# Substitution box

In cryptography, a S-Box (Substitution-box) is a basic component of symmetric-key algorithms. In block ciphers, the S-Boxes are used to make the relation between the key and the ciphertext (coded text) difficult to understand–Shannon's property of confusion. The S-Boxes are carefully chosen to resist cryptanalysis (decoding).

In general, an S-Box takes some number of input bits, m, and transforms them into some number of output bits, n: an m×n S-Box can be implemented as a lookup table with 2m words of n bits each. Fixed tables are normally used, as in the Data Encryption Standard (DES), but in some ciphers the tables are generated dynamically from the key; e.g. the Blowfish and the Twofish encryption algorithms. Bruce Schneier describes IDEA's modular multiplication step as a key-dependent S-Box.

One good example is this 6×4-bit S-Box from DES (S5):

S5 Middle (inner) 4 bits of input
0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111
Outer bits 00 0010 1100 0100 0001 0111 1010 1011 0110 1000 0101 0011 1111 1101 0000 1110 1001
01 1110 1011 0010 1100 0100 0111 1101 0001 0101 0000 1111 1010 0011 1001 1000 0110
10 0100 0010 0001 1011 1010 1101 0111 1000 1111 1001 1100 0101 0110 0011 0000 1110
11 1011 1000 1100 0111 0001 1110 0010 1101 0110 1111 0000 1001 1010 0100 0101 0011

Given a 6-bit input, the 4-bit output is found by selecting the row using the outer two bits, and the column using the inner four bits. For example, an input "011011" has outer bits "01" and inner bits "1101"; the corresponding output would be "1001".

The 8 S-Boxes of DES were the subject of intensive studies for many years cause of a concern that a method of bypassing the DES cipher to obtaining access to the plaintext–a vulnerability (susceptibility) known only to its designers–might have been planted (inserted) in the cipher. In 1994, the S-Box design criteria were finally published by its designers after the public rediscovery of differential cryptanalysis, showing that they had been carefully tuned the design to increase resistance against differential cryptanalysis attacks. Other research had already indicated that even a very small modification to one of the 8 S-Box used by the DES could weaken it very much.

The design of good S-Boxes was the subject of a great amount of research; now much more is understood about their use in block ciphers than when the DES S-Boxes were released.

## References

• Kaisa Nyberg (1991). "Perfect nonlinear S-boxes" (PDF). Advances in Cryptology - EUROCRYPT '91. Brighton. pp. 378–386. Archived from the original (PDF) on 2007-09-26. Retrieved 2007-02-20.
• Don Coppersmith (1994). "The Data Encryption Standard (DES) and its strength against attacks" (PDF). IBM Journal of Research and Development. 38 (3): 243–250. doi:10.1147/rd.383.0243. Retrieved 2007-02-20.
• S. Mister and C. Adams (1996). "Practical S-Box Design". Workshop on Selected Areas in Cryptography (SAC '96) Workshop Record. Queens University. pp. 61–76. Archived from the original on 2006-08-31. Retrieved 2007-02-20.
• Schneier, Bruce (1996). Applied Cryptography, Second Edition. John Wiley & Sons. pp. 296–298, 349. ISBN 0-471-11709-9.