# S-box

(Redirected from Substitution box)

In cryptography, an S-box (substitution-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, thus ensuring Shannon's property of confusion. Mathematically, an S-box is a nonlinear[1] vectorial Boolean function.[2]

In general, an S-box takes some number of input bits, m, and transforms them into some number of output bits, n, where n is not necessarily equal to m.[3] 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).

## Example

One good example of a fixed table is the S-box from DES (S5), mapping 6-bit input into a 4-bit output:

S5 Middle 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 (the first and last 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".[4]

## Analysis and properties

When DES was first published in 1977, the design criteria of its S-boxes were kept secret to avoid compromising the technique of differential cryptanalysis (which was not yet publicly known). As a result, research in what made a good S-boxes was sparse at the time. Rather, the eight S-boxes of DES were the subject of intense study for many years out of a concern that a backdoor (a vulnerability known only to its designers) might have been planted in the cipher. As the S-boxes are the only nonlinear part of the cipher, compromising those would compromise the entire cipher.[5]

The S-box design criteria were eventually published (in Coppersmith 1994) after the public rediscovery of differential cryptanalysis, showing that they had been carefully tuned to increase resistance against this specific attack such that it was no better than brute force. Biham and Shamir found that even small modifications to an S-box could significantly weaken DES.[6]

Any S-box where any linear combination of output bits is produced by a bent function of the input bits is termed a perfect S-box.[7]

S-boxes can be analyzed using linear cryptanalysis and differential cryptanalysis in the form of a Linear approximation table (LAT) or Walsh transform and Difference Distribution Table (DDT) or autocorrelation table and spectrum. Its strength may be summarized by the nonlinearity (bent, almost bent) and differential uniformity (perfectly nonlinear, almost perfectly nonlinear).[8][9][10][2]

## References

1. ^ Daemen & Rijmen 2013, p. 22.
2. ^ a b Carlet, Claude (2010), Hammer, Peter L.; Crama, Yves (eds.), "Vectorial Boolean Functions for Cryptography", Boolean Models and Methods in Mathematics, Computer Science, and Engineering, Encyclopedia of Mathematics and its Applications, Cambridge: Cambridge University Press, pp. 398–470, ISBN 978-0-521-84752-0, retrieved 2021-04-30
3. ^ Chandrasekaran, J.; et al. (2011). "A Chaos Based Approach for Improving Non Linearity in the S-box Design of Symmetric Key Cryptosystems". In Meghanathan, N.; et al. (eds.). Advances in Networks and Communications: First International Conference on Computer Science and Information Technology, CCSIT 2011, Bangalore, India, January 2-4, 2011. Proceedings, Part 2. Springer. p. 516. ISBN 978-3-642-17877-1.
4. ^ Buchmann, Johannes A. (2001). "5. DES". Introduction to cryptography (Corr. 2. print. ed.). New York, NY [u.a.]: Springer. pp. 119–120. ISBN 978-0-387-95034-1.
5. ^ Coppersmith, D. (May 1994). "The Data Encryption Standard (DES) and its strength against attacks". IBM Journal of Research and Development. 38 (3): 243–250. doi:10.1147/rd.383.0243. ISSN 0018-8646.
6. ^
7. ^ RFC 4086. Section 5.3 "Using S-boxes for Mixing"
8. ^ Heys, Howard M. "A Tutorial on Linear and Differential Cryptanalysis" (PDF).`{{cite web}}`: CS1 maint: url-status (link)
9. ^ "S-Boxes and Their Algebraic Representations — Sage 9.2 Reference Manual: Cryptography". doc.sagemath.org. Retrieved 2021-04-30.
10. ^ Saarinen, Markku-Juhani O. (2012). Miri, Ali; Vaudenay, Serge (eds.). "Cryptographic Analysis of All 4 × 4-Bit S-Boxes". Selected Areas in Cryptography. Lecture Notes in Computer Science. Berlin, Heidelberg: Springer. 7118: 118–133. doi:10.1007/978-3-642-28496-0_7. ISBN 978-3-642-28496-0.