In cryptography, an S-Box is a basic component of symmetric key algorithms which performs substitution. In block ciphers, they are typically used to obscure the relationship between the key and the ciphertext — Shannon's property of confusion...
Advanced Encryption Standard is a specification for the encryption of electronic data. It has been adopted by the U.S. government and is now used worldwide. It supersedes DES...
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...
for a given number in GF(28) = GF(2)[x]/(x8 + x4 + x3 + x + 1), Rijndael's finite field (zero,
which has no inverse, is set to zero). The multiplicative inverse is then transformed using the following affine transformation
Affine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
:
where [x0, ..., x7] is the multiplicative inverse as a vector.
The matrix multiplication can be calculated by the following algorithm:
Store the multiplicative inverse of the input number in two 8-bit unsigned temporary variables: s and x.
Rotate the value s one bit to the left; if the value of s had a high bit (eighth bit from the right) of one, make the low bit of s one; otherwise the low bit of s is zero.
Exclusive or the value of x with the value of s, storing the value in x
For three more iterations, repeat steps two and three; steps two and three are done a total of four times.
The value of x will now have the result of the multiplication.
After the matrix multiplication is done, exclusive or the value by the decimal number 99 (the hexadecimal number 0x63, the binary number 1100011, and the bit string 11000110 representing the number in LSb first notation).
This will generate the following S-box, which is represented here with hexadecimal
Hexadecimal
In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...
notation:
| 0 1 2 3 4 5 6 7 8 9 a b c d e f
---|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
00 |63 7c 77 7b f2 6b 6f c5 30 01 67 2b fe d7 ab 76
10 |ca 82 c9 7d fa 59 47 f0 ad d4 a2 af 9c a4 72 c0
20 |b7 fd 93 26 36 3f f7 cc 34 a5 e5 f1 71 d8 31 15
30 |04 c7 23 c3 18 96 05 9a 07 12 80 e2 eb 27 b2 75
40 |09 83 2c 1a 1b 6e 5a a0 52 3b d6 b3 29 e3 2f 84
50 |53 d1 00 ed 20 fc b1 5b 6a cb be 39 4a 4c 58 cf
60 |d0 ef aa fb 43 4d 33 85 45 f9 02 7f 50 3c 9f a8
70 |51 a3 40 8f 92 9d 38 f5 bc b6 da 21 10 ff f3 d2
80 |cd 0c 13 ec 5f 97 44 17 c4 a7 7e 3d 64 5d 19 73
90 |60 81 4f dc 22 2a 90 88 46 ee b8 14 de 5e 0b db
a0 |e0 32 3a 0a 49 06 24 5c c2 d3 ac 62 91 95 e4 79
b0 |e7 c8 37 6d 8d d5 4e a9 6c 56 f4 ea 65 7a ae 08
c0 |ba 78 25 2e 1c a6 b4 c6 e8 dd 74 1f 4b bd 8b 8a
d0 |70 3e b5 66 48 03 f6 0e 61 35 57 b9 86 c1 1d 9e
e0 |e1 f8 98 11 69 d9 8e 94 9b 1e 87 e9 ce 55 28 df
f0 |8c a1 89 0d bf e6 42 68 41 99 2d 0f b0 54 bb 16
Here the column is determined by the least significant nibble
Nibble
In computing, a nibble is a four-bit aggregation, or half an octet...
, and the row is determined by the most significant nibble. For example, the value 0x9a is converted in to 0xb8 by Rijndael's S-box. Note that the multiplicative inverse of 0x00 is defined as itself.
C is a general-purpose computer programming language developed between 1969 and 1973 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system....
C++ is a statically typed, free-form, multi-paradigm, compiled, general-purpose programming language. It is regarded as an intermediate-level language, as it comprises a combination of both high-level and low-level language features. It was developed by Bjarne Stroustrup starting in 1979 at Bell...
The inverse S-box is simply the S-box run in reverse. For example, the inverse S-box of 0xdb is 0x9f. It is calculated by first calculating the inverse affine transformation of the input value, followed by the multiplicative inverse. The inverse affine transformation is as follows:
The following table represents Rijndael's inverse S-box:
| 0 1 2 3 4 5 6 7 8 9 a b c d e f
---|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
00 |52 09 6a d5 30 36 a5 38 bf 40 a3 9e 81 f3 d7 fb
10 |7c e3 39 82 9b 2f ff 87 34 8e 43 44 c4 de e9 cb
20 |54 7b 94 32 a6 c2 23 3d ee 4c 95 0b 42 fa c3 4e
30 |08 2e a1 66 28 d9 24 b2 76 5b a2 49 6d 8b d1 25
40 |72 f8 f6 64 86 68 98 16 d4 a4 5c cc 5d 65 b6 92
50 |6c 70 48 50 fd ed b9 da 5e 15 46 57 a7 8d 9d 84
60 |90 d8 ab 00 8c bc d3 0a f7 e4 58 05 b8 b3 45 06
70 |d0 2c 1e 8f ca 3f 0f 02 c1 af bd 03 01 13 8a 6b
80 |3a 91 11 41 4f 67 dc ea 97 f2 cf ce f0 b4 e6 73
90 |96 ac 74 22 e7 ad 35 85 e2 f9 37 e8 1c 75 df 6e
a0 |47 f1 1a 71 1d 29 c5 89 6f b7 62 0e aa 18 be 1b
b0 |fc 56 3e 4b c6 d2 79 20 9a db c0 fe 78 cd 5a f4
c0 |1f dd a8 33 88 07 c7 31 b1 12 10 59 27 80 ec 5f
d0 |60 51 7f a9 19 b5 4a 0d 2d e5 7a 9f 93 c9 9c ef
e0 |a0 e0 3b 4d ae 2a f5 b0 c8 eb bb 3c 83 53 99 61
f0 |17 2b 04 7e ba 77 d6 26 e1 69 14 63 55 21 0c 7d
For C, C++ implementation, here is the initialization of the table:
The Rijndael S-Box was specifically designed to be resistant to linear
Linear cryptanalysis
In cryptography, linear cryptanalysis is a general form of cryptanalysis based on finding affine approximations to the action of a cipher. Attacks have been developed for block ciphers and stream ciphers...
Differential cryptanalysis is a general form of cryptanalysis applicable primarily to block ciphers, but also to stream ciphers and cryptographic hash functions. In the broadest sense, it is the study of how differences in an input can affect the resultant difference at the output...
. This was done by minimizing the correlation between linear transformations of input/output bits, and at the same time minimizing the difference propagation probability.
In addition, to strengthen the S-Box against algebraic attacks, the affine transformation was added. In the case of suspicion of a trapdoor being built into the cipher, the current S-box might be replaced by another one. The authors claim that the Rijndael cipher structure should provide enough resistance against differential and linear cryptanalysis, even if an S-Box with "average" correlation / difference propagation properties is used.
An alternate equation for the affine transformation
An equivalent equation for the affine transformation is
where b' b and c are 8 bit arrays and c is 01100011.
Java is a programming language originally developed by James Gosling at Sun Microsystems and released in 1995 as a core component of Sun Microsystems' Java platform. The language derives much of its syntax from C and C++ but has a simpler object model and fewer low-level facilities...
