Changing Points in APN Functions Nikolay S. Kaleyski (joint work - PowerPoint PPT Presentation

Changing Points in APN Functions Nikolay S. Kaleyski (joint work with Lilya Budaghyan, Claude Carlet and Tor Helleseth) University of Bergen June 18, 2018 Nikolay S. Kaleyski (University of Bergen) Changing Points in APN Functions June 18, 2018 1 / 17

Introduction Preliminaries Preliminaries consider F : F 2 n → F 2 n ; Nikolay S. Kaleyski (University of Bergen) Changing Points in APN Functions June 18, 2018 2 / 17

Introduction Preliminaries Preliminaries consider F : F 2 n → F 2 n ; derivative D a F ( x ) = F ( x ) + F ( a + x ); Nikolay S. Kaleyski (University of Bergen) Changing Points in APN Functions June 18, 2018 2 / 17

Introduction Preliminaries Preliminaries consider F : F 2 n → F 2 n ; derivative D a F ( x ) = F ( x ) + F ( a + x ); ∆ F ( a , b ) = # { x ∈ F 2 n : D a F ( x ) = b } ; Nikolay S. Kaleyski (University of Bergen) Changing Points in APN Functions June 18, 2018 2 / 17

Introduction Preliminaries Preliminaries consider F : F 2 n → F 2 n ; derivative D a F ( x ) = F ( x ) + F ( a + x ); ∆ F ( a , b ) = # { x ∈ F 2 n : D a F ( x ) = b } ; differential uniformity ∆ F = max a ∈ F ∗ 2 n , b ∈ F 2 n ∆ F ( a , b ); Nikolay S. Kaleyski (University of Bergen) Changing Points in APN Functions June 18, 2018 2 / 17

Introduction Preliminaries Preliminaries consider F : F 2 n → F 2 n ; derivative D a F ( x ) = F ( x ) + F ( a + x ); ∆ F ( a , b ) = # { x ∈ F 2 n : D a F ( x ) = b } ; differential uniformity ∆ F = max a ∈ F ∗ 2 n , b ∈ F 2 n ∆ F ( a , b ); measures resistance to differential cryptanalysis; Nikolay S. Kaleyski (University of Bergen) Changing Points in APN Functions June 18, 2018 2 / 17

Introduction Preliminaries Preliminaries consider F : F 2 n → F 2 n ; derivative D a F ( x ) = F ( x ) + F ( a + x ); ∆ F ( a , b ) = # { x ∈ F 2 n : D a F ( x ) = b } ; differential uniformity ∆ F = max a ∈ F ∗ 2 n , b ∈ F 2 n ∆ F ( a , b ); measures resistance to differential cryptanalysis; always even; Nikolay S. Kaleyski (University of Bergen) Changing Points in APN Functions June 18, 2018 2 / 17

Introduction Preliminaries Preliminaries consider F : F 2 n → F 2 n ; derivative D a F ( x ) = F ( x ) + F ( a + x ); ∆ F ( a , b ) = # { x ∈ F 2 n : D a F ( x ) = b } ; differential uniformity ∆ F = max a ∈ F ∗ 2 n , b ∈ F 2 n ∆ F ( a , b ); measures resistance to differential cryptanalysis; always even; F is Almost Perfect Nonlinear (APN) if ∆ F = 2. Nikolay S. Kaleyski (University of Bergen) Changing Points in APN Functions June 18, 2018 2 / 17

Introduction Preliminaries Preliminaries (2) unique univariate representation of any ( n , n )-function as 2 n − 1 � c i x i , c i ∈ F 2 n . F ( x ) = i =0 Nikolay S. Kaleyski (University of Bergen) Changing Points in APN Functions June 18, 2018 3 / 17

Introduction Preliminaries Preliminaries (2) unique univariate representation of any ( n , n )-function as 2 n − 1 � c i x i , c i ∈ F 2 n . F ( x ) = i =0 algebraic degree of F is deg( F ) = max i : c i � =0 w 2 ( i ) where w 2 ( i ) is the two-weight of i . Nikolay S. Kaleyski (University of Bergen) Changing Points in APN Functions June 18, 2018 3 / 17

Introduction Preliminaries Preliminaries (2) unique univariate representation of any ( n , n )-function as 2 n − 1 � c i x i , c i ∈ F 2 n . F ( x ) = i =0 algebraic degree of F is deg( F ) = max i : c i � =0 w 2 ( i ) where w 2 ( i ) is the two-weight of i . Walsh Transform of F is the function W : F 2 2 n → Z defined as � ( − 1) Tr ( bF ( x )+ ax ) W ( a , b ) = x ∈ F 2 n where Tr : F 2 n → F 2 is the absolute trace function. Nikolay S. Kaleyski (University of Bergen) Changing Points in APN Functions June 18, 2018 3 / 17

Introduction Preliminaries Preliminaries (3) F b : F 2 n → F 2 defined as F b ( x ) = Tr ( bF ( x )) for b ∈ F 2 n are the component functions of F ; Nikolay S. Kaleyski (University of Bergen) Changing Points in APN Functions June 18, 2018 4 / 17

Introduction Preliminaries Preliminaries (3) F b : F 2 n → F 2 defined as F b ( x ) = Tr ( bF ( x )) for b ∈ F 2 n are the component functions of F ; the Hamming distance between two functions F and G is d ( F , G ) = # { x : F ( x ) � = G ( x ) } ; Nikolay S. Kaleyski (University of Bergen) Changing Points in APN Functions June 18, 2018 4 / 17

Introduction Preliminaries Preliminaries (3) F b : F 2 n → F 2 defined as F b ( x ) = Tr ( bF ( x )) for b ∈ F 2 n are the component functions of F ; the Hamming distance between two functions F and G is d ( F , G ) = # { x : F ( x ) � = G ( x ) } ; the nonlinearity of F is NL ( F ) = max 2 n min a ∈ F 2 n d ( F b , a ) b ∈ F ∗ with the last minimum over all affine a : F 2 n → F 2 ; Nikolay S. Kaleyski (University of Bergen) Changing Points in APN Functions June 18, 2018 4 / 17

Introduction Preliminaries Preliminaries (3) F b : F 2 n → F 2 defined as F b ( x ) = Tr ( bF ( x )) for b ∈ F 2 n are the component functions of F ; the Hamming distance between two functions F and G is d ( F , G ) = # { x : F ( x ) � = G ( x ) } ; the nonlinearity of F is NL ( F ) = max 2 n min a ∈ F 2 n d ( F b , a ) b ∈ F ∗ with the last minimum over all affine a : F 2 n → F 2 ; Useful formula: NL ( F ) = 2 n − 1 − 1 max 2 n , a ∈ F 2 n | W F ( a , b ) | . 2 b ∈ F ∗ Nikolay S. Kaleyski (University of Bergen) Changing Points in APN Functions June 18, 2018 4 / 17

Introduction Changing One Point Changing a Single Point in a Function research on constructions of i.a. APN functions is ongoing; Nikolay S. Kaleyski (University of Bergen) Changing Points in APN Functions June 18, 2018 5 / 17

Introduction Changing One Point Changing a Single Point in a Function research on constructions of i.a. APN functions is ongoing; maximum algebraic degree of APN function is an open problem: Nikolay S. Kaleyski (University of Bergen) Changing Points in APN Functions June 18, 2018 5 / 17

Introduction Changing One Point Changing a Single Point in a Function research on constructions of i.a. APN functions is ongoing; maximum algebraic degree of APN function is an open problem: is it possible to have deg( F ) = n for F over F 2 n APN? Nikolay S. Kaleyski (University of Bergen) Changing Points in APN Functions June 18, 2018 5 / 17

Introduction Changing One Point Changing a Single Point in a Function research on constructions of i.a. APN functions is ongoing; maximum algebraic degree of APN function is an open problem: is it possible to have deg( F ) = n for F over F 2 n APN? “On upper bounds for algebraic degrees of APN functions” (Budaghyan, Carlet, Helleseth, Li, Sun): changing one point in a given function F by � F ( x ) x � = u G ( x ) = F ( x ) + (1 + ( x + u ) 2 n − 1 ) v = F ( u ) + v x = u . Nikolay S. Kaleyski (University of Bergen) Changing Points in APN Functions June 18, 2018 5 / 17

Introduction Changing Multiple Points Changing Multiple Points in a Function Given natural K ≥ 1 and F : F 2 n → F 2 n , construct G by changing K points: Nikolay S. Kaleyski (University of Bergen) Changing Points in APN Functions June 18, 2018 6 / 17

Introduction Changing Multiple Points Changing Multiple Points in a Function Given natural K ≥ 1 and F : F 2 n → F 2 n , construct G by changing K points: select u 1 , u 2 , . . . , u k from F 2 n with # { u 1 , u 2 , . . . , u K } = K ; Nikolay S. Kaleyski (University of Bergen) Changing Points in APN Functions June 18, 2018 6 / 17

Introduction Changing Multiple Points Changing Multiple Points in a Function Given natural K ≥ 1 and F : F 2 n → F 2 n , construct G by changing K points: select u 1 , u 2 , . . . , u k from F 2 n with # { u 1 , u 2 , . . . , u K } = K ; select v 1 , v 2 , . . . , v k from F ∗ 2 n ; Nikolay S. Kaleyski (University of Bergen) Changing Points in APN Functions June 18, 2018 6 / 17

Introduction Changing Multiple Points Changing Multiple Points in a Function Given natural K ≥ 1 and F : F 2 n → F 2 n , construct G by changing K points: select u 1 , u 2 , . . . , u k from F 2 n with # { u 1 , u 2 , . . . , u K } = K ; select v 1 , v 2 , . . . , v k from F ∗ 2 n ; define G as K (1 + ( x + u 1 ) 2 n − 1 ) v i � G ( x ) = F ( x ) + i =1 � F ( x ) x / ∈ U = F ( u i ) + v i x = u i , i ∈ { 1 , 2 , . . . , K } . Nikolay S. Kaleyski (University of Bergen) Changing Points in APN Functions June 18, 2018 6 / 17

Introduction Changing Multiple Points Changing Multiple Points in a Function Given natural K ≥ 1 and F : F 2 n → F 2 n , construct G by changing K points: select u 1 , u 2 , . . . , u k from F 2 n with # { u 1 , u 2 , . . . , u K } = K ; select v 1 , v 2 , . . . , v k from F ∗ 2 n ; define G as K (1 + ( x + u 1 ) 2 n − 1 ) v i � G ( x ) = F ( x ) + i =1 � F ( x ) x / ∈ U = F ( u i ) + v i x = u i , i ∈ { 1 , 2 , . . . , K } . What can be said about the properties of F and G ? Nikolay S. Kaleyski (University of Bergen) Changing Points in APN Functions June 18, 2018 6 / 17

Introduction General Observations General Observations (Algebraic Degree) in front of x 2 n − k ; F ( x ) + G ( x ) has coefficient � K i =1 v i u k − 1 i Nikolay S. Kaleyski (University of Bergen) Changing Points in APN Functions June 18, 2018 7 / 17