## Cryptology ePrint Archive: Report 2017/1055

Cellular Automata Based S-boxes

Luca Mariot and Stjepan Picek and Alberto Leporati and Domagoj Jakobovic

Abstract: Cellular Automata (CA) represent an interesting approach to design Substitution Boxes (S-boxes) having good cryptographic properties and low implementation costs. From the cryptographic perspective, up to now there have been only ad-hoc studies about specific kinds of CA, the best known example being the $\chi$ nonlinear transformation used in Keccak. In this paper, we undertake a systematic investigation of the cryptographic properties of S-boxes defined by CA, proving some upper bounds on their nonlinearity and differential uniformity. Next, we extend some previous published results about the construction of CA-based S-boxes by means of a heuristic technique, namely Genetic Programming (GP). In particular, we propose a "reverse engineering" method based on De Bruijn graphs to determine whether a specific S-box is expressible through a single CA rule. Then, we use GP to assess if some CA-based S-box with optimal cryptographic properties can be described by a smaller CA. The results show that GP is able to find much smaller CA rules defining the same reference S-boxes up to size $7\times 7$, suggesting that our method could be used to find more efficient representations of CA-based S-boxes for hardware implementations. Finally, we classify up to affine equivalence all $3\times 3$ and $4\times 4$ CA-based S-boxes.

Category / Keywords: secret-key cryptography / ellular Automata, S-box, Cryptographic properties, Heuristics

Date: received 28 Oct 2017, last revised 17 Feb 2018

Contact author: luca mariot at disco unimib it

Available format(s): PDF | BibTeX Citation

Note: General revision fixing several errors and typos. Structure changed to better reflect the original contributions of the paper and summarize the previous work.

Short URL: ia.cr/2017/1055

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