AES加密算法

AES加密算法

Rust实现

目录

• AES加密算法
  • AES算法流程
  • SubBytes变换
  • ShiftRows变换
  • MixColumns变换
  • AddRoundKey变换
  • RoundKey计算
  • 附录
    • 非线性字节替换表
  • 参考资料

记有如下定义:

• State: 对明文加密的中间状态, 是一个\(4*N_b\)的矩阵, 初始状态等于输入;
• CipherKey: 密钥, 可以看成一个\(4*N_k\)的矩阵;
• \(N_b\): State矩阵的列数, \(N_b=4\);
• \(N_k\): 密钥字长(1字=32bits), \(N_k=4,6,8\);
• \(N_r\): 加密的轮数, \(N_r=10,12,14\);
• RoundKey: 由CipherKey产生, 每一轮加密计算对应一个RoundKey;
• KeyExpansion: 将CipherKey转为RoundKey;
• AddRoundKey: 使用RoundKey对State进行加密计算(RoundKey和State异或操作);
• SubBytes: 使用非线性字节替换表sbox对State进行替换操作;
• ShiftRows: 对State矩阵后三行进行偏移操作;
• MixColumns: 对State矩阵的所有列进行列混合操作;

密钥长度和\(N_b, N_k, N_r\)对应关系如下表:

	\(N_k\)	\(N_b\)	\(N_r\)
AES-128	4	4	10
AES-192	6	4	12
AES-256	8	4	14

AES算法流程

AES算法伪代码如下:

Cipher(byte in[4*Nb], byte out[4*Nb], word w[Nb*(Nr+1)])
begin
    byte state[4, Nb]
    state = in
    AddRoundkey(state, w[0, Nb-1])

    for round = 1 step 1 to Nr-1
        SubBytes(state)
        ShiftRows(state)
        MixColumns(state)
        AddRoundKey(state, w[round*Nb, (round+1)*Nb - 1])
    end

    SubBytes(state)
    ShiftRows(state)
    AddRoundKey(state, w[Nr*Nb, (Nr+1)*Nb - 1])

    out = state
end

InvCipher(byte in[4*Nb], byte out[4*Nb], word w[Nb*(Nr_1)])
begin

    byte stae[4, Nb]
    state = in

    AddRoundKey(state, w[Nr*Nb, (Nr+1)*Nb - 1])

    for round = Nr-1 step -1 downto 1
        InvShiftRows(state)
        InvSubBytes(state)
        AddRoundKey(state, w[round*Nb, (round+1)*Nb - 1])
        InvMixColumns(state)
    end

    InvShiftRows(state)
    InvSubBytes(state)
    AddRoundKey(state, w[0, Nb-1])

    out = state
end


// 另一种等效的加解密步骤, Intel AES指令集采用的是这种步骤(两者加密密钥扩展是一样的)
// SubBytes和ShiftRows可以交换
// InvMixColumns(AddRoundKey(state, RoundKey)) = AddRoundKey(InvMixColumns(state), InvMixColumns(RoundKey)

Cipher(byte in[4*Nb], byte out[4*Nb], word w[Nb*(Nr+1)])
begin
    byte state[4, Nb]
    state = in
    AddRoundkey(state, w[0, Nb-1])

    for round = 1 step 1 to Nr-1
        ShiftRows(state)
        SubBytes(state)
        MixColumns(state)
        AddRoundKey(state, w[round*Nb, (round+1)*Nb - 1])
    end

    ShiftRows(state)
    SubBytes(state)
    AddRoundKey(state, w[Nr*Nb, (Nr+1)*Nb - 1])

    out = state
end

InvCipher(byte in[4*Nb], byte out[4*Nb], word w[Nb*(Nr_1)])
begin

    byte stae[4, Nb]
    state = in

    AddRoundKey(state, w[Nr*Nb, (Nr+1)*Nb - 1])

    for round = Nr-1 step -1 downto 1
        InvSubBytes(state)
        InvShiftRows(state)
        InvMixColumns(state)
        AddRoundKey(state, w[round*Nb, (round+1)*Nb - 1])
    end

    InvSubBytes(state)
    InvShiftRows(state)
    AddRoundKey(state, w[0, Nb-1])

    out = state
end

// 解密密钥扩展需要增加如下步骤
for i = 0 step 1 to (Nb*(Nr+1)-1)
    dw[i] = w[i]
end
// 增加的运算
for round=1 step 1 to Nr-1
    InvMixColumns(dw[round*Nb, (round+1)*Nb-1])
end for

SubBytes变换

记待替换字节为0xmn, 那么替换后的字节为sbox[16 * 0xm + 0xn], 对应的逆变换为isbox[16 * 0xm + 0xn];

假设0xab对应的多项式为\(f(x)\), sbox的计算步骤如下:

1. 求\(f(x)\)的在有限域\(GF(2^8)\)中的逆: \(f(x)*g(x)=1 \mod(x^8+x^4+x^3+x+1)\);
2. 记\(n(x)\)对应的字节为\(b_0b_1 \cdots b_7\), 按照如下公式依次求\([0,255]\)在\(GF(2)\)上的放射变换值;

\[b_{i}^{'} = b_{i} \oplus b_{(i+4)mod8} \oplus b_{(i+5)mod8} \oplus b_{(i+6)mod8} \oplus b_{(i+7)mod8} \oplus c_{i}; \quad c=0x63 \]

ShiftRows变换

记state矩阵中待替换元素为\(s_{r,c}\), 替换后的元素为\(s_{r,c}^{'}\). 其中, \(r,c\)表示其在矩阵中的行和列.
那么, 替换前后的元素关系为:

\[s_{r,c}^{'} = s_{r, (c+(r \% N_b)) \% N_b} \]

上式对应的逆变换为:

\[s_{r,c} = s_{r, (c+(N_b - r)) \% N_b}^{'} \]

MixColumns变换

AES使用\(a(x)\)对列进行变换, \(a(x)\)定义如下:

\[a(x) = \{03\}x^3 + \{01\}x^2 + \{01\}x + \{02\} \\ a^{-1}(x) = \{0b\}x^3 + \{0d\}x^2 + \{09\}x + \{0e\} \]

则, MixColumns对state变换前后的关系如下:

\[\begin{aligned} & s^{'}(x) = a(x) \otimes s(x) \\ & \left[ \begin{matrix} s^{'}_{0,c} \\ s^{'}_{1,c} \\ s^{'}_{2,c} \\ s^{'}_{3,c} \end{matrix} \right] = \left[ \begin{matrix} 02 & 03 & 01 & 01 \\ 01 & 02 & 03 & 01 \\ 01 & 01 & 02 & 03 \\ 03 & 01 & 01 & 02 \end{matrix} \right] \left[ \begin{matrix} s_{0,c} \\ s_{1,c} \\ s_{2,c} \\ s_{3,c} \end{matrix} \right] \end{aligned} \]

对应的逆变换为:

\[\begin{aligned} & s^{'}(x) = a^{-1}(x) \otimes s(x) \\ & \left[ \begin{matrix} s^{'}_{0,c} \\ s^{'}_{1,c} \\ s^{'}_{2,c} \\ s^{'}_{3,c} \end{matrix} \right] = \left[ \begin{matrix} 0e & 0b & 0d & 09 \\ 09 & 0e & 0b & 0d \\ 0d & 09 & 0e & 0b \\ 0b & 0d & 09 & 0e \end{matrix} \right] \left[ \begin{matrix} s_{0,c} \\ s_{1,c} \\ s_{2,c} \\ s_{3,c} \end{matrix} \right] \end{aligned} \]

AddRoundKey变换

\[\left[ \begin{matrix} s^{'}_{0,c} & s^{'}_{1,c} & s^{'}_{2,c} & s^{'}_{3,c} \end{matrix} \right] = \left[ \begin{matrix} s_{0,c} & s_{1,c} & s_{2,c} & s_{3,c} \end{matrix} \right] \oplus \left[ \begin{matrix} w_{round*N_b+0} & w_{round*N_b+1} & w_{round*N_b+2} & w_{round*N_b+3} \end{matrix} \right] \]

异或运算是可逆的, 对应的逆变换公式不变;

RoundKey计算

RoundKey计算算法流程如下:

RoundKey(byte key[4*Nk], word w[Nb*(Nr+1)], Nk)
begin

word temp
i = 0

while i < Nk
    w[i] = word(key[4*i], key[4*i+1], key[4*i+2], key[4*i+3])
    i += 1
end

i = Nk

while i < Nb * (Nr+1)
    temp = w[i-1]
    if i % Nk == 0
        temp = SubWord(RotWord(temp)) ^ Rcon[i/Nk]
    else if Nk > 6 && i % Nk == 4
        temp = SubWord(temp)
    end
    w[i] = w[i-Nk] ^ temp
    i = i + 1
end

end

• RotWord: 将\([a_0,a_1,a_2,a_3]\)翻转为\([a_1,a_2,a_3,a_0]\);
• SubWord: 通过sbox完成非线性变换;
• Rcon: [\(x^{i-1} mod(x^8+x^4+x^3+x+1)\), {00}, {00}, {00}], i = [1-10];

附录

非线性字节替换表

const sbox: [u8; 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,
];

const isbox: [u8; 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,
];

const rcon: [u32; 10] = [
    0x01000000, 0x02000000, 0x04000000, 0x08000000, 0x10000000, 0x20000000,
    0x40000000, 0x80000000, 0x1b000000, 0x36000000,
];

参考资料

• FIPS-197 AES;