphics trick* The key to this is that we need to xor with 0x1b if the top bit is set.* a 1xxx xxxx   0xxx 0xxx First we mask the 7bit,* b 1000 0000   0000 0000 then we shift right by 7 putting the 7bit in 0bit,* c 0000 0001   0000 0000 we then subtract (c) from (b)* d 0111 1111   0000 0000 and now we and with our mask* e 0001 1011   0000 0000*/
#define mt  0x80808080
#define ml  0x7f7f7f7f
#define mh  0xfefefefe
#define mm  0x1b1b1b1b
#define mul2(x,t)   ((t)=((x)&mt), \((((x)+(x))&mh)^(((t)-((t)>>7))&mm)))#define inv_mix_col(x,f2,f4,f8,f9) (\(f2)=mul2(x,f2), \(f4)=mul2(f2,f4), \(f8)=mul2(f4,f8), \(f9)=(x)^(f8), \(f8)=((f2)^(f4)^(f8)), \(f2)^=(f9), \(f4)^=(f9), \(f8)^=rot3(f2), \(f8)^=rot2(f4), \(f8)^rot1(f9))/** AES S-box*/
static const uint8_t aes_sbox[256] =
{0x63,0x7C,0x77,0x7B,0xF2,0x6B,0x6F,0xC5,0x30,0x01,0x67,0x2B,0xFE,0xD7,0xAB,0x76,0xCA,0x82,0xC9,0x7D,0xFA,0x59,0x47,0xF0,0xAD,0xD4,0xA2,0xAF,0x9C,0xA4,0x72,0xC0,0xB7,0xFD,0x93,0x26,0x36,0x3F,0xF7,0xCC,0x34,0xA5,0xE5,0xF1,0x71,0xD8,0x31,0x15,0x04,0xC7,0x23,0xC3,0x18,0x96,0x05,0x9A,0x07,0x12,0x80,0xE2,0xEB,0x27,0xB2,0x75,0x09,0x83,0x2C,0x1A,0x1B,0x6E,0x5A,0xA0,0x52,0x3B,0xD6,0xB3,0x29,0xE3,0x2F,0x84,0x53,0xD1,0x00,0xED,0x20,0xFC,0xB1,0x5B,0x6A,0xCB,0xBE,0x39,0x4A,0x4C,0x58,0xCF,0xD0,0xEF,0xAA,0xFB,0x43,0x4D,0x33,0x85,0x45,0xF9,0x02,0x7F,0x50,0x3C,0x9F,0xA8,0x51,0xA3,0x40,0x8F,0x92,0x9D,0x38,0xF5,0xBC,0xB6,0xDA,0x21,0x10,0xFF,0xF3,0xD2,0xCD,0x0C,0x13,0xEC,0x5F,0x97,0x44,0x17,0xC4,0xA7,0x7E,0x3D,0x64,0x5D,0x19,0x73,0x60,0x81,0x4F,0xDC,0x22,0x2A,0x90,0x88,0x46,0xEE,0xB8,0x14,0xDE,0x5E,0x0B,0xDB,0xE0,0x32,0x3A,0x0A,0x49,0x06,0x24,0x5C,0xC2,0xD3,0xAC,0x62,0x91,0x95,0xE4,0x79,0xE7,0xC8,0x37,0x6D,0x8D,0xD5,0x4E,0xA9,0x6C,0x56,0xF4,0xEA,0x65,0x7A,0xAE,0x08,0xBA,0x78,0x25,0x2E,0x1C,0xA6,0xB4,0xC6,0xE8,0xDD,0x74,0x1F,0x4B,0xBD,0x8B,0x8A,0x70,0x3E,0xB5,0x66,0x48,0x03,0xF6,0x0E,0x61,0x35,0x57,0xB9,0x86,0xC1,0x1D,0x9E,0xE1,0xF8,0x98,0x11,0x69,0xD9,0x8E,0x94,0x9B,0x1E,0x87,0xE9,0xCE,0x55,0x28,0xDF,0x8C,0xA1,0x89,0x0D,0xBF,0xE6,0x42,0x68,0x41,0x99,0x2D,0x0F,0xB0,0x54,0xBB,0x16,
};/** AES is-box*/
static const uint8_t aes_isbox[256] =
{0x52,0x09,0x6a,0xd5,0x30,0x36,0xa5,0x38,0xbf,0x40,0xa3,0x9e,0x81,0xf3,0xd7,0xfb,0x7c,0xe3,0x39,0x82,0x9b,0x2f,0xff,0x87,0x34,0x8e,0x43,0x44,0xc4,0xde,0xe9,0xcb,0x54,0x7b,0x94,0x32,0xa6,0xc2,0x23,0x3d,0xee,0x4c,0x95,0x0b,0x42,0xfa,0xc3,0x4e,0x08,0x2e,0xa1,0x66,0x28,0xd9,0x24,0xb2,0x76,0x5b,0xa2,0x49,0x6d,0x8b,0xd1,0x25,0x72,0xf8,0xf6,0x64,0x86,0x68,0x98,0x16,0xd4,0xa4,0x5c,0xcc,0x5d,0x65,0xb6,0x92,0x6c,0x70,0x48,0x50,0xfd,0xed,0xb9,0xda,0x5e,0x15,0x46,0x57,0xa7,0x8d,0x9d,0x84,0x90,0xd8,0xab,0x00,0x8c,0xbc,0xd3,0x0a,0xf7,0xe4,0x58,0x05,0xb8,0xb3,0x45,0x06,0xd0,0x2c,0x1e,0x8f,0xca,0x3f,0x0f,0x02,0xc1,0xaf,0xbd,0x03,0x01,0x13,0x8a,0x6b,0x3a,0x91,0x11,0x41,0x4f,0x67,0xdc,0xea,0x97,0xf2,0xcf,0xce,0xf0,0xb4,0xe6,0x73,0x96,0xac,0x74,0x22,0xe7,0xad,0x35,0x85,0xe2,0xf9,0x37,0xe8,0x1c,0x75,0xdf,0x6e,0x47,0xf1,0x1a,0x71,0x1d,0x29,0xc5,0x89,0x6f,0xb7,0x62,0x0e,0xaa,0x18,0xbe,0x1b,0xfc,0x56,0x3e,0x4b,0xc6,0xd2,0x79,0x20,0x9a,0xdb,0xc0,0xfe,0x78,0xcd,0x5a,0xf4,0x1f,0xdd,0xa8,0x33,0x88,0x07,0xc7,0x31,0xb1,0x12,0x10,0x59,0x27,0x80,0xec,0x5f,0x60,0x51,0x7f,0xa9,0x19,0xb5,0x4a,0x0d,0x2d,0xe5,0x7a,0x9f,0x93,0xc9,0x9c,0xef,0xa0,0xe0,0x3b,0x4d,0xae,0x2a,0xf5,0xb0,0xc8,0xeb,0xbb,0x3c,0x83,0x53,0x99,0x61,0x17,0x2b,0x04,0x7e,0xba,0x77,0xd6,0x26,0xe1,0x69,0x14,0x63,0x55,0x21,0x0c,0x7d
};static const unsigned char Rcon[30]=
{0x01,0x02,0x04,0x08,0x10,0x20,0x40,0x80,0x1b,0x36,0x6c,0xd8,0xab,0x4d,0x9a,0x2f,0x5e,0xbc,0x63,0xc6,0x97,0x35,0x6a,0xd4,0xb3,0x7d,0xfa,0xef,0xc5,0x91,
};/* ----- static functions ----- */
static void AES_encrypt(const AES_CTX *ctx, uint32_t *data);
static void AES_decrypt(const AES_CTX *ctx, uint32_t *data);/* Perform doubling in Galois Field GF(2^8) using the irreducible polynomialx^8+x^4+x^3+x+1 */
static unsigned char AES_xtime(uint32_t x)
{return x = (x&0x80) ? (x<<1)^0x1b : x<<1;
}/*** Set up AES with the key/iv and cipher size.*/
void AES_set_key(AES_CTX *ctx, const uint8_t *key, const uint8_t *iv, AES_MODE mode)
{int i, ii;uint32_t *W, tmp, tmp2;const unsigned char *ip;int words;switch (mode){case AES_MODE_128:i = 10;words = 4;break;case AES_MODE_256:i = 14;words = 8;break;default:        /* fail silently */return;}ctx->rounds = i;ctx->key_size = words;W = ctx->ks;for (i = 0; i < words; i+=2){W[i+0]=   ((uint32_t)key[ 0]<<24)|((uint32_t)key[ 1]<<16)|((uint32_t)key[ 2]<< 8)|((uint32_t)key[ 3]    );W[i+1]= ((uint32_t)key[ 4]<<24)|((uint32_t)key[ 5]<<16)|((uint32_t)key[ 6]<< 8)|((uint32_t)key[ 7]    );key += 8;}ip = Rcon;ii = 4 * (ctx->rounds+1);for (i = words; i<ii; i++){tmp = W[i-1];if ((i % words) == 0){tmp2 =(uint32_t)aes_sbox[(tmp    )&0xff]<< 8;tmp2|=(uint32_t)aes_sbox[(tmp>> 8)&0xff]<<16;tmp2|=(uint32_t)aes_sbox[(tmp>>16)&0xff]<<24;tmp2|=(uint32_t)aes_sbox[(tmp>>24)     ];tmp=tmp2^(((unsigned int)*ip)<<24);ip++;}if ((words == 8) && ((i % words) == 4)){tmp2 =(uint32_t)aes_sbox[(tmp    )&0xff]    ;tmp2|=(uint32_t)aes_sbox[(tmp>> 8)&0xff]<< 8;tmp2|=(uint32_t)aes_sbox[(tmp>>16)&0xff]<<16;tmp2|=(uint32_t)aes_sbox[(tmp>>24)     ]<<24;tmp=tmp2;}W[i]=W[i-words]^tmp;}/* copy the iv across */memcpy(ctx->iv, iv, 16);
}/*** Change a key for decryption.*/
void AES_convert_key(AES_CTX *ctx)
{int i;uint32_t *k,w,t1,t2,t3,t4;k = ctx->ks;k += 4;for (i= ctx->rounds*4; i > 4; i--){w= *k;w = inv_mix_col(w,t1,t2,t3,t4);*k++ =w;}
}/*** Encrypt a byte sequence (with a block size 16) using the AES cipher.*/
void AES_cbc_encrypt(AES_CTX *ctx, const uint8_t *msg, uint8_t *out, int length)
{int i;uint32_t tin[4], tout[4], iv[4];memcpy(iv, ctx->iv, AES_IV_SIZE);for (i = 0; i < 4; i++)tout[i] = ntohl(iv[i]);for (length -= AES_BLOCKSIZE; length >= 0; length -= AES_BLOCKSIZE){uint32_t msg_32[4];uint32_t out_32[4];memcpy(msg_32, msg, AES_BLOCKSIZE);msg += AES_BLOCKSIZE;for (i = 0; i < 4; i++)tin[i] = ntohl(msg_32[i])^tout[i];AES_encrypt(ctx, tin);for (i = 0; i < 4; i++){tout[i] = tin[i]; out_32[i] = htonl(tout[i]);}memcpy(out, out_32, AES_BLOCKSIZE);out += AES_BLOCKSIZE;}for (i = 0; i < 4; i++)iv[i] = htonl(tout[i]);memcpy(ctx->iv, iv, AES_IV_SIZE);
}/*** Decrypt a byte sequence (with a block size 16) using the AES cipher.*/
void AES_cbc_decrypt(AES_CTX *ctx, const uint8_t *msg, uint8_t *out, int length)
{int i;uint32_t tin[4], xor[4], tout[4], data[4], iv[4];memcpy(iv, ctx->iv, AES_IV_SIZE);for (i = 0; i < 4; i++)xor[i] = ntohl(iv[i]);for (length -= 16; length >= 0; length -= 16){uint32_t msg_32[4];uint32_t out_32[4];memcpy(msg_32, msg, AES_BLOCKSIZE);msg += AES_BLOCKSIZE;for (i = 0; i < 4; i++){tin[i] = ntohl(msg_32[i]);data[i] = tin[i];}AES_decrypt(ctx, data);for (i = 0; i < 4; i++){tout[i] = data[i]^xor[i];xor[i] = tin[i];out_32[i] = htonl(tout[i]);}memcpy(out, out_32, AES_BLOCKSIZE);out += AES_BLOCKSIZE;}for (i = 0; i < 4; i++)iv[i] = htonl(xor[i]);memcpy(ctx->iv, iv, AES_IV_SIZE);
}/*** Encrypt a single block (16 bytes) of data*/
static void AES_encrypt(const AES_CTX *ctx, uint32_t *data)
{/* To make this code smaller, generate the sbox entries on the fly.* This will have a really heavy effect upon performance.*/uint32_t tmp[4];uint32_t tmp1, old_a0, a0, a1, a2, a3, row;int curr_rnd;int rounds = ctx->rounds; const uint32_t *k = ctx->ks;/* Pre-round key addition */for (row = 0; row < 4; row++)data[row] ^= *(k++);/* Encrypt one block. */for (curr_rnd = 0; curr_rnd < rounds; curr_rnd++){/* Perform ByteSub and ShiftRow operations together */for (row = 0; row < 4; row++){a0 = (uint32_t)aes_sbox[(data[row%4]>>24)&0xFF];a1 = (uint32_t)aes_sbox[(data[(row+1)%4]>>16)&0xFF];a2 = (uint32_t)aes_sbox[(data[(row+2)%4]>>8)&0xFF]; a3 = (uint32_t)aes_sbox[(data[(row+3)%4])&0xFF];/* Perform MixColumn iff not last round */if (curr_rnd < (rounds - 1)){tmp1 = a0 ^ a1 ^ a2 ^ a3;old_a0 = a0;a0 ^= tmp1 ^ AES_xtime(a0 ^ a1);a1 ^= tmp1 ^ AES_xtime(a1 ^ a2);a2 ^= tmp1 ^ AES_xtime(a2 ^ a3);a3 ^= tmp1 ^ AES_xtime(a3 ^ old_a0);}tmp[row] = ((a0 << 24) | (a1 << 16) | (a2 << 8) | a3);}/* KeyAddition - note that it is vital that this loop is separate fromthe MixColumn operation, which must be atomic...*/ for (row = 0; row < 4; row++)data[row] = tmp[row] ^ *(k++);}
}/*** Decrypt a single block (16 bytes) of data*/
static void AES_decrypt(const AES_CTX *ctx, uint32_t *data)
{ uint32_t tmp[4];uint32_t xt0,xt1,xt2,xt3,xt4,xt5,xt6;uint32_t a0, a1, a2, a3, row;int curr_rnd;int rounds = ctx->rounds;const uint32_t *k = ctx->ks + ((rounds+1)*4);/* pre-round key addition */for (row=4; row > 0;row--)data[row-1] ^= *(--k);/* Decrypt one block */for (curr_rnd = 0; curr_rnd < rounds; curr_rnd++){/* Perform ByteSub and ShiftRow operations together */for (row = 4; row > 0; row--){a0 = aes_isbox[(data[(row+3)%4]>>24)&0xFF];a1 = aes_isbox[(data[(row+2)%4]>>16)&0xFF];a2 = aes_isbox[(data[(row+1)%4]>>8)&0xFF];a3 = aes_isbox[(data[row%4])&0xFF];/* Perform MixColumn iff not last round */if (curr_rnd<(rounds-1)){/* The MDS cofefficients (0x09, 0x0B, 0x0D, 0x0E)are quite large compared to encryption; this operation slows decryption down noticeably. */xt0 = AES_xtime(a0^a1);xt1 = AES_xtime(a1^a2);xt2 = AES_xtime(a2^a3);xt3 = AES_xtime(a3^a0);xt4 = AES_xtime(xt0^xt1);xt5 = AES_xtime(xt1^xt2);xt6 = AES_xtime(xt4^xt5);xt0 ^= a1^a2^a3^xt4^xt6;xt1 ^= a0^a2^a3^xt5^xt6;xt2 ^= a0^a1^a3^xt4^xt6;xt3 ^= a0^a1^a2^xt5^xt6;tmp[row-1] = ((xt0<<24)|(xt1<<16)|(xt2<<8)|xt3);}elsetmp[row-1] = ((a0<<24)|(a1<<16)|(a2<<8)|a3);}for (row = 4; row > 0; row--)data[row-1] = tmp[row-1] ^ *(--k);}
}

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