# Mutually orthogonal latin squares based on cellular automata

@article{Mariot2019MutuallyOL, title={Mutually orthogonal latin squares based on cellular automata}, author={Luca Mariot and Maximilien Gadouleau and Enrico Formenti and Alberto Leporati}, journal={Designs, Codes and Cryptography}, year={2019}, volume={88}, pages={391-411} }

We investigate sets of mutually orthogonal latin squares (MOLS) generated by cellular automata (CA) over finite fields. After introducing how a CA defined by a bipermutive local rule of diameter d over an alphabet of q elements generates a Latin square of order $$q^{d-1}$$ q d - 1 , we study the conditions under which two CA generate a pair of orthogonal Latin squares. In particular, we prove that the Latin squares induced by two Linear Bipermutive CA (LBCA) over the finite field $$\mathbb {F…

## 12 Citations

### Hip to Be (Latin) Square: Maximal Period Sequences from Orthogonal Cellular Automata

- Mathematics, Computer Science2021 Ninth International Symposium on Computing and Networking (CANDAR)
- 2021

An algorithm based on Lagrange's theorem is devised to efficiently enumerate all linear OCA pairs that induce Sylvester matrices of maximal order up to diameter $d=11$, and characterize an upper bound on the periods of the sequences in terms of the order of the subgroup generated by an invertible Sylveste matrix.

### Bent Functions from Cellular Automata

- Mathematics, Computer ScienceIACR Cryptol. ePrint Arch.
- 2020

This work presents a primary construction of bent functions based on cellular automata (CA) and proves that the functions generated by the construction belong to the partial spread class PS−, a particular case of the Desarguesian spread construction.

### Latin Hypercubes and Cellular Automata

- Computer Science, MathematicsAutomata
- 2020

It is proved that linear bipermutive CA (LBCA) yielding Latin hypercubes of dimension $k>2$ are defined by sequences of invertible Toeplitz matrices with partially overlapping coefficients, which can be described by a specific kind of regular de Bruijn graph induced by the support of the determinant function.

### Enumeration of Maximal Cycles Generated by Orthogonal Cellular Automata

- Computer Science, MathematicsNatural Computing
- 2022

This paper considers an alternative approach to generate pseudorandom sequences through orthogonal CA (OCA), which guarantees a better amount of diffusion.

### An Enumeration Algorithm for Binary Coprime Polynomials with Nonzero Constant Term

- Mathematics, Computer ScienceArXiv
- 2022

This work addresses the enumeration of coprime polynomial pairs over F 2 where both polynomials have a nonzero constant term, motivated by the construction of orthogonal Latin squares via cellular automata and devise a combinatorial algorithm to enumerate all suchCoprime pairs of a given degree.

### On the Linear Components Space of S-boxes Generated by Orthogonal Cellular Automata

- Computer Science, MathematicsACRI
- 2022

An exhaustive search of all nonlinear OCA pairs of diameter d = 4 and d = 5, which generate S-boxes of size 6 × 6 and 8 × 8 turns out to be linear, and thus they are not useful for the design of confusion layers in block ciphers.

### Exploring Semi-bent Boolean Functions Arising from Cellular Automata

- Computer Science, MathematicsACRI
- 2020

A combinatorial algorithm is devised to enumerate all quadratic Boolean functions through a construction based on cellular automata, and it is remarked that the semi-bent functions generated through the authors' construction by the remaining rules have a constant number of linear structures.

### Heuristic search of (semi-)bent functions based on cellular automata

- Computer Science, MathematicsNatural Computing
- 2022

This work continues the investigation of a secondary construction based on cellular automata (CA), focusing on the classes of bent and semi-bent functions, and proves that the construction preserves the algebraic degree of the local rule.

### Building Correlation Immune Functions from Sets of Mutually Orthogonal Cellular Automata

- Computer Science, MathematicsArXiv
- 2022

This paper shows that the orthogonal array associated to a family of MOCA can be expanded to a binary OA of strength at least 2 and observes that their correlation immunity order is actually greater, always at least 3.

### Bent functions in the partial spread class generated by linear recurring sequences

- Mathematics, Computer ScienceDesigns, Codes and Cryptography
- 2022

Many of these functions defined by polynomials of degree d=2 are not EA-equivalent to any Maiorana–McFarland or Desarguesian partial spread function.

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