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On the Periods of Spatially Periodic Preimages in Linear Bipermutive CA Automata 2015 - June 8-10 - Turku Luca Mariot, Alberto Leporati Dipartimento di Informatica, Sistemistica e Comunicazione Università degli Studi Milano - Bicocca l.mariot@campus.unimib.it, alberto.leporati@unimib.it June 10, 2015 Luca Mariot, Alberto Leporati On the Periods of Spatially Periodic Preimages in Linear Bipermutive CA

Problem Statement Preimages Periods in Generic BCA Linear BCA Preimages and Concatenated LRS Conclusions and Future Directions of Research Outline Problem Statement Preimages Periods in Generic BCA Linear BCA Preimages and Concatenated LRS Conclusions and Future Directions of Research Luca Mariot, Alberto Leporati On the Periods of Spatially Periodic Preimages in Linear Bipermutive CA

Problem Statement Preimages Periods in Generic BCA Linear BCA Preimages and Concatenated LRS Conclusions and Future Directions of Research Spatially Periodic Preimages in Surjective CAs ◮ Let F : A Z → A Z be a (CA) with | A | = q , and let y ∈ A Z be a spatially periodic configuration of period p ∈ N defined by a finite word u ∈ A p , i.e. y = ω u ω ◮ If F is surjective, it is known that each preimage x of y under F is spatially periodic as well [Hedlund73, Cattaneo00] x = ··· ··· z z z ↓ F y = ··· ··· u u u u ◮ What are the periods of preimages x ∈ F − 1 ( y ) ? Luca Mariot, Alberto Leporati On the Periods of Spatially Periodic Preimages in Linear Bipermutive CA

Problem Statement Preimages Periods in Generic BCA Linear BCA Preimages and Concatenated LRS Conclusions and Future Directions of Research Assumptions and Problem Statement ◮ We focus our attention on the class of bipermutive CA (BCA) ◮ A CA F : A Z → A Z induced by a local rule f : A 2 r + 1 → A is bipermutive if, by fixing the first (the last) 2 r coordinates of f , the resulting restriction f R , z : A → A ( f L , z : A → A ) is a permutation on A Problem PBCAP - Periods of BCA Preimages Let y ∈ A Z be a spatially periodic configuration of period p ∈ N . Given a BCA F : A Z → A Z , find the relation between p and the spatial periods of the preimages x ∈ F − 1 ( y ) . Luca Mariot, Alberto Leporati On the Periods of Spatially Periodic Preimages in Linear Bipermutive CA

Problem Statement Preimages Periods in Generic BCA Linear BCA Preimages and Concatenated LRS Conclusions and Future Directions of Research Motivation: BCA-based Secret Sharing Scheme ◮ Motivation for solving PBCAP: find the maximum number of players in a BCA-based Secret Sharing Scheme [Mariot14] P k + 1 P 1 P k ↑ ↑ ↑ ··· B 1 B k B k + 1 ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· F − 2 ↑ F − 1 ↑ S S Luca Mariot, Alberto Leporati On the Periods of Spatially Periodic Preimages in Linear Bipermutive CA

Problem Statement Preimages Periods in Generic BCA Linear BCA Preimages and Concatenated LRS Conclusions and Future Directions of Research Outline Problem Statement Preimages Periods in Generic BCA Linear BCA Preimages and Concatenated LRS Conclusions and Future Directions of Research Luca Mariot, Alberto Leporati On the Periods of Spatially Periodic Preimages in Linear Bipermutive CA

Problem Statement Preimages Periods in Generic BCA Linear BCA Preimages and Concatenated LRS Conclusions and Future Directions of Research Preimage Computation in BCA ◮ Let F : A Z → A Z be a BCA with local rule f : A 2 r + 1 → A , and let y ∈ A Z be a configuration ◮ Additionally, let x [ i , i + 2 r − 1 ] ∈ A 2 r be the 2 r -cell block placed at position i ∈ Z of a preimage x ∈ F − 1 ( y ) ◮ The remainder of x is determined by the following equation:  f − 1 R , z ( n ) ( y n − r ) , where z ( n ) = x [ n − 2 r , n − 1 ] , if n ≥ i + 2 r (a)       x n =     f − 1  L , z ( n ) ( y n + r ) , where z ( n ) = x [ n + 1 , n + 2 r ] , if n < i (b)    Luca Mariot, Alberto Leporati On the Periods of Spatially Periodic Preimages in Linear Bipermutive CA

Problem Statement Preimages Periods in Generic BCA Linear BCA Preimages and Concatenated LRS Conclusions and Future Directions of Research Preimages Periods in Generic BCA (1/2) Lemma Let F : A Z → A Z be a BCA with local rule f : A 2 r + 1 → A. Given a configuration y ∈ A Z and i , j ∈ Z , for all x ∈ F − 1 ( y ) there exists a permutation ϕ y between the blocks x [ i , i + 2 r − 1 ] and x [ j , j + 2 r − 1 ] . ϕ y is bijective x [ i , i + 2 r − 1 ] x [ j , j + 2 r − 1 ] ··· ··· ··· 2 r cells 2 r cells ··· y ··· Luca Mariot, Alberto Leporati On the Periods of Spatially Periodic Preimages in Linear Bipermutive CA

Problem Statement Preimages Periods in Generic BCA Linear BCA Preimages and Concatenated LRS Conclusions and Future Directions of Research Preimages Periods in Generic BCA (2/2) Proposition Let F : A Z → A Z be a BCA with local rule f : A 2 r + 1 → A and let y ∈ A Z be a spatially periodic configuration of period p ∈ N . Given a preimage x ∈ F − 1 ( y ) , the period of x is m = p · h, where h ∈ { 1 , ··· , q 2 r } . ϕ u ··· ϕ u ϕ u ··· w 1 v 1 w 2 ··· w h − 1 v h − 1 w 1 v 1 w 2 ··· ··· ··· ··· u u u h ≤ q 2 r copies of u Luca Mariot, Alberto Leporati On the Periods of Spatially Periodic Preimages in Linear Bipermutive CA

Problem Statement Preimages Periods in Generic BCA Linear BCA Preimages and Concatenated LRS Conclusions and Future Directions of Research Outline Problem Statement Preimages Periods in Generic BCA Linear BCA Preimages and Concatenated LRS Conclusions and Future Directions of Research Luca Mariot, Alberto Leporati On the Periods of Spatially Periodic Preimages in Linear Bipermutive CA

Problem Statement Preimages Periods in Generic BCA Linear BCA Preimages and Concatenated LRS Conclusions and Future Directions of Research Linear BCA ◮ We now assume that the alphabet is a finite field, that is, A = F q where q is a power of a prime q is linear if its local rule f : F 2 r + 1 ◮ A CA F : F Z q → F Z → F q is a q linear combination of the neighborhood x ∈ F 2 r + 1 : q f ( x 0 , ··· , x 2 r ) = c 0 · x 0 + ··· + c 2 r · x 2 r , for a certain vector c = ( c 0 , c 1 , ··· , c 2 r ) ∈ F 2 r + 1 q ◮ Remark: if c 0 , c 2 r � 0, then a linear CA is also bipermutive (LBCA) Luca Mariot, Alberto Leporati On the Periods of Spatially Periodic Preimages in Linear Bipermutive CA

Problem Statement Preimages Periods in Generic BCA Linear BCA Preimages and Concatenated LRS Conclusions and Future Directions of Research Linear Recurring Sequences ◮ Given a 0 , a 1 , ··· , a k − 1 ∈ F q , a linear recurring sequence (LRS) of order k is a sequence s = s 0 , s 1 , ··· of elements in F q satisfying s n + k = a 0 s n + a 1 s n + 1 + ··· + a k − 1 s n + k − 1 ∀ n ∈ N ◮ A LRS is generated by a Linear Feedback Shift Register (LFSR) ◮ The characteristic polynomial of s is defined as a ( X ) = X k − a k − 1 X k − 1 − a k − 2 X k − 2 −···− a 0 ◮ The period of s equals the order of the minimal polynomial m ( X ) , which depends on a ( X ) and the initial terms of s Luca Mariot, Alberto Leporati On the Periods of Spatially Periodic Preimages in Linear Bipermutive CA

Problem Statement Preimages Periods in Generic BCA Linear BCA Preimages and Concatenated LRS Conclusions and Future Directions of Research Characterising LBCA Preimages as Concatenated LRS (1/2) ◮ Given a LBCA F , a preimage x ∈ F − 1 ( y ) of y can be considered as a LRS of order k = 2 r “disturbed” by y ◮ Let c 0 , ··· , c 2 r be the coefficients of the local rule f , and set ◮ d = c − 1 2 r ◮ a i = − d · c i for i ∈ { 0 , ··· , 2 r − 1 } ◮ Moreover, define sequence v as the r-shift of y , that is, v n = y n + r for n ∈ N ◮ Case (a) of the preimage recurrence equation becomes x n + k = a 0 x n + a 1 x n + 1 + ··· + a k − 1 x n + k − 1 + dv n ∀ n ≥ 2 r Luca Mariot, Alberto Leporati On the Periods of Spatially Periodic Preimages in Linear Bipermutive CA

Problem Statement Preimages Periods in Generic BCA Linear BCA Preimages and Concatenated LRS Conclusions and Future Directions of Research Characterising LBCA Preimages as Concatenated LRS (2/2) ◮ Remark: If y is spatially periodic of period p , then sequence v = { v n } n ∈ N is a LRS of a certain order l ∈ N : v n + l = b 0 v n + b 1 v n + 1 + ··· + b l − 1 v n + l − 1 ∀ n ∈ N ◮ In the worst case, v will be generated by the “trivial” LRS of order l = p which cyclically shifts a word of length p ◮ We define x as the concatenation s � v of the LRS s induced by the local rule f and the LRS v which is the r -shift of y Luca Mariot, Alberto Leporati On the Periods of Spatially Periodic Preimages in Linear Bipermutive CA

Problem Statement Preimages Periods in Generic BCA Linear BCA Preimages and Concatenated LRS Conclusions and Future Directions of Research LBCA Preimage Generation By Concatenated LFSR + ··· + + Disturbing LFSR b 0 b 1 b l − 2 b l − 1 ··· y E 0 E 1 E l − 2 E l − 1 d ··· + + + a 0 a 1 a k − 2 a k − 1 Disturbed LFSR ··· + x D 0 D 1 D k − 2 D k − 1 Luca Mariot, Alberto Leporati On the Periods of Spatially Periodic Preimages in Linear Bipermutive CA