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Context Main result Example Possible exension A Spectral Approach for Characterizing the Self-Synchronization of Stream Ciphers emy Parriaux 1 Philippe Guillot 2 erioux 1 J´ er´ Gilles Mill´ Nancy University, CNRS, Research Center for Automatic Control of Nancy (CRAN UMR 7039), France, jeremy.parriaux@esstin.uhp-nancy.fr, gilles.millerioux@esstin.uhp-nancy.fr , Paris 8 University Laboratoire Analyse, G´ eom´ etrie et Applications (LAGA UMR 7539), France philippe.guillot@univ-paris8.fr February 16, 2011 1 / 17 J´ er´ emy Parriaux, Philippe Guillot, Gilles Mill´ erioux A Spectral Approach for Characterizing the Self-Synchronization of Stream Ciphers

Context Main result Example Possible exension Context 1 Main result 2 Example 3 Possible exension 4 2 / 17 J´ er´ emy Parriaux, Philippe Guillot, Gilles Mill´ erioux A Spectral Approach for Characterizing the Self-Synchronization of Stream Ciphers

Context Main result Example Possible exension Outline Context 1 Main result 2 Example 3 Possible exension 4 3 / 17 J´ er´ emy Parriaux, Philippe Guillot, Gilles Mill´ erioux A Spectral Approach for Characterizing the Self-Synchronization of Stream Ciphers

Context Main result Example Possible exension Self-synchronizing Stream Ciphers Canonical form ⊕ ⊖ u k y k y k u k � z k � z k h θ h θ n n y k − n · · · y k − 1 y k − 1 · · · y k − n x k � x k θ key y k cipher-text m k plain-text m k recovered plain-text � x k state of the cipher x k state of the decipher � z k complex sequence � z k complex sequence f θ next-state function h θ output function 4 / 17 J´ er´ emy Parriaux, Philippe Guillot, Gilles Mill´ erioux A Spectral Approach for Characterizing the Self-Synchronization of Stream Ciphers

Context Main result Example Possible exension Self-synchronizing Stream Ciphers Canonical form ⊕ ⊖ u k y k y k u k � z k � z k h θ h θ n n y k − n · · · y k − 1 y k − 1 · · · y k − n x k � x k Advantages Synchronization of cipher and decipher is structural property Does not require any external synchronization protocol 4 / 17 J´ er´ emy Parriaux, Philippe Guillot, Gilles Mill´ erioux A Spectral Approach for Characterizing the Self-Synchronization of Stream Ciphers

Context Main result Example Possible exension Self-synchronizing Stream Ciphers Recursive form ⊕ ⊖ u k y k y k � u k z k � z k h θ h θ n n x k f θ f θ x k � n n n n 5 / 17 J´ er´ emy Parriaux, Philippe Guillot, Gilles Mill´ erioux A Spectral Approach for Characterizing the Self-Synchronization of Stream Ciphers

Context Main result Example Possible exension Self-synchronizing Stream Ciphers Recursive form ⊕ ⊖ u k y k y k � u k z k � z k h θ h θ n n x k f θ f θ � x k n n n n Question How to characterize the functions f θ so that ∀ k > k t the state � x k does not depend on the initial state � x 0 ? Is there any non strict T function f θ that can be used ? 5 / 17 J´ er´ emy Parriaux, Philippe Guillot, Gilles Mill´ erioux A Spectral Approach for Characterizing the Self-Synchronization of Stream Ciphers

Context Main result Example Possible exension strict T-function (parameter) y x 0 f 0 ( y ) f 0 f 1 ( y , x 0 ) . . . x 1 f 1 f n − 2 ( y , x 0 , . . . , x n − 4 , x n − 3 ) f n − 1 ( y , x 0 , . . . , . . . , x n − 3 , x n − 2 ) Non strict T-function f 0 ( y , x 0 , . . . , x n − 2 , x n − 1 ) fn − 2 f 1 ( y , x 0 , . . . , x n − 2 , x n − 1 ) xn − 2 . . . f n − 2 ( y , x 0 , . . . , x n − 2 , x n − 1 ) fn − 1 xn − 1 f n − 1 ( y , x 0 , . . . , x n − 2 , x n − 1 ) 6 / 17 J´ er´ emy Parriaux, Philippe Guillot, Gilles Mill´ erioux A Spectral Approach for Characterizing the Self-Synchronization of Stream Ciphers

Context Main result Example Possible exension strict T-function (parameter) y x 0 f 0 ( y ) f 0 f 1 ( y , x 0 ) . . . x 1 f 1 f n − 2 ( y , x 0 , . . . , x n − 4 , x n − 3 ) f n − 1 ( y , x 0 , . . . , . . . , x n − 3 , x n − 2 ) Non strict T-function f 0 ( y , x 0 , . . . , x n − 2 , x n − 1 ) fn − 2 f 1 ( y , x 0 , . . . , x n − 2 , x n − 1 ) xn − 2 . . . f n − 2 ( y , x 0 , . . . , x n − 2 , x n − 1 ) fn − 1 xn − 1 f n − 1 ( y , x 0 , . . . , x n − 2 , x n − 1 ) 6 / 17 J´ er´ emy Parriaux, Philippe Guillot, Gilles Mill´ erioux A Spectral Approach for Characterizing the Self-Synchronization of Stream Ciphers

Context Main result Example Possible exension Self-synchronization Definition (Self-Synchronizing sequence) A sequence ( y ) is self-synchronizing with respect to f if there exists an integer k y so that for all initial state x 0 and � x 0 ∀ k ≥ k y , x k = � x k Definition (Finite time self-synchronization) The function f is finite time self-synchronizing if the minimum value k y is upper bounded when ( y ) stands in the set of all input sequences. The upper bound is called the self-synchronization delay of f . 7 / 17 J´ er´ emy Parriaux, Philippe Guillot, Gilles Mill´ erioux A Spectral Approach for Characterizing the Self-Synchronization of Stream Ciphers

Context Main result Example Possible exension Self-Synchronizing Stream Ciphers Equations y k x k +1 f n x k n Decomposition of the next-state function f 0 , f 1 : F n → F n 2 − 2 � f 0 ( x k ) if y k = 0 f ( y k , x k ) = (1) f 1 ( x k ) if y k = 1 Iterated function φ i ( y , x 0 ) = f ( y i , f ( y i − 1 , f ( . . . , f ( y 0 , x 0 ) · · · ))) (2) = f y i ◦ f y i − 1 ◦ · · · ◦ f y 1 ◦ f y 0 ( x 0 ) 8 / 17 J´ er´ emy Parriaux, Philippe Guillot, Gilles Mill´ erioux A Spectral Approach for Characterizing the Self-Synchronization of Stream Ciphers

Context Main result Example Possible exension Spectral Analysis Walsh Transform (of a Boolean function f : F n 2 − → F 2 ) � 2 , � ∀ v ∈ F n ( − 1) f ( x )+ x · v f χ ( v ) = (3) x ∈ F n 2 Walsh Matrix (of a vectorial Boolean function f : F n → F m 2 − 2 ) � ∀ u ∈ F m 2 , v ∈ F n 2 , w f ( − 1) u · f ( x )+ v · x u , v = (4) x ∈ F n 2 Composition of vectorial Boolean functions W f ◦ g = 1 2 n W f × W g (5) 9 / 17 J´ er´ emy Parriaux, Philippe Guillot, Gilles Mill´ erioux A Spectral Approach for Characterizing the Self-Synchronization of Stream Ciphers

Context Main result Example Possible exension Outline Context 1 Main result 2 Example 3 Possible exension 4 10 / 17 J´ er´ emy Parriaux, Philippe Guillot, Gilles Mill´ erioux A Spectral Approach for Characterizing the Self-Synchronization of Stream Ciphers

Context Main result Example Possible exension The system is self-synchronizing with synchronization delay i + 1 ⇐ ⇒ The function φ i ( y , x 0 ) is constant with respect to x 0 (or the function φ y i ( x 0 ) is constant) Walsh matrix of φ i restricted to a sequence y ∈ F i +1 2 1 W φ y i = 2 n · i W f yi × · · · × W f y 0 (6) Walsh matrix of a constant function   2 n 0 0 · · ·   ± 2 n 0 0 · · ·     . . . . . .   . . . ± 2 n 0 · · · 0 11 / 17 J´ er´ emy Parriaux, Philippe Guillot, Gilles Mill´ erioux A Spectral Approach for Characterizing the Self-Synchronization of Stream Ciphers

Context Main result Example Possible exension Finite time self-synchronization     2 n 2 n 0 0 0 0 · · · · · ·     w 2 , 1 w 2 , 2 · · · w 2 , 2 n w 2 , 1 w 2 , 2 · · · w 2 , 2 n         W f 0 =   W f 1 =   . . . . . .     . . . . . . . . . . . .     w 2 n , 1 w 2 n , 2 · · · w 2 n , 2 n w 2 n , 1 w 2 n , 2 · · · w 2 n , 2 n W ∗ W ∗ f 0 f 1 Conditions on W f 0 and W f 1 Finite time self-synchronization ⇐ ⇒ W ∗ f 0 and W ∗ f 1 generate a nilpotent semigroup. 12 / 17 J´ er´ emy Parriaux, Philippe Guillot, Gilles Mill´ erioux A Spectral Approach for Characterizing the Self-Synchronization of Stream Ciphers

Context Main result Example Possible exension Nilpotent reduced Walsh matrix Nilpotent deduced Walsh matrix Triangular reduced Walsh matrix ⇔ strict T-function Levitzky: Any semigroup of nilpotent operators is triangularizable Three kinds of nilpotent Walsh matrices those which are already triangular f T 1 those that can be triangularized by a change of basis whose matrix is 2 a Walsh matrix ( b ◦ f T ◦ b − 1 ) those that cannot be triangularized with such a matrix 3 Remark If two reduced Walsh matrices W ∗ f 0 , W ∗ f 1 span a nilpotent semigroup of nilpotency class greater than n , it necessary corresponds to Case 3. 13 / 17 J´ er´ emy Parriaux, Philippe Guillot, Gilles Mill´ erioux A Spectral Approach for Characterizing the Self-Synchronization of Stream Ciphers

Context Main result Example Possible exension Outline Context 1 Main result 2 Example 3 Possible exension 4 14 / 17 J´ er´ emy Parriaux, Philippe Guillot, Gilles Mill´ erioux A Spectral Approach for Characterizing the Self-Synchronization of Stream Ciphers