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… 

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References

SHOWING 1-10 OF 34 REFERENCES

Enumerating Orthogonal Latin Squares Generated by Bipermutive Cellular Automata

The general case of nonlinear rules in bipermutive cellular automata, which could be interesting for cryptographic applications such as the design of cheater-immune secret sharing schemes, is addressed.

Complexity of some arithmetic problems for binary polynomials

An exponential lower bound on the size of a decision tree for this function is obtained, and an asymptotic formula is derived, having a linear main term, for its average sensitivity is derived.

Computing the periods of preimages in surjective cellular automata

An equivalence is shown between preimages of LBCA and concatenated linear recurring sequences (LRS) that allows for a complete characterization of their periods and this paper considers the case of linear and bipermutive cellular automata defined over a finite field as state alphabet.

Cellular automata based S-boxes

A systematic investigation of the cryptographic properties of S-boxes defined by CA, proving some upper bounds on their nonlinearity and differential uniformity and proposing a “reverse engineering” method based on De Bruijn graphs to determine whether a specific S-box is expressible through a single CA rule.

A cryptographic and coding-theoretic perspective on the global rules of cellular automata

It is shown that linear CA are equivalent to linear cyclic codes, and a formula is derived from which the Walsh spectrum of CA induced by permutive local rules is derived, from which a formula for the nonlinearity of such CA is derived.

Rook domains, Latin squares, affine planes, and error-distributing codes

It is shown how various concepts in the theory of Latin squares are best expressed in the form of questions about the placing of rooks on k -dimensional hyperchessboards of side n, and that the optimal colorings in certain cases correspond to duals of desarguian projective planes.

Partially permutive cellular automata

A comprehensive scheme emerges that unifies the analysis of topological defects in cellular automata that can generate random walks as well as their degenerate forms.

Algebraic Properties of the Block Transformation on Cellular Automata

If the blocked rule satisfies an identity which holds for a broad class of algebras, then the underlying rule must have essentially the same structure, and must depend only on its leftmost and rightmost inputs; roughly speaking, that the block transformation cannot turn a nonlinear rule into a linear one.

Endomorphisms and automorphisms of the shift dynamical system

It is shown that closed subset of X(SQ which is invariant under a defines a subdynamical system, and that these mappings, composed with powers of the shift, constitute the entire class of continuous transformations which commute with the shift.