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B-Q-Fcts G-R Construction D-B-Fcts Example Quaternary Cryptographic Bent Functions and their Binary projection JADDA Zoubida 1 QARBOUA Soukayna 2 PARRAUD Patrice 1 1 Ecoles Militaires de Saint-Cyr Cotquidan, FRANCE CREC, UR - MACCLIA

  1. B-Q-Fcts G-R Construction D-B-Fcts Example Quaternary Cryptographic Bent Functions and their Binary projection JADDA Zoubida 1 QARBOUA Soukayna 2 PARRAUD Patrice 1 1 Ecoles Militaires de Saint-Cyr Coëtquidan, FRANCE CREC, UR - MACCLIA zoubida.jadda@st-cyr.terre-net.defense.gouv.fr patrice.parraud@st-cyr.terre-net.defense.gouv.fr 2 LMIA, Faculty of Sciences, University of Mohamed V Agdal, Rabat, Morocco soukayna.qarboua@gmail.com J OURNÉES C2, O CTOBER 2012 / D INARD J ADDA & P ARRAUD & Q ARBOUA 2012

  2. B-Q-Fcts G-R Construction D-B-Fcts Example The aim of this work m-variables quaternary functions ( F , m ) n − variables boolean functions ( f , n ) F : GR ( 4 , m ) → Z 4 Z 4 = { 0 , 1 , 2 , 3 } f : F n ( F 2 = { 0 , 1 } ) 2 → F 2 The quaternary approach ( CONSTRUCTION ): G ALOIS rings R = GR ( 4 , m ) . C RYPTOGRAPHIC properties . Characterization of a family of quaternary C RYPTOGRAPHIC functions. From Z 4 to F 2 (A PPLICATION ): The binary projection. Drived boolean cryptographic functions. J ADDA & P ARRAUD & Q ARBOUA 2012

  3. B-Q-Fcts G-R Construction D-B-Fcts Example The aim of this work m-variables quaternary functions ( F , m ) n − variables boolean functions ( f , n ) F : GR ( 4 , m ) → Z 4 Z 4 = { 0 , 1 , 2 , 3 } f : F n ( F 2 = { 0 , 1 } ) 2 → F 2 The quaternary approach ( CONSTRUCTION ): G ALOIS rings R = GR ( 4 , m ) . C RYPTOGRAPHIC properties . Characterization of a family of quaternary C RYPTOGRAPHIC functions. From Z 4 to F 2 (A PPLICATION ): The binary projection. Drived boolean cryptographic functions. J ADDA & P ARRAUD & Q ARBOUA 2012

  4. B-Q-Fcts G-R Construction D-B-Fcts Example The aim of this work m-variables quaternary functions ( F , m ) n − variables boolean functions ( f , n ) F : GR ( 4 , m ) → Z 4 Z 4 = { 0 , 1 , 2 , 3 } f : F n ( F 2 = { 0 , 1 } ) 2 → F 2 The quaternary approach ( CONSTRUCTION ): G ALOIS rings R = GR ( 4 , m ) . C RYPTOGRAPHIC properties . Characterization of a family of quaternary C RYPTOGRAPHIC functions. From Z 4 to F 2 (A PPLICATION ): CONTEXT : The binary projection. Drived boolean cryptographic functions. J ADDA & P ARRAUD & Q ARBOUA 2012

  5. B-Q-Fcts G-R Construction D-B-Fcts Example The aim of this work m-variables quaternary functions ( F , m ) n − variables boolean functions ( f , n ) F : GR ( 4 , m ) → Z 4 Z 4 = { 0 , 1 , 2 , 3 } f : F n ( F 2 = { 0 , 1 } ) 2 → F 2 The quaternary approach ( CONSTRUCTION ): G ALOIS rings R = GR ( 4 , m ) . C RYPTOGRAPHIC properties . Characterization of a family of quaternary C RYPTOGRAPHIC functions. From Z 4 to F 2 (A PPLICATION ): CONTEXT : The binary projection. Drived boolean cryptographic functions. J ADDA & P ARRAUD & Q ARBOUA 2012

  6. B-Q-Fcts G-R Construction D-B-Fcts Example Outline Boolean and Quaternary functions 1 Galois Rings 2 The construction 3 Derived boolean functions 4 Complete example of construction 5 J ADDA & P ARRAUD & Q ARBOUA 2012

  7. B-Q-Fcts G-R Construction D-B-Fcts Example Outline Boolean and Quaternary functions 1 Galois Rings 2 The construction 3 Derived boolean functions 4 Complete example of construction 5 J ADDA & P ARRAUD & Q ARBOUA 2012

  8. B-Q-Fcts G-R Construction D-B-Fcts Example Outline Boolean and Quaternary functions 1 Galois Rings 2 The construction 3 Derived boolean functions 4 Complete example of construction 5 J ADDA & P ARRAUD & Q ARBOUA 2012

  9. B-Q-Fcts G-R Construction D-B-Fcts Example Outline Boolean and Quaternary functions 1 Galois Rings 2 The construction 3 Derived boolean functions 4 Complete example of construction 5 J ADDA & P ARRAUD & Q ARBOUA 2012

  10. B-Q-Fcts G-R Construction D-B-Fcts Example Outline Boolean and Quaternary functions 1 Galois Rings 2 The construction 3 Derived boolean functions 4 Complete example of construction 5 J ADDA & P ARRAUD & Q ARBOUA 2012

  11. B-Q-Fcts G-R Construction D-B-Fcts Example outline Boolean and Quaternary functions 1 Galois Rings 2 The construction 3 Derived boolean functions 4 Complete example of construction 5 J ADDA & P ARRAUD & Q ARBOUA 2012

  12. B-Q-Fcts G-R Construction D-B-Fcts Example Boolean functions Let f : F n 2 �− → F 2 Truth table : [ f ( 0 , · · · , 0 ) , · · · , f ( 1 , · · · , 1 )] 2 n Support : supp ( f ) = { u ∈ F n 2 | f ( u ) � = 0 } Weight : w H ( f ) = | supp ( f ) | H AMMING metric : d H ( f , g ) = w H ( f ⊕ g ) ( ⊕ = + mod ( 2 )) J ADDA & P ARRAUD & Q ARBOUA 2012

  13. B-Q-Fcts G-R Construction D-B-Fcts Example Boolean functions Let f : F n 2 �− → F 2 Truth table : [ f ( 0 , · · · , 0 ) , · · · , f ( 1 , · · · , 1 )] 2 n Support : supp ( f ) = { u ∈ F n 2 | f ( u ) � = 0 } Weight : w H ( f ) = | supp ( f ) | H AMMING metric : d H ( f , g ) = w H ( f ⊕ g ) ( ⊕ = + mod ( 2 )) C RYPTOGRAPHIC PROPERTIES . Walsh transform : W f ( u ) = � ( − 1 ) u . v ( − 1 ) f ( v ) , u ∈ F n 2 v ∈ F n 2 w H ( f ) = 2 n − 1 Balancedness : W f ( 0 ) = 0 ⇐ ⇒ nl 2 ( f ) = 2 n − 1 − 1 Nonlinearity : nl 2 ( f ) = min 2 max g affine d H ( f , g ) ⇐ ⇒ | W f ( u ) | u ∈ F n 2 J ADDA & P ARRAUD & Q ARBOUA 2012

  14. B-Q-Fcts G-R Construction D-B-Fcts Example Boolean functions Let f : F n 2 �− → F 2 Truth table : [ f ( 0 , · · · , 0 ) , · · · , f ( 1 , · · · , 1 )] 2 n Support : supp ( f ) = { u ∈ F n 2 | f ( u ) � = 0 } Weight : w H ( f ) = | supp ( f ) | H AMMING metric : d H ( f , g ) = w H ( f ⊕ g ) ( ⊕ = + mod ( 2 )) C RYPTOGRAPHIC PROPERTIES . Walsh transform : W f ( u ) = � ( − 1 ) u . v ( − 1 ) f ( v ) , u ∈ F n 2 v ∈ F n 2 w H ( f ) = 2 n − 1 Balancedness : W f ( 0 ) = 0 ⇐ ⇒ nl 2 ( f ) = 2 n − 1 − 1 Nonlinearity : nl 2 ( f ) = min 2 max g affine d H ( f , g ) ⇐ ⇒ | W f ( u ) | u ∈ F n 2 n 2 . f is bent ⇐ ⇒ ∀ u ∈ F n | W f ( u ) | = 2 2 , n Maximal nonlinearity ( 2 n − 1 − 2 2 − 1 ) for n even but (not balanced) . J ADDA & P ARRAUD & Q ARBOUA 2012

  15. B-Q-Fcts G-R Construction D-B-Fcts Example Quaternary functions Let F , G ∈ F ( Z m 4 , Z 4 ) : Z m 4 �− → Z 4 The group of 4 th root of unity in C The ring of integers modulus 4 group i 2 = − 1 Z 4 = Z / 4 Z = { 0 , 1 , 2 , 3 } U 4 = {± 1 , ± i } ∼ J ADDA & P ARRAUD & Q ARBOUA 2012

  16. B-Q-Fcts G-R Construction D-B-Fcts Example Quaternary functions Let F , G ∈ F ( Z m 4 , Z 4 ) : Z m 4 �− → Z 4 The group of 4 th root of unity in C The ring of integers modulus 4 group i 2 = − 1 Z 4 = Z / 4 Z = { 0 , 1 , 2 , 3 } U 4 = {± 1 , ± i } ∼ Truth table : [ F ( 0 , · · · , 0 ) , · · · , F ( 1 , · · · , 1 )] 4 m Relative support : supp j ( F ) = { u ∈ Z m 4 | F ( u ) = j } 0 ≤ j ≤ 3 Relative cardinal : η j ( F ) = | supp j ( F ) | L EE METRIC 0 1 3 z ∈ Z 4 2 0 1 1 w L ( z ) 2 L EE Weight : w L ( F ) = η 1 ( F ) + 2 η 2 ( F ) + η 3 ( F ) L EE Distance : d L ( F , G ) = w L ( F − G ) [” − ” mod ( 4 )] J ADDA & P ARRAUD & Q ARBOUA 2012

  17. B-Q-Fcts G-R Construction D-B-Fcts Example Quaternary functions Let F , G ∈ F ( Z m 4 , Z 4 ) : Z m 4 �− → Z 4 The group of 4 th root of unity in C The ring of integers modulus 4 group i 2 = − 1 Z 4 = Z / 4 Z = { 0 , 1 , 2 , 3 } U 4 = {± 1 , ± i } ∼ Truth table : [ F ( 0 , · · · , 0 ) , · · · , F ( 1 , · · · , 1 )] 4 m Relative support : supp j ( F ) = { u ∈ Z m 4 | F ( u ) = j } 0 ≤ j ≤ 3 Relative cardinal : η j ( F ) = | supp j ( F ) | L EE METRIC 0 1 3 z ∈ Z 4 2 0 1 1 w L ( z ) 2 L EE Weight : w L ( F ) = η 1 ( F ) + 2 η 2 ( F ) + η 3 ( F ) L EE Distance : d L ( F , G ) = w L ( F − G ) [” − ” mod ( 4 )] W ALSH TRANSFORM . i u . v ( − 1 ) F ( v ) , u ∈ Z n W 2 � i u . v i F ( v ) , u ∈ Z n � W F ( u ) = F ( u ) = 4 4 v ∈ Z n v ∈ Z m 4 4 J ADDA & P ARRAUD & Q ARBOUA 2012

  18. B-Q-Fcts G-R Construction D-B-Fcts Example C RYPTOGRAPHIC PROPERTIES . Let F ∈ F ( Z m 4 , Z 4 ) η j ( F ) = 4 m − 1 The function F is balanced ∀ j ∈ { 0 , 1 , 2 , 3 } ⇐ ⇒ W F ( 0 ) = W 2 F ( 0 ) = 0 ⇐ ⇒ The nonlinearity of F : nl L = min 4 ( F ) G affine d L ( F , G ) 4 m − � � = max Re ( i b W F ( a )) a ∈ Z m 4 , b ∈ Z 4 J ADDA & P ARRAUD & Q ARBOUA 2012

  19. B-Q-Fcts G-R Construction D-B-Fcts Example C RYPTOGRAPHIC PROPERTIES . Let F ∈ F ( Z m 4 , Z 4 ) η j ( F ) = 4 m − 1 The function F is balanced ∀ j ∈ { 0 , 1 , 2 , 3 } ⇐ ⇒ W F ( 0 ) = W 2 F ( 0 ) = 0 ⇐ ⇒ The nonlinearity of F : nl L = min 4 ( F ) G affine d L ( F , G ) 4 m − � � = max Re ( i b W F ( a )) a ∈ Z m 4 , b ∈ Z 4 F is bent ⇐ ⇒ ∀ x ∈ Z m 4 , | W F ( x ) | = 2 m 4 ( F ) = 4 m − 2 m . The nonlinearity of a bent function F is nl L J ADDA & P ARRAUD & Q ARBOUA 2012

  20. B-Q-Fcts G-R Construction D-B-Fcts Example outline Boolean and Quaternary functions 1 Galois Rings 2 The construction 3 Derived boolean functions 4 Complete example of construction 5 J ADDA & P ARRAUD & Q ARBOUA 2012

  21. B-Q-Fcts G-R Construction D-B-Fcts Example G ALOIS Rings T HE M ULTIPLICATIVE REPRESENTATION AND CYCLOTOMIC CLASSES R ≃ Z 4 [ x ] / ( g ( x )) ( b _ poly of deg m ) ≃ Z 4 [ β ] ( 2 m − 1 th root of unity ) ≃ Z m 4 J ADDA & P ARRAUD & Q ARBOUA 2012

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