| Crypto Forum Research Group | R. Tse |
| Internet-Draft | Ribose |
| Intended status: Informational | W. Wong |
| Expires: June 3, 2018 | Hang Seng Management College |
| November 30, 2017 |
The SM4 Blockcipher Algorithm And Its Modes Of Operations
draft-ribose-cfrg-sm4-06
This document describes the SM4 symmetric blockcipher algorithm published as GB/T 32907-2016 by the State Cryptography Administration of China (SCA).
This document is a product of the Crypto Forum Research Group (CFRG).
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SM4 [GBT.32907-2016] [ISO.IEC.18033-3.AMD2] is a cryptographic standard issued by the State Cryptography Administration (SCA) of China [SCA] (formerly the Office of State Commercial Cryptography Administration, OSCCA) as an authorized cryptographic algorithm for the use within China. The algorithm is published in public.
SM4 is a symmetric encryption algorithm, specifically a blockcipher, designed for data encryption.
This document does not aim to introduce a new algorithm, but to provide a clear and open description of the SM4 algorithm in English, and also to serve as a stable reference for IETF documents that utilize this algorithm.
While this document is similar to [SM4-En] in nature, [SM4-En] is a textual translation of the "SMS4" algorithm [SM4] published in 2006. Instead, this document follows the updated description and structure of [GBT.32907-2016] published in 2016. Sections 1 to 7 of this document directly map to the corresponding sections numbers of the [GBT.32907-2016] standard for convenience of the reader.
This document also provides additional information on the design considerations of the SM4 algorithm [SM4-Details], its modes of operations that are currently being used (see Section 8), and the offical SM4 OIDs (see Section 9).
The "SMS4" algorithm (the former name of SM4) was invented by Shu-Wang Lu [LSW-Bio]. It was first published in 2003 as part of [GB.15629.11-2003], then published independently in 2006 by SCA (OSCCA at that time) [SM4], published as an industry cryptographic standard and renamed to "SM4" in 2012 by SCA (OSCCA at that time) [GMT-0002-2012], and finally formalized in 2016 as a Chinese National Standard (GB Standard) [GBT.32907-2016]. SM4 has also been standardized in [ISO.IEC.18033-3.AMD2] by the International Organization for Standardization in 2017.
SMS4 was originally created for use in protecting wireless networks [SM4], and is mandated in the Chinese National Standard for Wireless LAN WAPI (Wired Authentication and Privacy Infrastructure) [GB.15629.11-2003]. A proposal was made to adopt SMS4 into the IEEE 802.11i standard, but the algorithm was eventually not included due to concerns of introducing inoperability with existing ciphers.
The latest SM4 standard [GBT.32907-2016] was proposed by the SCA (OSCCA at that time), standardized through TC 260 of the Standardization Administration of the People’s Republic of China (SAC), and was drafted by the following individuals at the Data Assurance and Communication Security Research Center (DAS Center) of the Chinese Academy of Sciences, the China Commercial Cryptography Testing Center and the Beijing Academy of Information Science & Technology (BAIST):
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in [RFC2119].
The following terms and definitions apply to this document.
The SM4 algorithm is a blockcipher, with block size of 128 bits and a key length of 128 bits.
Both encryption and key expansion use 32 rounds of a nonlinear key schedule per block. Each round processes one of the four 32-bit words that constitute the block.
The structure of encryption and decryption are identical, except that the round key schedule has its order reversed during decryption.
Using a 8-bit S-box, it only uses exclusive-or, cyclic bit shifts and S-box lookups to execute.
The SM4 encryption key is 128 bits long and represented below, where each MK_i, (i = 0, 1, 2, 3) is 32 bits long.
MK = (MK_0, MK_1, MK_2, MK_3)
The round key schedule is derived from the encryption key, represented as below where each rk_i (i = 0, ..., 31) is 32 bits long:
(rk_0, rk_1, ... , rk_31)
The family key used for key expansion is represented as FK, where each FK_i (i = 0, ..., 3) is 32 bits long:
FK = (FK_0, FK_1, FK_2, FK_3)
The constant key used for key expansion is represented as CK, where each CK_i (i = 0, ..., 31) is 32 bits long:
CK = (CK_0, CK_1, ... , CK_31)
The round function F is defined as:
F(X_0, X_1, X_2, X_3, rk) = X_0 xor T(X_1 xor X_2 xor X_3 xor rk)
Where:
T is a reversible permutation that outputs 32 bits from a 32-bit input.
It consists of a nonlinear transform tau and linear transform L.
T(.) = L(tau(.))
The permutation T' is created from T by replacing the linear transform function L with L'.
T'(.) = L'(tau(.))
tau is composed of four parallel S-boxes.
Given a 32-bit input A, where each a_i is a 8-bit string:
A = (a_0, a_1, a_2, a_3)
The output is a 32-bit B, where each b_i is a 8-bit string:
B = (b_0, b_1, b_2, b_3)
B is calculated as follows:
(b_0, b_1, b_2, b_3) = tau(A)
tau(A) = (S(a_0), S(a_1), S(a_2), S(a_3))
The output of nonlinear transformation function tau is used as input to linear transformation function L.
Given B, a 32-bit input.
The linear transformation L' is defined as follows.
L(B) = B xor (B <<< 2) xor (B <<< 10) xor (B <<< 18) xor (B <<< 24)
The linear transformation L' is defined as follows.
L'(B) = B xor (B <<< 13) xor (B <<< 23)
The S-box S used in nonlinear transformation tau is given in the lookup table shown in Figure 1 with hexadecimal values.
| 0 1 2 3 4 5 6 7 8 9 A B C D E F ---|------------------------------------------------- 0 | D6 90 E9 FE CC E1 3D B7 16 B6 14 C2 28 FB 2C 05 1 | 2B 67 9A 76 2A BE 04 C3 AA 44 13 26 49 86 06 99 2 | 9C 42 50 F4 91 EF 98 7A 33 54 0B 43 ED CF AC 62 3 | E4 B3 1C A9 C9 08 E8 95 80 DF 94 FA 75 8F 3F A6 4 | 47 07 A7 FC F3 73 17 BA 83 59 3C 19 E6 85 4F A8 5 | 68 6B 81 B2 71 64 DA 8B F8 EB 0F 4B 70 56 9D 35 6 | 1E 24 0E 5E 63 58 D1 A2 25 22 7C 3B 01 21 78 87 7 | D4 00 46 57 9F D3 27 52 4C 36 02 E7 A0 C4 C8 9E 8 | EA BF 8A D2 40 C7 38 B5 A3 F7 F2 CE F9 61 15 A1 9 | E0 AE 5D A4 9B 34 1A 55 AD 93 32 30 F5 8C B1 E3 A | 1D F6 E2 2E 82 66 CA 60 C0 29 23 AB 0D 53 4E 6F B | D5 DB 37 45 DE FD 8E 2F 03 FF 6A 72 6D 6C 5B 51 C | 8D 1B AF 92 BB DD BC 7F 11 D9 5C 41 1F 10 5A D8 D | 0A C1 31 88 A5 CD 7B BD 2D 74 D0 12 B8 E5 B4 B0 E | 89 69 97 4A 0C 96 77 7E 65 B9 F1 09 C5 6E C6 84 F | 18 F0 7D EC 3A DC 4D 20 79 EE 5F 3E D7 CB 39 48
Figure 1: SM4 S-box Values
For example, input "EF" will produce an output read from the S-box table row E and column F, giving the result S(EF) = 84.
The encryption algorithm consists of 32 rounds and 1 reverse transform R.
Given a 128-bit plaintext input, where each X_i is 32-bit wide:
(X_0, X_1, X_2, X_3)
The output is a 128-bit ciphertext, where each Y_i is 32-bit wide:
(Y_0, Y_1, Y_2, Y_3)
Each round key is designated as rk_i, where each rk_i is 32-bit wide and i = 0, 1, 2, ..., 31.
i = 0, 1, ..., 31
X_{i+4} = F(X_i, X_{i+1}, X_{i+2}, X_{i+3}, rk_i)
(Y_0, Y_1, Y_2, Y_3) = R(X_32, X_33, X_34, X_35) R(X_32, X_33, X_34, X_35) = (X_35, X_34, X_33, X_32)
Please refer to Appendix A for sample calculations.
A flow of the calculation is given in Figure 2.
128-bits plaintext
\___________________________/
|
v
X_0 X_1 X_2 X_3
| | | |
v v v v
+---------------------------+
Round 1 | Round Function | <--rk_0
+---------------------------+
| | | |
X_1 X_2 X_3 X_4
| | | |
v v v v
+---------------------------+
Round 2 | Round Function | <--rk_1
+---------------------------+
| | | |
X_2 X_3 X_4 X_5
| | | |
v v v v
...
X_31 X_32 X_33 X_34
| | | |
v v v v
+---------------------------+
Round 32 | Round Function | <--rk_31
+---------------------------+
| | | |
X_32 X_33 X_34 X_35
| | | |
v v v v
+---------------------------+
| Reverse Transformation R |
+---------------------------+
| | | |
Y_0 Y_1 Y_2 Y_3
\___________________________/
|
v
128-bits ciphertext
Figure 2: SM4 Encryption Flow
Decryption takes an identical process as encryption, with the only difference the order of the round key sequence.
During decryption, the round key sequence is:
(rk_31, rk_30, ..., rk_0)
Round keys used during encryption are derived from the encryption key.
Specifically, given the encryption key MK, where each MK_i is 32-bit wide:
MK = (MK_0, MK_1, MK_2, MK_3)
Each round key rk_i is created as follows, where i = 0, 1, ..., 31.
(K_0, K_1, K_2, K_3) =
(MK_0 xor FK_0, MK_1 xor FK_1, MK_2 xor FK_2, MK_3 xor FK_3)
rk_i = K_{i + 4}
K_{i + 4} =
K_i xor T' (K_{i + 1} xor K_{i + 2} xor K_{i + 3} xor CK_i)
Since the decryption key is identical to the encryption key, the round keys used in the decryption process are derived from the decryption key through the identical process to that of during encryption.
Figure 3 depicts the i-th round of SM4.
X_i rk_i X_{i+1} X_{i+2} X_{i+3}
| | | | |
| | | | |
| v | | |
+---+ +---+ +---+ | | |
| X | | | | X | <--+ | |
| O | <- | T | <- | O | <----------+ |
| R | | | | R | <------------------+
+---+ +---+ +---+ | | |
| / / /
| / / /
| / / /
\--------------------------------------=
/ / / \
/ / / |
/---------------/ / / |
| | | |
X_{i+1} X_{i+2} X_{i+3} X_{i+4}
Figure 3: SM4 Round Function For the i-th Round
Family key FK given in hexadecimal notation, is:
FK_0 = A3B1BAC6 FK_1 = 56AA3350 FK_2 = 677D9197 FK_3 = B27022DC
The method to retrieve values from the constant key CK is as follows.
Let ck_{i, j} be the j-th byte (i = 0, 1, ..., 31; j = 0, 1, 2, 3) of CK_i.
Therefore, each ck_{i, j} is a 8-bit string, and each CK_i a 32-bit word.
CK_i = (ck_{i, 0}, ck_{i, 1}, ck_{i, 2}, ck_{i, 3})
ck_{i, j} = (4i + j) x 7 (mod 256)
The values of the constant key CK_i, where (i = 0, 1, ..., 31), in hexadecimal, are:
CK_0 = 00070E15 CK_16 = C0C7CED5 CK_1 = 1C232A31 CK_17 = DCE3EAF1 CK_2 = 383F464D CK_18 = F8FF060D CK_3 = 545B6269 CK_19 = 141B2229 CK_4 = 70777E85 CK_20 = 30373E45 CK_5 = 8C939AA1 CK_21 = 4C535A61 CK_6 = A8AFB6BD CK_22 = 686F767D CK_7 = C4CBD2D9 CK_23 = 848B9299 CK_8 = E0E7EEF5 CK_24 = A0A7AEB5 CK_9 = FC030A11 CK_25 = BCC3CAD1 CK_10 = 181F262D CK_26 = D8DFE6ED CK_11 = 343B4249 CK_27 = F4FB0209 CK_12 = 50575E65 CK_28 = 10171E25 CK_13 = 6C737A81 CK_29 = 2C333A41 CK_14 = 888F969D CK_30 = 484F565D CK_15 = A4ABB2B9 CK_31 = 646B7279
This document defines multiple modes of operation for the SM4 blockcipher algorithm.
The CBC (Cipher Block Chaining), ECB (Electronic CodeBook), CFB (Cipher FeedBack), OFB (Output FeedBack) and CTR (Counter) modes are defined in [NIST.SP.800-38A] and utilized with the SM4 algorithm in the following sections.
Hereinafter we define:
The CBC, CFB and OFB modes require an additional input to the encryption process, called the initialization vector (IV). The identical IV is used in the input of encryption as well as the decryption of the corresponding ciphertext.
Generation of IV values MUST take into account of the considerations in Section 12 recommended by [BC-EVAL].
In SM4-ECB, the same key is utilized to create a fixed assignment for a plaintext block with a ciphertext block, meaning that a given plaintext block always gets encrypted to the same ciphertext block. As described in [NIST.SP.800-38A], this mode should be avoided if this property is undesirable.
This mode requires input plaintext to be a multiple of the block size, which in this case of SM4 it is 128-bit. It also allows multiple blocks to be computed in parallel.
Inputs:
Output:
C is defined as follows.
_____________________________________________________________________ n = NBlocks(P, b) for i = 1 to n C_i = SM4Encrypt(P_i, K) end for C = C_1 || ... || C_n _____________________________________________________________________
Inputs:
Output:
P is defined as follows.
_____________________________________________________________________ n = NBlocks(C, b) for i = 1 to n P_i = SM4Decrypt(C_i, K) end for P = P_1 || ... || P_n _____________________________________________________________________
SM4-CBC is similar to SM4-ECB that the input plaintext MUST be a multiple of the block size, which is 128-bit in SM4. SM4-CBC requires an additional input, the IV, that is unpredictable for a particular execution of the encryption process.
Since CBC encryption relies on a forward cipher operation that depend on results of the previous operation, it cannot be parallelized. However, for decryption, since ciphertext blocks are already available, CBC parallel decryption is possible.
Inputs:
Output:
C is defined as follows.
_____________________________________________________________________
n = NBlocks(P, b)
C_1 = SM4Encrypt(P_1 xor IV, K)
for i = 2 to n
C_i = SM4Encrypt(P_i xor C_{i - 1}, K)
end for
C = C_1 || ... || C_n
_____________________________________________________________________
Inputs:
Output:
P is defined as follows.
_____________________________________________________________________
n = NBlocks(C, b)
P_1 = SM4Decrypt(C_1, K) xor IV
for i = 2 to n
P_i = SM4Decrypt(C_i, K) xor C_{i - 1}
end for
P = P_1 || ... || P_n
_____________________________________________________________________
SM4-CFB relies on feedback provided by successive ciphertext segments to generate output blocks. The plaintext given must be a multiple of the block size.
Similar to SM4-CBC, SM4-CFB requires an IV that is unpredictable for a particular execution of the encryption process.
SM4-CFB further allows setting a positive integer parameter s, that is less than or equal to the block size, to specify the size of each data segment. The same segment size must be used in encryption and decryption.
In SM4-CFB, since the input block to each forward cipher function depends on the output of the previous block (except the first that depends on the IV), encryption is not parallelizable. Decryption, however, can be parallelized.
SM4-CFB takes an integer s to determine segment size in its encryption and decryption routines. We define the following variants of SM4-CFB for various s:
Inputs:
Output:
C# is defined as follows.
_____________________________________________________________________
n = NBlocks(P#, s)
I_1 = IV
for i = 2 to n
I_i = LSB(b - s, I_{i - 1}) || C#_{j - 1}
end for
for i = 1 to n
O_j = SM4Encrypt(I_i, K)
end for
for i = 1 to n
C#_i = P#_1 xor MSB(s, O_j)
end for
C# = C#_1 || ... || C#_n
_____________________________________________________________________
Inputs:
Output:
P# is defined as follows.
_____________________________________________________________________
n = NBlocks(P#, s)
I_1 = IV
for i = 2 to n
I_i = LSB(b - s, I_{i - 1}) || C#_{j - 1}
end for
for i = 1 to n
O_j = SM4Encrypt(I_i, K)
end for
for i = 1 to n
P#_i = C#_1 xor MSB(s, O_j)
end for
P# = P#_1 || ... || P#_n
_____________________________________________________________________
SM4-OFB is the application of SM4 through the Output Feedback mode. This mode requires that the IV is a nonce, meaning that the IV MUST be unique for each execution for an input key. OFB does not require the input plaintext to be a multiple of the block size.
In OFB, the routines for encryption and decryption are identical. As each forward cipher function (except the first) depends on previous results, both routines cannot be parallelized. However given a known IV, output blocks could be generated prior to the input of plaintext (encryption) or ciphertext (decryption).
Inputs:
Output:
C is defined as follows.
_____________________________________________________________________
n = NBlocks(P, b)
I_1 = IV
for i = 1 to (n - 1)
O_i = SM4Encrypt(I_i)
I_{i + 1} = O_i
end for
for i = 1 to (n - 1)
C_i = P_i xor O_i
end for
C_n = P_n xor MSB(u, O_n)
C = C_1 || ... || C_n
_____________________________________________________________________
Inputs:
Output:
C is defined as follows.
_____________________________________________________________________
n = NBlocks(C, b)
I_1 = IV
for i = 1 to (n - 1)
O_i = SM4Encrypt(I_i)
I_{i + 1} = O_i
end for
for i = 1 to (n - 1)
P_i = C_i xor O_i
end for
P_n = C_n xor MSB(u, O_n)
P = P_1 || ... || P_n
_____________________________________________________________________
SM4-CTR is an implementation of a stream cipher through a blockcipher primitive. It generates a "keystream" of keys that are used to encrypt successive blocks, with the keystream created from the input key, a nonce (the IV) and an incremental counter. The counter could be any sequence that does not repeat within the block size.
Both SM4-CTR encryption and decryption routines could be parallelized, and random access is also possible.
Inputs:
Output:
C is defined as follows.
_____________________________________________________________________ n = NBlocks(P, b) for i = 1 to n O_i = SM4Encrypt(T_i) end for for i = 1 to (n - 1) C_i = P_i xor O_i end for C_n = P_n xor MSB(u, O_n) C = C_1 || ... || C_n _____________________________________________________________________
Inputs:
Output:
P is defined as follows.
_____________________________________________________________________ n = NBlocks(C, b) for i = 1 to n O_i = SM4Encrypt(T_i) end for for i = 1 to (n - 1) P_i = C_i xor O_i end for P_n = C_n xor MSB(u, O_n) C = C_1 || ... || C_n _____________________________________________________________________
The Object Identifier for SM4 is identified through these OIDs.
"1.2.156.10197.1.104" for "SM4 Algorithm" [GMT-0006-2012].
"1.0.18033.3.2.4" for "id-bc128-sm4" [ISO.IEC.18033-3.AMD2], described below.
The chaos principle and the diffusion principle are two basic principles of block cipher design. A well-designed blockcipher algorithm should be based on a cryptographically sound basic transformation structure, with its round calculation based on a cryptographically sound basic transformation.
The cryptographic properties of the basic transformation determines the efficiency of the resulting encryption transformation.
The SM4 algorithm is structured on orthomorphic permutation. Its round transformation is an orthomorphic permutation, and its cryptographic properties can be deduced from the characteristics of orthomorphic permutations.
Let the single round of the SM4 block cipher algorithm be P, for any given plaintext X, P (X, K ')! = P (X, K) if the key K'! = K.
The conclusion shows that if X is a row variable and K is a column variable, the square P(X, K) forms a Latin square. There are two conclusions about the nature of cryptography:
An S-box can be viewed as a bijection:
S(X) = (f_1(X), f_2(X), ... , f_m(X)) : F_2^n -> F_2^m.
S(x): F_2^n -> F_2^m can be represented as a multi-output boolean function with n-bit input and m-bit output, or a n x m S-box (an S-box with n inputs and m outputs), usually realized as a substitution that takes an n-bit input and produces a m-bit output. In SM4, the S-box takes n = m = 8.
In many blockciphers, the S-box is the sole element providing nonlinearity, for the purpose of mixing, in order to reduce linearity and to hide its variable structure.
The cryptographic properties of the S-box directly affects the resulting cryptographic strength of the blockcipher. When designing a blockcipher, the cryptographic strength of the S-box must be taken into account. The cryptographic strength of an S-box can be generally measured by factors such as its nonlinearity and differential distribution.
In order to prevent insertion attacks, the algebraic formula used for cryptographic substitution should be a high degree polynomial and contain a large number of terms.
The algebraic expression of the SM4 S-box [SM4-Sbox] is determined through Lagrange’s interpolation to be a polynomial of the 254th degree with 255 terms, providing the highest level of complexity based on its size:
f(x) : sum_{i=0}^{255} y_i
PI_{j!=i, j=0}^255 ((x - x_j) / (x_i - x_j))
Any n boolean function f(x): F_2^n -> F_2 can be represented uniquely in its algebraic normal form shown below:
f(X) = a_0 + sum_{1<=i_i<...<i_k<=n, 1<=k<=n}
a_{i_1 i_2 ... i_k} x_{i_1} x_{i_2} ... x_{i_k}
X = (x_1, x_2, ..., x_n)
a_0, a_{i_1, i_2, ... i_k} element-of F_2
The "algebraic degree" of the n-boolean function f(X) is defined to be the algebraic degree of the highest algebraic degree of its terms with a nonzero coefficient in its ANF representation. The constant of the i-th term of f(x) in ANF representation is called the i-th term of f(X), the total number of all i-th (0<=i<=n) terms is called the "number of terms" of f(X).
S(X) can be represented as a m-component function S(X) = (f_1(X), f_2(X), ... f_m(X)): F_2^n -> F_2^m. Consider S(X) to be a random substitution, each of its component functions would be best to have algebraic degree of n-1, each component function i-th coefficient should be near C_n^i/2. If the algebraic degree is too low, for example, each component function has a degree of 2, then the algorithm can be easily attacked by advanced differential cryptanalysis. If the number of terms are insufficient, then it may improve the success probability of insert attacks.
The algebraic degrees and number of terms of the SM4 S-box are described in Figure 4.
+--------------------+--------------------------------------------+ | | Algebraic Degree | | Component Function +-----+---+----+----+----+----+----+---+-----+ | | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | +--------------------+-----+---+----+----+----+----+----+---+-----+ | Y_0 | 0 | 3 | 15 | 31 | 28 | 29 | 14 | 3 | 1 | | Y_1 | 0 | 3 | 12 | 34 | 40 | 33 | 12 | 4 | 1 | | Y_2 | 0 | 5 | 17 | 24 | 40 | 24 | 11 | 3 | 0 | | Y_3 | 0 | 2 | 11 | 31 | 34 | 27 | 15 | 5 | 1 | | Y_4 | 0 | 5 | 15 | 28 | 33 | 24 | 13 | 5 | 0 | | Y_5 | 0 | 5 | 11 | 25 | 41 | 25 | 16 | 4 | 1 | | Y_6 | 0 | 4 | 15 | 29 | 27 | 32 | 18 | 4 | 1 | | Y_7 | 0 | 4 | 14 | 32 | 35 | 30 | 16 | 3 | 0 | +--------------------+-----+---+----+----+----+----+----+---+-----+ | Expected Value | 1/2 | 4 | 14 | 28 | 35 | 28 | 14 | 4 | 1/2 | +--------------------+-----+---+----+----+----+----+----+---+-----+
Figure 4: SM4 S-box Component Functions Algebraic Degree And Terms
The definition of differential distribution has been given in [BC-Design].
Differential cryptanalysis is a chosen-plaintext attack, with the understanding that analysis of selected plaintexts of differentials can retrive the most probable key. Differential distribution is an attribute to measure the resistance of a cryptographic function against differential cryptanalysis.
delta_S = 1/2^n max_{a in F_2^n, a!=0} max_{beta in F_2^m} |
{ X in F_2^n : S(X and alpha) - S(X) = beta } |
delta_S is the differential distribution of the S-box S.
According to the definition of differential distribution, 2^{-m} <= delta_S <= 2^{m-n}, if there is a delta_S = 2^{m-n} then S is considered a fully nonlinear function from F_2^n to F_2^m. For resistance against differential cryptanalysis, the differential distribution should be as low as possible.
The highest differential distribution of the SM4 S-box is 2^{-6}, meaning it has a good resistance against differential cryptanalysis.
The nonlinearity of an S-box is described by [BC-Design].
Let S(X) = (f_1(X), f_2(X), ... , f_m(X)) : F_2^n -> F_2^m be a multi-output function. The nonlinearity of S(X) is defined as N_S = min_{l in L_n, 0 != u in F_2^m} d_H (u . S(X), l(X)).
L_n is the group of all n-boolean functions, d_H(f, l) is the Hamming distance between f and l. The nonlinearity of the S-box is in fact the minimum Hamming distance between all the Boolean functions and all affine functions.
The upper-bound of nonlinearity is known to be 2^{n-1} - 2^{n/2 - 1}, where a Boolean function that reaches this bound is called a "bent function".
The nonlinearity of a Boolean function is used to measure resistance against linear attacks. The higher the nonlinearity, the higher resistance that the Boolean function f(x) has against linear attacks. On the contrary, the lower the nonlinearity, the Boolean function f(x) has lower resistance against linear attacks.
The nonlinearity of the SM4 S-box is 112.
Linear approximation of a S-box is defined in [BC-Design]. Given a S-box with n inputs and m outputs, any linear approximation can be represented as : a . X = b . Y, where a in F_2^n, b in F_2^m.
The probability p that satisfies a . X = b . Y is
| p - 1/2 | <= 1/2 - N_S / 2^n
where | p - 1/2 | is the advantage of the linear approximation equation, lambda_S = 1/2 - N_s / 2^n is the maximum advantage of the S-box.
The maximum advantage of the SM4 S-box is 2^{-4}.
A S-box S(X) = (f_1(X), f_2(X), ... , f_m(X)) : F_2^n -> F_2^m is considered "balanced" if for any beta in F_2^m, there are 2^{n-m} x in F_2^n, such that S(x) = beta.
The SM4 S-box is balanced.
A S-box S(X) = (f_1(X), f_2(X), ... , f_m(X)) : F_2^n -> F_2^m is considered "complete" if every input bit directly correlates to an output bit.
In algebraic expression, each component function contains the unknown variables x_1, x_2, ... x_n, such that for any (s, t) in { (i, j) | 1 <= i <= n, 1 <= j <= m}, there is an X that S(X) and S(X and e_s) would contain a different bit t.
Avalanche effect refers to a single bit change in the input would correspond to a change of half of the output bits.
The SM4 S-box satisfies completness and the avalanche effect.
Linear transformation is used to provide diffusion in SM4. A blockcipher algorithm often adopts m x m S-boxes to form an obfuscation layer.
Since the m-bits output by one S-box are only related to the m bits of its input and are irrelevant to the input of other S boxes, the introduction of a linear transform would disrupt and mix the output m-bits so that they seem correlating to the other S-box inputs.
A sound linear transform design will diffuse the S-box output, allowing the blockcipher to resist differential and linear cryptanalysis.
An important measure of the diffusivity of a linear transform is its branch number.
The "branch number" of a linear transform is defined in [BC-Design]:
B(theta) = min_{x!=0} w_b(x) + w_b(theta(x))
Where B(theta) is the branch number of transform theta, w_b(x) is a non-zero integer x_i (1 ⇐ i ⇐ m), and x_i is called the "bundle weight".
The branch number can be used to quantify the resistance of the block cipher algorithm to differential cryptanalysis and linear cryptanalysis.
Similar to differential cryptanalysis and linear cryptanalysis, the differential branch number and linear branch number of theta can be defined as follows.
The differential branch number of theta is:
B_d(theta) = min_{x, x!= x*}
(w_b(x and x*) + w_b(theta(x)) and theta(x*))
The linear branch number of theta is:
B_l(theta) = min_{a, b, c (x . alpha^t , theta(x) . beta) != 0}
(w_b(alpha) + w_b(beta))
where,
c (x . a^t , theta(x) . beta) =
2 X Pr(x . alpha^t = theta(x) . beta) - 1
x . alpha^t is a matrix multiplication.
The branch number in a linear transformation reflects its diffusivity. The higher the branch number, the better the diffusion effect.
This means that the larger the differential branch number or linear branch number, the more known plaintexts will be required for differential or linear cryptanalysis respectively.
The linear transform differential branch number and linear branch number of SM4 are both 5.
The SM4 key schedule is designed to fulfill the security requirements of the encryption algorithm and achieve ease of implementation for performance reasons.
All subkeys are derived from the encryption key, and therefore, subkeys are always statistically relevant. In the context of a blockcipher, it is not possible to have non-statistical-correlated subkeys, but the designer can only aim to have subkeys achieve near statistical independence [BC-Design].
The purpose of the key schedule, generated through the key expansion algorithm, is to mask the statistical correlation between subkeys to make this relationship difficult to exploit.
The SM4 key expansion algorithm satisfies the following design criteria:
SM4 has been heavily cryptanalyzed by international researchers since it was first published. Nearly all currently known cryptanalysis techniques have been applied to SM4.
At the time of publishing this document, there are no known practical attacks against the full SM4 blockcipher. However, there are side-channel concerns [SideChannel] when the algorithm is implemented in a hardware device.
A summary of cryptanalysis results are presented in the following sections.
In 2008, Zhang et al. [SM4-DiffZhang1] gave a 21-round differential analysis with data complexity 2^188, time complexity 2^126.8 encryptions.
In 2008, Kim et al. [SM4-LDA] gave a 22-round differential attack that requires 2^118 chosen plaintexts, 2^123 memory and 2^125.71 encryptions.
In 2009, Zhang et al. (differing author but overlapping team) [SM4-DiffZhang2] gave a 18-round differential characteristics with an attack that reaches the 22nd round, with data complexity 2^117 and time complexity 2^112.3.
In 2010, Zhang et al. (with no relation to above) [SM4-DiffZhang3] utilized 18-round differential characteristics for the 22nd round with 2^117 chosen plaintexts with time complexity 2^123 encryptions, memory complexity of 2^112.
In 2011, Su et al. [SM4-DiffSu] gave a 19 round differential characteristics and pushed their attack to the 23rd round, with data complexity of 2^118 chosen plaintexts, time complexity 2^126.7 encryptions, and memory complexity 2^120.7.
In 2008 Etrog et al. [SM4-LinearEtrog] provided a linear cryptanalysis result for 22 rounds of SM4, the data complexity is given as 2^188.4 known plaintexts, time complexity 2^117 encrypt operations.
In the same year, Kim et al. [SM4-LDA] improved on the linear cryptanalysis result for 22 rounds of SM4 with data complexity of 2^117 known plaintexts, memory complexity of 2^109 and time complexity of 2^109.86.
In 2011 Dong [SM4-LinearDong] presented a linear cryptanalysis result for 20 rounds, 2^110.4 known ciphertexts, 2^106.8 encryption operations, memory complexity 2^90.
In 2014 Liu et al. [SM4-LinearLiu] presented their linear cryptanalysis for 23-rounds of SM4, time complexity 2^112 encryption operations, data complexity 2^126.54 known ciphertexts, memory complexity 2^116.
In 2017 Liu et al. [SM4-NLC] presented an attack based on linear cryptanalysis on 24-rounds of SM4, with time complexity of 2^122.6 encryptions, data complexity of 2^122.6 known ciphertexts, and memory complexity of 2^85.
In 2010, Liu et al. [SM4-MLLiu] constructed a series of 18 rounds of linear traces based on a 5-round circular linear trace, capable of attacking 22 rounds of SM4. The required data complexity was 2^112 known plaintexts, time complexity 2^124.21 encryption operations, with memory complexity of 2^118.83.
In 2010 Cho et al. [SM4-MLCho] gave a linear analysis of 23 rounds of SM4 with a data complexity of 2^126.7 known plaintexts and a time complexity of 2^127, memory complexity of 2^120.7.
In 2014, Liu et al. [SM4-LinearLiu] gave the results of multi-dimensional linear analysis of 23 rounds of SM4 algorithm. The time complexity was 2^122.7, data complexity was 2^122.6 known plaintext with memory complexity 2^120.6.
In 2007 Lu et al. [SM4-IDCLu] first presented 16 rounds of impossible differential analysis of SM4 with the required data complexity 2^105 chosen plaintexts, time complexity 2^107 encryption operations.
In 2008 Toz et al. [SM4-IDCToz] revised the results of [SM4-IDCLu], that the data complexity is actually 2^117.05 chosen plaintexts, time complexity 2^132.06 encryptions, but its complexity is already beyond the 2^128 limit.
In 2010 Wang et al. [SM4-IDCWang] pushed the impossible differential cryptanalysis to 17 rounds of SM4, the data complexity is 2^117 chosen ciphertexts, time complexity 2^132 memory queries.
In 2015 Ma et al. [SM4-ZCLC] gives the results of multi-dimensional zero-correlation linear cryptanalysis of a 14-round SM4 algorithm. The required data complexity is 2^123.5 known plaintexts, time complexity is 2^120.7 encryption operations and memory complexity of 2^73 blocks.
In 2007 Liu et al. [SM4-ICLiu] first gave a 13-round integral analysis of SM4, which required 2^16 chosen plaintexts and time complexity of 2^114 encryption operations.
In 2008 Zhong et al. [SM4-ICZhong] constructed a 12-round distinguisher of SM4 to attack 14-round SM4, with data complexity of 2^32 chosen plaintexts and time complexity 2^96.5 encryptions.
In 2009 Ji et al. [SM4-AAJi] and in 2010 Erickson et al. [SM4-AAEr] utilized algebraic methods such as XL, Groebner base and SAT to analyze the resistance of SM4 against algebraic attacks. The results demonstrate that SM4 is safe against algebraic attacks, and specifically, has a higher resistance against algebraic attacks than AES.
In 2007 Lu et al. [SM4-IDCLu] provided a matrix attack against 14-round SM4, with data complexity 2^121.82 chosen plaintexts, time complexity 2^116.66 encryptions.
In 2008 Toz et al. [SM4-IDCToz] lowered both data and time complexity of the aformentioned attack to 2^106.89 chosen ciphertexts and time complexity of 2^107.89.
In 2008, Zhang et al. [SM4-DiffZhang1] provided a matrix attack against 16-round SM4, which required a data complexity of 2^125 chosen plaintexts and time complexity of 2^116 encryptions.
She’s Master dissertation [SM4-MatrixShe] provided a SM4 16-round matrix distinguisher that can attack 18-round SM4, with data complexity of 2^127 chosen plaintexts and time complexity 2^110.77 encryptions with memory complexity of 2^130.
In 2012 Wei et al. [SM4-MatrixWei] applied differential analysis and algebraic attack techniques on 20-round SM4 and discovered that the combined attack results on 20-round SM4 are superior than using pure differential cryptanalysis.
SM4 uses a novel structure differing from the general Feistel and SP structures.
[SM4-Random] has proven that the SM4 non-balanced Feistel structure is pseudo-random.
[SM4-SLDC] analyzes the SM4 non-balanced Feistel structure on its resistance against differential and linear cryptanalysis techniques. Under SP type round functions with branch number 5, it is proven that in a 27-round SM4 guarantees at least 22 active S-boxes, therefore SM4 is secure against differential attacks.
[SM4-SLC] has analyzed resistance of SM4 against linear cryptanalysis.
Related-key differential cryptanalysis is related to the encryption algorithm and key schedule. When performing a related-key attack, the attacker simultaneously insert differences in both the key and the message.
In [AutoDC], Sun et al. proposed an automated differential route search method based on MILP (mixed-integer linear programming) that can be used to assess the security bounds of a blockcipher under (related-key) differential cryptanalysis.
[SM4-RKDC] describes the lower bounds of active S-boxes within SM4 and is shown in Table 1.
| Round | Single Key | Related Key |
|---|---|---|
| 3 | 0 | 0 |
| 4 | 1 | 1 |
| 5 | 2 | 2 |
| 6 | 2 | 4 |
| 7 | 5 | 6 |
| 8 | 6 | 8 |
| 9 | 7 | 9 |
| 10 | 8 | 10 |
| 11 | 9 | 11 |
| 12 | 10 | 13 |
| 13 | 10 | 14 |
| 14 | 10 | 14 |
| 15 | 13 | 16 |
| 16 | 14 | 18 |
| 17 | 15 | 19 |
| 18 | 16 | 20 |
| 19 | 18 | 22 |
| 20 | 18 | - |
| 21 | 19 | - |
| 22 | 20 | - |
| 23 | 22 | - |
| 24 | 23 | - |
| 25 | 23 | - |
| 26 | 24 | - |
As the maximal probability of the SM4 S-box is 2^−6, when the minimum active S-boxes reach 22 the differential characteristics will have probability 2^132, which is higher than enumeration (2^128).
This indicates that 19 rounds and 23 rounds under related key and single key settings will provide a minimum of 22 active S-boxes and is able to resist related-key differential attacks.
Table 2 provides a summary on the strongest attacks on SM4 at the time of publishing.
| Method | Rounds | Complexity | Reference |
|---|---|---|---|
| Linear | 24 | Time: 2^{122.6}, Data: 2^{122.6}, Memory: 2^{85} | [SM4-NLC] |
| Multi-dimensional Linear | 23 | Time: 2^{122.7}, Data: 2^{122.6}, Memory: 2^{120.6} | [SM4-LinearLiu] |
| Differential | 23 | Time: 2^{126.7}, Data: 2^{117}, Memory: 2^{120.7} | [SM4-DiffSu] |
| Matrix | 18 | Time: 2^{110.77}, Data: 2^{127}, Memory 2^{130} | [SM4-MatrixShe] |
| Impossible Differential | 17 | Time: 2^{132}, Data: 2^{117}, Memory: — | [SM4-IDCWang] |
| Zero-correlation Linear | 14 | Time: 2^{120.7}, Data: 2^{123.5}, Memory: 2^{73} | [SM4-ZCLC] |
| Integral | 14 | Time: 2^{96.5}, Data: 2^{32}, Memory: — | [SM4-ICZhong] |
As of the publication of this document, no open research results have provided a method to successfully attack beyond 24 rounds of SM4.
The traditional view suggests that SM4 provides an extra safety margin compared to blockciphers adopted in [ISO.IEC.18033-3] that already have full-round attacks, including MISTY1 [MISTY1-IC] [MISTY1-270] and AES [AES-CA] [AES-BC] [AES-RKC].
This document does not require any action by IANA.
| [GBT.32907-2016] | Standardization Administration of the People's Republic of China, "GB/T 32907-2016: Information security technology -- SM4 block cipher algorithm", August 2016. |
| [ISO.IEC.18033-3.AMD2] | International Organization for Standardization, "ISO/IEC WD1 18033-3/AMD2 -- Encryption algorithms -- Part 3: Block ciphers -- Amendment 2", June 2017. |
| [RFC2119] | Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, March 1997. |
This is example 1 provided by [GBT.32907-2016] to demonstrate encryption of a plaintext.
Plaintext:
01 23 45 67 89 AB CD EF FE DC BA 98 76 54 32 10
Encryption key:
01 23 45 67 89 AB CD EF FE DC BA 98 76 54 32 10
Status of the round key (rk_i) and round output (X_i) per round:
rk_0 = F12186F9 X_4 = 27FAD345 rk_1 = 41662B61 X_5 = A18B4CB2 rk_2 = 5A6AB19A X_6 = 11C1E22A rk_3 = 7BA92077 X_7 = CC13E2EE rk_4 = 367360F4 X_8 = F87C5BD5 rk_5 = 776A0C61 X_9 = 33220757 rk_6 = B6BB89B3 X_10 = 77F4C297 rk_7 = 24763151 X_11 = 7A96F2EB rk_8 = A520307C X_12 = 27DAC07F rk_9 = B7584DBD X_13 = 42DD0F19 rk_10 = C30753ED X_14 = B8A5DA02 rk_11 = 7EE55B57 X_15 = 907127FA rk_12 = 6988608C X_16 = 8B952B83 rk_13 = 30D895B7 X_17 = D42B7C59 rk_14 = 44BA14AF X_18 = 2FFC5831 rk_15 = 104495A1 X_19 = F69E6888 rk_16 = D120B428 X_20 = AF2432C4 rk_17 = 73B55FA3 X_21 = ED1EC85E rk_18 = CC874966 X_22 = 55A3BA22 rk_19 = 92244439 X_23 = 124B18AA rk_20 = E89E641F X_24 = 6AE7725F rk_21 = 98CA015A X_25 = F4CBA1F9 rk_22 = C7159060 X_26 = 1DCDFA10 rk_23 = 99E1FD2E X_27 = 2FF60603 rk_24 = B79BD80C X_28 = EFF24FDC rk_25 = 1D2115B0 X_29 = 6FE46B75 rk_26 = 0E228AEB X_30 = 893450AD rk_27 = F1780C81 X_31 = 7B938F4C rk_28 = 428D3654 X_32 = 536E4246 rk_29 = 62293496 X_33 = 86B3E94F rk_30 = 01CF72E5 X_34 = D206965E rk_31 = 9124A012 X_35 = 681EDF34
Ciphertext:
68 1E DF 34 D2 06 96 5E 86 B3 E9 4F 53 6E 42 46
This demonstrates the decryption process of the Example 1 ciphertext provided by [GBT.32907-2016].
Ciphertext:
68 1E DF 34 D2 06 96 5E 86 B3 E9 4F 53 6E 42 46
Encryption key:
01 23 45 67 89 AB CD EF FE DC BA 98 76 54 32 10
Status of the round key (rk_i) and round output (X_i) per round:
rk_31 = 9124A012 X_35 = 7B938F4C rk_30 = 01CF72E5 X_34 = 893450AD rk_29 = 62293496 X_33 = 6FE46B75 rk_28 = 428D3654 X_32 = EFF24FDC rk_27 = F1780C81 X_31 = 2FF60603 rk_26 = 0E228AEB X_30 = 1DCDFA10 rk_25 = 1D2115B0 X_29 = F4CBA1F9 rk_24 = B79BD80C X_28 = 6AE7725F rk_23 = 99E1FD2E X_27 = 124B18AA rk_22 = C7159060 X_26 = 55A3BA22 rk_21 = 98CA015A X_25 = ED1EC85E rk_20 = E89E641F X_24 = AF2432C4 rk_19 = 92244439 X_23 = F69E6888 rk_18 = CC874966 X_22 = 2FFC5831 rk_17 = 73B55FA3 X_21 = D42B7C59 rk_16 = D120B428 X_20 = 8B952B83 rk_15 = 104495A1 X_19 = 907127FA rk_14 = 44BA14AF X_18 = B8A5DA02 rk_13 = 30D895B7 X_17 = 42DD0F19 rk_12 = 6988608C X_16 = 27DAC07F rk_11 = 7EE55B57 X_15 = 7A96F2EB rk_10 = C30753ED X_14 = 77F4C297 rk_9 = B7584DBD X_13 = 33220757 rk_8 = A520307C X_12 = F87C5BD5 rk_7 = 24763151 X_11 = CC13E2EE rk_6 = B6BB89B3 X_10 = 11C1E22A rk_5 = 776A0C61 X_9 = A18B4CB2 rk_4 = 367360F4 X_8 = 27FAD345 rk_3 = 7BA92077 X_7 = 76543210 rk_2 = 5A6AB19A X_6 = FEDCBA98 rk_1 = 41662B61 X_5 = 89ABCDEF rk_0 = F12186F9 X_4 = 01234567
Plaintext:
01 23 45 67 89 AB CD EF FE DC BA 98 76 54 32 10
This example is provided by [GBT.32907-2016] to demonstrate encryption of a plaintext 1,000,000 times repeatedly, using a fixed encryption key.
Plaintext:
01 23 45 67 89 AB CD EF FE DC BA 98 76 54 32 10
Encryption Key:
01 23 45 67 89 AB CD EF FE DC BA 98 76 54 32 10
Ciphertext:
59 52 98 C7 C6 FD 27 1F 04 02 F8 04 C3 3D 3F 66
The following example demonstrates encryption of a different message using a different key from the above examples.
Plaintext:
01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F
Encryption key:
FE DC BA 98 76 54 32 10 01 23 45 67 89 AB CD EF
Status of the round key (rk_i) and round output (X_i) per round:
rk_0 = 0D8CC1B4 X_4 = F7EAEB6A rk_1 = AC44F213 X_5 = B4967C0F rk_2 = 188C0C40 X_6 = 5B9B2419 rk_3 = 7537585E X_7 = F46BECBA rk_4 = 627646F5 X_8 = A8013E25 rk_5 = 54D785AD X_9 = B38E2ABE rk_6 = 51B96DEE X_10 = 3E7C99A1 rk_7 = 0C385958 X_11 = 6DD5F47F rk_8 = 5E494992 X_12 = B286430C rk_9 = 32F3FE04 X_13 = AB997DE3 rk_10 = 3A3A733D X_14 = 80F8F21F rk_11 = 0EDFB91D X_15 = 4EF7052E rk_12 = 6823CD6B X_16 = 4462FFAF rk_13 = 40F7D825 X_17 = 14DFD5EA rk_14 = 4BD68EE5 X_18 = 6D33EFED rk_15 = 165A36C8 X_19 = 3A4F8B3C rk_16 = 56608984 X_20 = 1A435088 rk_17 = 23F35FF4 X_21 = 4E64B153 rk_18 = 8B592B3E X_22 = 0415CEDA rk_19 = 80F7388A X_23 = ADD88955 rk_20 = 0415C409 X_24 = 73964EF1 rk_21 = AFDF1370 X_25 = B0085092 rk_22 = CF444772 X_26 = 554A1293 rk_23 = 9AF9901F X_27 = 4BC6D6A8 rk_24 = C457578C X_28 = 7BB650E1 rk_25 = 95701C60 X_29 = DDFB8A61 rk_26 = 2B0F4EE1 X_30 = 5C4DFD78 rk_27 = 7F826139 X_31 = FD9066FD rk_28 = FA37F8D9 X_32 = 55ADB594 rk_29 = D18AF8CE X_33 = AC1B3EA9 rk_30 = 5BD5D8C6 X_34 = 13F01ADE rk_31 = 711138B7 X_35 = F766678F
Ciphertext:
F7 66 67 8F 13 F0 1A DE AC 1B 3E A9 55 AD B5 94
The following example demonstrates decryption of Example 4.
Ciphertext:
F7 66 67 8F 13 F0 1A DE AC 1B 3E A9 55 AD B5 94
Encryption key:
FE DC BA 98 76 54 32 10 01 23 45 67 89 AB CD EF
Status of the round key (rk_i) and round output (X_i) per round:
rk_31 = 711138B7 X_35 = FD9066FD rk_30 = 5BD5D8C6 X_34 = 5C4DFD78 rk_29 = D18AF8CE X_33 = DDFB8A61 rk_28 = FA37F8D9 X_32 = 7BB650E1 rk_27 = 7F826139 X_31 = 4BC6D6A8 rk_26 = 2B0F4EE1 X_30 = 554A1293 rk_25 = 95701C60 X_29 = B0085092 rk_24 = C457578C X_28 = 73964EF1 rk_23 = 9AF9901F X_27 = ADD88955 rk_22 = CF444772 X_26 = 0415CEDA rk_21 = AFDF1370 X_25 = 4E64B153 rk_20 = 0415C409 X_24 = 1A435088 rk_19 = 80F7388A X_23 = 3A4F8B3C rk_18 = 8B592B3E X_22 = 6D33EFED rk_17 = 23F35FF4 X_21 = 14DFD5EA rk_16 = 56608984 X_20 = 4462FFAF rk_15 = 165A36C8 X_19 = 4EF7052E rk_14 = 4BD68EE5 X_18 = 80F8F21F rk_13 = 40F7D825 X_17 = AB997DE3 rk_12 = 6823CD6B X_16 = B286430C rk_11 = 0EDFB91D X_15 = 6DD5F47F rk_10 = 3A3A733D X_14 = 3E7C99A1 rk_9 = 32F3FE04 X_13 = B38E2ABE rk_8 = 5E494992 X_12 = A8013E25 rk_7 = 0C385958 X_11 = F46BECBA rk_6 = 51B96DEE X_10 = 5B9B2419 rk_5 = 54D785AD X_9 = B4967C0F rk_4 = 627646F5 X_8 = F7EAEB6A rk_3 = 7537585E X_7 = 0C0D0E0F rk_2 = 188C0C40 X_6 = 08090A0B rk_1 = AC44F213 X_5 = 04050607 rk_0 = 0D8CC1B4 X_4 = 00010203
Plaintext:
01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F
This example is based on Example 4 to demonstrate encryption of a plaintext 1,000,000 times repeatedly, using a fixed encryption key.
Plaintext:
01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F
Encryption Key:
FE DC BA 98 76 54 32 10 01 23 45 67 89 AB CD EF
Ciphertext:
37 9A 96 D0 A6 A5 A5 06 0F B4 60 C7 5D 18 79 ED
The following examples can be verified using open-source cryptographic libraries including:
Plaintext:
AA AA AA AA BB BB BB BB CC CC CC CC DD DD DD DD EE EE EE EE FF FF FF FF AA AA AA AA BB BB BB BB
Encryption Key:
01 23 45 67 89 AB CD EF FE DC BA 98 76 54 32 10
Ciphertext:
5E C8 14 3D E5 09 CF F7 B5 17 9F 8F 47 4B 86 19 2F 1D 30 5A 7F B1 7D F9 85 F8 1C 84 82 19 23 04 00 2A 8A 4E FA 86 3C CA D0 24 AC 03 00 BB 40 D2
Plaintext:
AA AA AA AA BB BB BB BB CC CC CC CC DD DD DD DD EE EE EE EE FF FF FF FF AA AA AA AA BB BB BB BB
Encryption Key:
FE DC BA 98 76 54 32 10 01 23 45 67 89 AB CD EF
Ciphertext:
C5 87 68 97 E4 A5 9B BB A7 2A 10 C8 38 72 24 5B 12 DD 90 BC 2D 20 06 92 B5 29 A4 15 5A C9 E6 00 A2 51 49 20 93 F8 F6 42 89 B7 8D 6E 8A 28 B1 C6
Plaintext:
AA AA AA AA BB BB BB BB CC CC CC CC DD DD DD DD EE EE EE EE FF FF FF FF AA AA AA AA BB BB BB BB
Encryption Key:
01 23 45 67 89 AB CD EF FE DC BA 98 76 54 32 10
IV:
00 01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F
Ciphertext:
78 EB B1 1C C4 0B 0A 48 31 2A AE B2 04 02 44 CB 4C B7 01 69 51 90 92 26 97 9B 0D 15 DC 6A 8F 6D 40 D8 41 32 E9 99 74 A4 A8 80 88 68 42 07 48 59
Plaintext:
AA AA AA AA BB BB BB BB CC CC CC CC DD DD DD DD EE EE EE EE FF FF FF FF AA AA AA AA BB BB BB BB
Encryption Key:
FE DC BA 98 76 54 32 10 01 23 45 67 89 AB CD EF
IV:
00 01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F
Ciphertext:
0D 3A 6D DC 2D 21 C6 98 85 72 15 58 7B 7B B5 9A 91 F2 C1 47 91 1A 41 44 66 5E 1F A1 D4 0B AE 38 67 C6 7F 15 25 1F 2A C5 21 9B B2 6E 80 7D A5 5B
Plaintext:
AA AA AA AA BB BB BB BB CC CC CC CC DD DD DD DD EE EE EE EE FF FF FF FF AA AA AA AA BB BB BB BB
Encryption Key:
01 23 45 67 89 AB CD EF FE DC BA 98 76 54 32 10
IV:
00 01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F
Ciphertext:
AC 32 36 CB 86 1D D3 16 E6 41 3B 4E 3C 75 24 B7 1D 01 AC A2 48 7C A5 82 CB F5 46 3E 66 98 53 9B
Plaintext:
AA AA AA AA BB BB BB BB CC CC CC CC DD DD DD DD EE EE EE EE FF FF FF FF AA AA AA AA BB BB BB BB
Encryption Key:
FE DC BA 98 76 54 32 10 01 23 45 67 89 AB CD EF
IV:
00 01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F
Ciphertext:
5D CC CD 25 A8 4B A1 65 60 D7 F2 65 88 70 68 49 33 FA 16 BD 5C D9 C8 56 CA CA A1 E1 01 89 7A 97
Plaintext:
AA AA AA AA BB BB BB BB CC CC CC CC DD DD DD DD EE EE EE EE FF FF FF FF AA AA AA AA BB BB BB BB
Encryption Key:
01 23 45 67 89 AB CD EF FE DC BA 98 76 54 32 10
IV:
00 01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F
Ciphertext:
AC 32 36 CB 86 1D D3 16 E6 41 3B 4E 3C 75 24 B7 69 D4 C5 4E D4 33 B9 A0 34 60 09 BE B3 7B 2B 3F
Plaintext:
AA AA AA AA BB BB BB BB CC CC CC CC DD DD DD DD EE EE EE EE FF FF FF FF AA AA AA AA BB BB BB BB
Encryption Key:
FE DC BA 98 76 54 32 10 01 23 45 67 89 AB CD EF
IV:
00 01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F
Ciphertext:
5D CC CD 25 A8 4B A1 65 60 D7 F2 65 88 70 68 49 0D 9B 86 FF 20 C3 BF E1 15 FF A0 2C A6 19 2C C5
Plaintext:
AA AA AA AA AA AA AA AA BB BB BB BB BB BB BB BB CC CC CC CC CC CC CC CC DD DD DD DD DD DD DD DD EE EE EE EE EE EE EE EE FF FF FF FF FF FF FF FF EE EE EE EE EE EE EE EE AA AA AA AA AA AA AA AA
Encryption Key:
01 23 45 67 89 AB CD EF FE DC BA 98 76 54 32 10
IV:
00 01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F
Ciphertext:
AC 32 36 CB 97 0C C2 07 91 36 4C 39 5A 13 42 D1 A3 CB C1 87 8C 6F 30 CD 07 4C CE 38 5C DD 70 C7 F2 34 BC 0E 24 C1 19 80 FD 12 86 31 0C E3 7B 92 6E 02 FC D0 FA A0 BA F3 8B 29 33 85 1D 82 45 14
Plaintext:
AA AA AA AA AA AA AA AA BB BB BB BB BB BB BB BB CC CC CC CC CC CC CC CC DD DD DD DD DD DD DD DD EE EE EE EE EE EE EE EE FF FF FF FF FF FF FF FF EE EE EE EE EE EE EE EE AA AA AA AA AA AA AA AA
Encryption Key:
FE DC BA 98 76 54 32 10 01 23 45 67 89 AB CD EF
IV:
00 01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F
Ciphertext:
5D CC CD 25 B9 5A B0 74 17 A0 85 12 EE 16 0E 2F 8F 66 15 21 CB BA B4 4C C8 71 38 44 5B C2 9E 5C 0A E0 29 72 05 D6 27 04 17 3B 21 23 9B 88 7F 6C 8C B5 B8 00 91 7A 24 88 28 4B DE 9E 16 EA 29 06
"sm4.h" is the header file for the SM4 function.
<CODE BEGINS>
#ifndef HEADER_SM4_H
# define HEADER_SM4_H
#include <inttypes.h>
# define SM4_BLOCK_SIZE 16
# define SM4_KEY_SCHEDULE 32
void sm4_encrypt(uint8_t key[],
unsigned char plaintext[],
unsigned char ciphertext[]);
void sm4_decrypt(uint8_t key[],
unsigned char ciphertext[],
unsigned char plaintext[]);
#endif
<CODE ENDS>
"sm4.c" contains the main implementation of SM4.
<CODE BEGINS>
/* A sample implementation of SM4 */
#include <stdlib.h>
#include <string.h>
#include "sm4.h"
#include "print.h"
/* Operations */
/* Rotate Left 32-bit number */
#define ROTL32(X, n) (((X) << (n)) | ((X) >> (32 - (n))))
static uint32_t sm4_ck[32] = {
0x00070E15, 0x1C232A31, 0x383F464D, 0x545B6269,
0x70777E85, 0x8C939AA1, 0xA8AFB6BD, 0xC4CBD2D9,
0xE0E7EEF5, 0xFC030A11, 0x181F262D, 0x343B4249,
0x50575E65, 0x6C737A81, 0x888F969D, 0xA4ABB2B9,
0xC0C7CED5, 0xDCE3EAF1, 0xF8FF060D, 0x141B2229,
0x30373E45, 0x4C535A61, 0x686F767D, 0x848B9299,
0xA0A7AEB5, 0xBCC3CAD1, 0xD8DFE6ED, 0xF4FB0209,
0x10171E25, 0x2C333A41, 0x484F565D, 0x646B7279
};
static uint8_t sm4_sbox[256] = {
0xD6, 0x90, 0xE9, 0xFE, 0xCC, 0xE1, 0x3D, 0xB7,
0x16, 0xB6, 0x14, 0xC2, 0x28, 0xFB, 0x2C, 0x05,
0x2B, 0x67, 0x9A, 0x76, 0x2A, 0xBE, 0x04, 0xC3,
0xAA, 0x44, 0x13, 0x26, 0x49, 0x86, 0x06, 0x99,
0x9C, 0x42, 0x50, 0xF4, 0x91, 0xEF, 0x98, 0x7A,
0x33, 0x54, 0x0B, 0x43, 0xED, 0xCF, 0xAC, 0x62,
0xE4, 0xB3, 0x1C, 0xA9, 0xC9, 0x08, 0xE8, 0x95,
0x80, 0xDF, 0x94, 0xFA, 0x75, 0x8F, 0x3F, 0xA6,
0x47, 0x07, 0xA7, 0xFC, 0xF3, 0x73, 0x17, 0xBA,
0x83, 0x59, 0x3C, 0x19, 0xE6, 0x85, 0x4F, 0xA8,
0x68, 0x6B, 0x81, 0xB2, 0x71, 0x64, 0xDA, 0x8B,
0xF8, 0xEB, 0x0F, 0x4B, 0x70, 0x56, 0x9D, 0x35,
0x1E, 0x24, 0x0E, 0x5E, 0x63, 0x58, 0xD1, 0xA2,
0x25, 0x22, 0x7C, 0x3B, 0x01, 0x21, 0x78, 0x87,
0xD4, 0x00, 0x46, 0x57, 0x9F, 0xD3, 0x27, 0x52,
0x4C, 0x36, 0x02, 0xE7, 0xA0, 0xC4, 0xC8, 0x9E,
0xEA, 0xBF, 0x8A, 0xD2, 0x40, 0xC7, 0x38, 0xB5,
0xA3, 0xF7, 0xF2, 0xCE, 0xF9, 0x61, 0x15, 0xA1,
0xE0, 0xAE, 0x5D, 0xA4, 0x9B, 0x34, 0x1A, 0x55,
0xAD, 0x93, 0x32, 0x30, 0xF5, 0x8C, 0xB1, 0xE3,
0x1D, 0xF6, 0xE2, 0x2E, 0x82, 0x66, 0xCA, 0x60,
0xC0, 0x29, 0x23, 0xAB, 0x0D, 0x53, 0x4E, 0x6F,
0xD5, 0xDB, 0x37, 0x45, 0xDE, 0xFD, 0x8E, 0x2F,
0x03, 0xFF, 0x6A, 0x72, 0x6D, 0x6C, 0x5B, 0x51,
0x8D, 0x1B, 0xAF, 0x92, 0xBB, 0xDD, 0xBC, 0x7F,
0x11, 0xD9, 0x5C, 0x41, 0x1F, 0x10, 0x5A, 0xD8,
0x0A, 0xC1, 0x31, 0x88, 0xA5, 0xCD, 0x7B, 0xBD,
0x2D, 0x74, 0xD0, 0x12, 0xB8, 0xE5, 0xB4, 0xB0,
0x89, 0x69, 0x97, 0x4A, 0x0C, 0x96, 0x77, 0x7E,
0x65, 0xB9, 0xF1, 0x09, 0xC5, 0x6E, 0xC6, 0x84,
0x18, 0xF0, 0x7D, 0xEC, 0x3A, 0xDC, 0x4D, 0x20,
0x79, 0xEE, 0x5F, 0x3E, 0xD7, 0xCB, 0x39, 0x48
};
static uint32_t sm4_fk[4] = {
0xA3B1BAC6, 0x56AA3350, 0x677D9197, 0xB27022DC
};
static uint32_t load_u32_be(const uint8_t *b, uint32_t n)
{
return ((uint32_t)b[4 * n + 3] << 24) |
((uint32_t)b[4 * n + 2] << 16) |
((uint32_t)b[4 * n + 1] << 8) |
((uint32_t)b[4 * n ]);
}
static void store_u32_be(uint32_t v, uint8_t *b)
{
b[3] = (uint8_t)(v >> 24);
b[2] = (uint8_t)(v >> 16);
b[1] = (uint8_t)(v >> 8);
b[0] = (uint8_t)(v);
}
static void sm4_key_schedule(uint8_t key[], uint32_t rk[])
{
uint32_t t, x, k[36];
int i;
for (i = 0; i < 4; i++)
{
k[i] = load_u32_be(key, i) ^ sm4_fk[i];
}
/* T' */
for (i = 0; i < SM4_KEY_SCHEDULE; ++i)
{
x = k[i + 1] ^ k[i + 2] ^ k[i + 3] ^ sm4_ck[i];
/* Nonlinear operation tau */
t = ((uint32_t)sm4_sbox[(uint8_t)(x >> 24)]) << 24 |
((uint32_t)sm4_sbox[(uint8_t)(x >> 16)]) << 16 |
((uint32_t)sm4_sbox[(uint8_t)(x >> 8)]) << 8 |
((uint32_t)sm4_sbox[(uint8_t)(x)]);
/* Linear operation L' */
k[i+4] = k[i] ^ (t ^ ROTL32(t, 13) ^ ROTL32(t, 23));
rk[i] = k[i + 4];
}
}
#define SM4_ROUNDS(k0, k1, k2, k3, F) \
do { \
X0 ^= F(X1 ^ X2 ^ X3 ^ rk[k0]); \
X1 ^= F(X0 ^ X2 ^ X3 ^ rk[k1]); \
X2 ^= F(X0 ^ X1 ^ X3 ^ rk[k2]); \
X3 ^= F(X0 ^ X1 ^ X2 ^ rk[k3]); \
debug_print("rk_%0.2i = %0.8x " \
" X_%0.2i = %0.8x\n", k0, rk[k0], k0+4, X0); \
debug_print("rk_%0.2i = %0.8x " \
" X_%0.2i = %0.8x\n", k1, rk[k1], k1+4, X1); \
debug_print("rk_%0.2i = %0.8x " \
" X_%0.2i = %0.8x\n", k2, rk[k2], k2+4, X2); \
debug_print("rk_%0.2i = %0.8x " \
" X_%0.2i = %0.8x\n", k3, rk[k3], k3+4, X3); \
} while(0)
static uint32_t sm4_t(uint32_t x)
{
uint32_t t = 0;
t |= ((uint32_t)sm4_sbox[(uint8_t)(x >> 24)]) << 24;
t |= ((uint32_t)sm4_sbox[(uint8_t)(x >> 16)]) << 16;
t |= ((uint32_t)sm4_sbox[(uint8_t)(x >> 8)]) << 8;
t |= sm4_sbox[(uint8_t)x];
/*
* L linear transform
*/
return t ^ ROTL32(t, 2) ^ ROTL32(t, 10) ^
ROTL32(t, 18) ^ ROTL32(t, 24);
}
void sm4_encrypt(uint8_t key[],
unsigned char plaintext[],
unsigned char ciphertext[])
{
uint32_t rk[SM4_KEY_SCHEDULE], X0, X1, X2, X3;
int i, j;
sm4_key_schedule(key, rk);
X0 = load_u32_be(plaintext, 0);
X1 = load_u32_be(plaintext, 1);
X2 = load_u32_be(plaintext, 2);
X3 = load_u32_be(plaintext, 3);
SM4_ROUNDS( 0, 1, 2, 3, sm4_t);
SM4_ROUNDS( 4, 5, 6, 7, sm4_t);
SM4_ROUNDS( 8, 9, 10, 11, sm4_t);
SM4_ROUNDS(12, 13, 14, 15, sm4_t);
SM4_ROUNDS(16, 17, 18, 19, sm4_t);
SM4_ROUNDS(20, 21, 22, 23, sm4_t);
SM4_ROUNDS(24, 25, 26, 27, sm4_t);
SM4_ROUNDS(28, 29, 30, 31, sm4_t);
store_u32_be(X3, ciphertext);
store_u32_be(X2, ciphertext + 4);
store_u32_be(X1, ciphertext + 8);
store_u32_be(X0, ciphertext + 12);
}
void sm4_decrypt(uint8_t key[],
unsigned char ciphertext[],
unsigned char plaintext[])
{
uint32_t rk[SM4_KEY_SCHEDULE], X0, X1, X2, X3;
int i, j;
sm4_key_schedule(key, rk);
X0 = load_u32_be(ciphertext, 0);
X1 = load_u32_be(ciphertext, 1);
X2 = load_u32_be(ciphertext, 2);
X3 = load_u32_be(ciphertext, 3);
SM4_ROUNDS(31, 30, 29, 28, sm4_t);
SM4_ROUNDS(27, 26, 25, 24, sm4_t);
SM4_ROUNDS(23, 22, 21, 20, sm4_t);
SM4_ROUNDS(19, 18, 17, 16, sm4_t);
SM4_ROUNDS(15, 14, 13, 12, sm4_t);
SM4_ROUNDS(11, 10, 9, 8, sm4_t);
SM4_ROUNDS( 7, 6, 5, 4, sm4_t);
SM4_ROUNDS( 3, 2, 1, 0, sm4_t);
store_u32_be(X3, plaintext);
store_u32_be(X2, plaintext + 4);
store_u32_be(X1, plaintext + 8);
store_u32_be(X0, plaintext + 12);
}
<CODE ENDS>
"sm4_main.c" is used to run the examples provided in this document and print out internal state for implementation reference.
<CODE BEGINS>
#include <stdlib.h>
#include <string.h>
#include <stdbool.h>
#include "sm4.h"
#include "print.h"
typedef struct {
unsigned char* key;
unsigned char* message;
unsigned char* expected;
int iterations;
bool encrypt;
} test_case;
int sm4_run_example(test_case tc)
{
unsigned char input[SM4_BLOCK_SIZE] = {0};
unsigned char output[SM4_BLOCK_SIZE] = {0};
int i;
debug_print("-----------------------"
" Message Input m Begin "
"-------------------------\n");
print_bytes((unsigned int*)tc.message, SM4_BLOCK_SIZE);
debug_print("----------------------- "
"Message Input m End "
"---------------------------\n");
if (tc.encrypt)
{
debug_print("----------------------- "
"Encrypt "
"---------------------------\n");
memcpy(input, tc.message, SM4_BLOCK_SIZE);
for (i = 0; i != tc.iterations; ++i)
{
sm4_encrypt(tc.key,
(unsigned char*)input,
(unsigned char*)output);
memcpy(input, output, SM4_BLOCK_SIZE);
}
}
else
{
debug_print("----------------------- "
"Decrypt "
"---------------------------\n");
memcpy(input, tc.message, SM4_BLOCK_SIZE);
for (i = 0; i != tc.iterations; ++i)
{
sm4_decrypt(tc.key,
(unsigned char*)input,
(unsigned char*)output);
memcpy(input, output, SM4_BLOCK_SIZE);
}
}
debug = 1;
debug_print("+++++++++++++++++++++++++++++++"
" RESULT "
"++++++++++++++++++++++++++++++++\n");
debug_print("RESULTS:\n");
debug_print(" Expected:\n");
print_bytes((unsigned int*)tc.expected, SM4_BLOCK_SIZE);
debug_print(" Output:\n");
print_bytes((unsigned int*)output, SM4_BLOCK_SIZE);
debug = 0;
return memcmp(
(unsigned char*)output,
(unsigned char*)tc.expected,
SM4_BLOCK_SIZE
);
}
int main(int argc, char **argv)
{
int i;
unsigned char key[SM4_BLOCK_SIZE];
unsigned char block[SM4_BLOCK_SIZE];
test_case tests[8] = {0};
/*
* This test vector comes from Example 1 of GB/T 32907-2016,
*/
static const unsigned int gbt32907k1[SM4_BLOCK_SIZE] = {
0x01234567, 0x89abcdef,
0xfedcba98, 0x76543210
};
static const unsigned int gbt32907m1[SM4_BLOCK_SIZE] = {
0x01234567, 0x89abcdef,
0xfedcba98, 0x76543210
};
static const unsigned int gbt32907e1[SM4_BLOCK_SIZE] = {
0x681edf34, 0xd206965e,
0x86b3e94f, 0x536e4246
};
test_case gbt32907t1 = {
(unsigned char*)gbt32907k1,
(unsigned char*)gbt32907m1,
(unsigned char*)gbt32907e1,
1,
true
};
tests[0] = gbt32907t1;
/*
* This test vector comes from Example 2 from GB/T 32907-2016.
* After 1,000,000 iterations.
*/
static const unsigned int gbt32907e2[SM4_BLOCK_SIZE] = {
0x595298c7, 0xc6fd271f,
0x0402f804, 0xc33d3f66
};
test_case gbt32907t2 = {
(unsigned char*)gbt32907k1,
(unsigned char*)gbt32907m1,
(unsigned char*)gbt32907e2,
1000000,
true
};
tests[1] = gbt32907t2;
/*
* This test vector reverses Example 1 of GB/T 32907-2016.
* After decrypting 1 iteration.
*/
test_case gbt32907t3 = {
(unsigned char*)gbt32907k1,
(unsigned char*)gbt32907e1,
(unsigned char*)gbt32907m1,
1,
false
};
tests[2] = gbt32907t3;
/*
* This test vector reverses Example 2 of GB/T 32907-2016.
* After decrypting 1,000,000 iterations.
*/
test_case gbt32907t4 = {
(unsigned char*)gbt32907k1,
(unsigned char*)gbt32907e2,
(unsigned char*)gbt32907m1,
1000000,
false
};
tests[3] = gbt32907t4;
/*
* Newly added examples to demonstrate key changes.
*/
static const unsigned int newexamplek1[SM4_BLOCK_SIZE] = {
0xfedcba98, 0x76543210,
0x01234567, 0x89abcdef
};
static const unsigned int newexamplem1[SM4_BLOCK_SIZE] = {
0x00010203, 0x04050607,
0x08090a0b, 0x0c0d0e0f
};
static const unsigned int newexamplee1[SM4_BLOCK_SIZE] = {
0xf766678f, 0x13f01ade,
0xac1b3ea9, 0x55adb594
};
/*
*/
test_case newexample1 = {
(unsigned char*)newexamplek1,
(unsigned char*)newexamplem1,
(unsigned char*)newexamplee1,
1,
true
};
tests[4] = newexample1;
test_case newexample2 = {
(unsigned char*)newexamplek1,
(unsigned char*)newexamplee1,
(unsigned char*)newexamplem1,
1,
false
};
tests[5] = newexample2;
/*
* After 1,000,000 iterations.
*/
static const unsigned int newexamplee2[SM4_BLOCK_SIZE] = {
0x379a96d0, 0xa6a5a506,
0x0fb460c7, 0x5d1879ed
};
test_case newexample3 = {
(unsigned char*)newexamplek1,
(unsigned char*)newexamplem1,
(unsigned char*)newexamplee2,
1000000,
true
};
tests[6] = newexample3;
for (i = 0; i < 7; ++i)
{
if (i == 1 || i == 3)
continue;
printf("sm4_example[%2i]: %s\n", i,
sm4_run_example(tests[i]) ? "FAIL" : "PASS");
}
return 0;
}
<CODE ENDS>
"print.c" and "print.h" are used to provide pretty formatting used to print out the examples for this document.
"print.h"
<CODE BEGINS>
#ifndef SM4PRINT_H
#define SM4PRINT_H
#define DEBUG 0
#define debug_print(...) \
do { if (DEBUG) fprintf(stderr, __VA_ARGS__); } while (0)
#include <stdio.h>
void print_bytes(unsigned* buf, int n);
#endif
<CODE ENDS>
"print.c"
<CODE BEGINS>
#include <stdio.h>
#include "print.h"
void print_bytes(unsigned int* buf, int n)
{
unsigned char* ptr = (unsigned char*)buf;
int i, j;
for (i = 0; i <= n/4; i++) {
if (i > 0 && i % 8 == 0) {
debug_print("\n");
}
for (j = 1; j <= 4; j++) {
if ((i*4+4-j) < n) {
debug_print("%.2X", ptr[(i*4)+4-j]);
}
}
debug_print(" ");
}
debug_print("\n");
}
<CODE ENDS>
The authors would like to thank the following persons for their valuable advice and input.