Outline Part 1 (uniqueness) special case general case Part 2 - PowerPoint PPT Presentation

Outline Preliminaries Part 1 Part 2 Q Non-extendable F q -quadratic Perfect Nonlinear Maps Ferruh ¨ Ozbudak (joint work with Alexander Pott) Department of Mathematics and Institute of Applied Mathematics, METU, Ankara, Turkey. (joint work with Alexander Pott.) Special Semester on Applications of Algebra and Number Theory, Emerging Applications of Finite Fields, RICAM, Linz, AUSTRIA. December 09-13, 2013 Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable F q -quadratic Perfect Nonlinear Maps 1/57

Outline Preliminaries Part 1 Part 2 Q Outline Part 1 (uniqueness) special case general case Part 2 (existence) notations and some results examples Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable F q -quadratic Perfect Nonlinear Maps 2/57

Outline Preliminaries Part 1 Part 2 Q Notations F q : Finite Field, where q is a power of an odd prime. n , m : positive integers with m | n . Tr F qn / F qm ( x ) = x + x q m + · · · + x q ( n m − 1 ) m Norm F qn / F qm ( x ) = x · x q m · · · · · x q ( n m − 1 ) m Tr = Tr F q 3 / F q and Norm = Norm F q 3 / F q K ∗ = K \ { 0 } Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable F q -quadratic Perfect Nonlinear Maps 3/57

Outline Preliminaries Part 1 Part 2 Q Preliminaries arbitrary F q -quadratic form f on F q 3 f : F q 3 → F q x �−→ Tr ( ax 2 + bx q + 1 ) a , b ∈ F q 3 . arbitrary F q -quadratic map from F q 3 to F 3 q . F : F q 3 → F 3 q  f 1 ( x )        x �−→ f 2 ( x )  ,           f 3 ( x )  where f 1 , f 2 , f 3 are F q -quadratic forms on F 3 q . Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable F q -quadratic Perfect Nonlinear Maps 4/57

Outline Preliminaries Part 1 Part 2 Q Preliminaries arbitrary F q -quadratic map F from F q 3 to F 2 q F : F q 3 → F 2 q � � f 1 ( x ) x �−→ , f 2 ( x ) where f 1 , f 2 are F q -quadratic forms on F q 3 . Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable F q -quadratic Perfect Nonlinear Maps 5/57

Outline Preliminaries Part 1 Part 2 Q Equivalence Definition 1 � � � � f 1 g 1 Let , be arbitrary F q -quadratic maps from F q 3 to F q 2 . f 2 g 2 We call that they are equivalent if there exists an F q -lineralized permutation polynomial L ( x ) ∈ F q 3 [ x ] and an invertible 2 × 2 matrix [ a i , j ] with entries from F q such that � � � � f 1 ( x ) g 1 ( L ( x )) [ a i , j ] · = f 2 ( x ) g 2 ( L ( x )) for all x ∈ F q 3 . Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable F q -quadratic Perfect Nonlinear Maps 6/57

Outline Preliminaries Part 1 Part 2 Q Remark Remark 1 Here we give notations and definitions for F q -quadratic maps from F q 3 to F q 2 . It is simple to generalize these to F q -quadratic maps from F q n to F q r with 1 ≤ r ≤ n . Remark 2 The equivalence above would seem rather restricted at first. However we show that our results also hold if we change the equivalence above with Extended Affine (EA) Equivalence or with Carlet-Charpin-Zinoviev (CCZ) Equivalence as well. Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable F q -quadratic Perfect Nonlinear Maps 7/57

Outline Preliminaries Part 1 Part 2 Q Definition Definition 2 Let 1 ≤ r ≤ 3, a ∈ F ∗ q 3 be. F : F q -quadratic maps from F q 3 to F q r D F , a : the difference map from F 3 q to F r q . D F , a : F q 3 → F q r x �−→ F ( x + a ) − F ( x ) − F ( a ) We call F perfect nonlinear or ( q 3 , q r ) -bent if the cardinality of the is the same and is equal to q 3 − r for all � � x ∈ F 3 set q : D F , a ( x ) = b a ∈ F ∗ q 3 and b ∈ F q r . If r = 3, F : Planar, If r = 1, F : Bent. Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable F q -quadratic Perfect Nonlinear Maps 8/57

Outline Preliminaries Part 1 Part 2 Q Problem Problem 1 Classification of F q -quadratic perfect nonlinear maps from F q n to F q r , where 1 ≤ r ≤ n. Classified completely if n ≥ 1 , r = 1 (non-degenerate F q -quadratic form) n = 2 , r = 2 (Dickson, finite field) n = 3 , r = 3 (Menichetti, finite field or twisted finite field) Our results We give a complete classification of the case n = 3 , r = 2. Namely, we prove that all F q -quadratic perfect nonlinear maps from F q 3 to F q 2 are equivalent. Also we give a geometric method to find an equivalence between two given F q -quadratic ( q 3 , q 2 ) -bent maps explicitly. Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable F q -quadratic Perfect Nonlinear Maps 9/57

Outline Preliminaries Part 1 Part 2 Q Content Part1 First consider some special cases using elementary techniques. The general case is more difficult, we use some results from Algebraic Geometry (in particular, Bezout’s Theorem). These results do not generalize to other ( n , r ) with 1 ≤ r ≤ n in general. Part2: a corollary: no non-extendable F q -quadratic ( 3 , 2 ) perfect nonlinear (PN) map. a proposition: no non-extendable F q -quadratic ( n , 1 ) PN map. existence of non-extendable F q -quadratic ( 4 , 3 ) PN map (in a sense an “atomic” structure). Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable F q -quadratic Perfect Nonlinear Maps 10/57

Outline Preliminaries Part 1 Part 2 Q Part 1: Some Special and Elementary Cases Lemma 1 Let { w 1 , w 2 } ⊆ F q 3 be a subset. � Tr ( w 1 x 2 ) � is ( q 3 , q 2 ) -bent ⇐⇒ { w 1 , w 2 } is linear Tr ( w 2 x 2 ) independent over F q . � Tr ( w 1 x q + 1 ) � is ( q 3 , q 2 ) -bent ⇐⇒ { w 1 , w 2 } is linear Tr ( w 2 x q + 1 ) independent over F q Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable F q -quadratic Perfect Nonlinear Maps 11/57

Outline Preliminaries Part 1 Part 2 Q Proposition 1 Let { w 1 , w 2 } ⊆ F q 3 be an F q -linearly independent subset and put w = w 2 w 1 . Then � Tr ( w 1 x 2 ) � Tr ( x 2 ) � � ∼ Tr ( w 2 x 2 ) Tr ( wx 2 ) � Tr ( w 1 x q + 1 ) � Tr ( x q + 1 ) � � ∼ Tr ( w 2 x q + 1 ) Tr ( wx q + 1 ) Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable F q -quadratic Perfect Nonlinear Maps 12/57

Outline Preliminaries Part 1 Part 2 Q Proposition 1 Let { w 1 , w 2 } ⊆ F q 3 be an F q -linearly independent subset and put w = w 2 w 1 . Then � Tr ( w 1 x 2 ) � Tr ( x 2 ) � � ∼ Tr ( w 2 x 2 ) Tr ( wx 2 ) � Tr ( w 1 x q + 1 ) � Tr ( x q + 1 ) � � ∼ Tr ( w 2 x q + 1 ) Tr ( wx q + 1 ) Proposition 2 Let w 1 , w 2 ∈ F q 3 \ F q . Then � Tr ( x 2 ) � � Tr ( x 2 ) � and both are ( q 3 , q 2 ) bent. ∼ Tr ( w 1 x 2 ) Tr ( w 2 x 2 ) Tr ( x q + 1 ) Tr ( x q + 1 ) � � � � and both are ( q 3 , q 2 ) -bent. ∼ Tr ( w 1 x q + 1 ) Tr ( w 2 x q + 1 ) Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable F q -quadratic Perfect Nonlinear Maps 12/57

Outline Preliminaries Part 1 Part 2 Q Sketch of the proof of Prop. 2: w 2 = a 0 + a 1 w 1 + a 2 w 2 1 Determine a 0 , a 1 , a 2 ∈ F q such that 1 . Here ( a 1 , a 2 ) � ( 0 , 0 ) as w 2 � F q . There exist a , b , c , d ∈ F q with ( c , d ) � ( 0 , 0 ) such that a + bw 1 = 1 . c + dw 1 w 2 Consider Tr (( a + bw 1 ) x 2 ) � � H ( x ) = . Tr (( c + dw 1 ) x 2 ) H ( x ) is ( q 3 , q 2 ) -bent. Using Proposition 1, Tr ( x 2 ) � � � Tr ( x 2 ) � H ( x ) ∼ = . Tr ( c + dw 1 a + bw 1 x 2 ) Tr ( w 2 x 2 ) Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable F q -quadratic Perfect Nonlinear Maps 13/57

Outline Preliminaries Part 1 Part 2 Q Theorem 1 Let { w 1 , w 2 } , { w 3 , w 4 } ⊆ F q 3 be linearly independent over F q . Then � Tr ( w 1 x 2 ) � Tr ( w 3 x q + 1 ) � � and they are ( q 3 , q 2 ) -bent. ∼ Tr ( w 4 x q + 1 ) Tr ( w 2 x 2 ) Sketch of the Proof: Note that x + x q − x q 2 is a linearized permutation polynomial over F q 3 as gcd ( 1 + t − t 2 , t 3 − 1 ) = 1. Hence L ( x ) = b ( x + x q − x q 2 ) ∈ F q 3 [ x ] is a linearized permutation polynomial for all b ∈ F ∗ q 3 . Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable F q -quadratic Perfect Nonlinear Maps 14/57

Outline Preliminaries Part 1 Part 2 Q Sketch of the Proof (Cont.) Claim: There exists b ∈ F ∗ q 3 and α 1 , α 2 ∈ F q 3 such that Tr ( L ( x ) 2 ) = Tr ( α 1 x q + 1 ) and Tr ( wL ( x ) 2 ) = Tr ( α 2 x q + 1 ) for all x ∈ F q 3 . In the proof of the claim we use existence of a basis � 1 , w , w 2 � which is trace orthogonal to the basis . Then � Tr ( α 1 x q + 1 ) � Tr ( w 1 x 2 ) � � ∼ . Tr ( α 2 x q + 1 ) Tr ( w 2 x 2 ) The equivalence of � Tr ( α 1 x q + 1 ) � Tr ( w 3 x q + 1 ) � � ∼ Tr ( α 2 x q + 1 ) Tr ( w 4 x q + 1 ) follows from Proposition 2. Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable F q -quadratic Perfect Nonlinear Maps 15/57

Outline Preliminaries Part 1 Part 2 Q The General Case Let h ( x ) be non-degenerate F q -quadratic form on F q 3 and η ∈ F ∗ q be non-square element in F q . Then it is well-known there exists a linearized permutation polynomial L ( x ) ∈ F q 3 [ x ] such that either h ( x ) = Tr ( x 2 ) for all x ∈ F q 3 or h ( x ) = η Tr ( x 2 ) for all x ∈ F q 3 . Hence, any F q -quadratic ( q 3 , q 2 ) -bent is of the form � Tr ( x 2 ) � F ( x ) = Tr ( θ x 2 + wx q + 1 ) without lost of generality. We fix such F ( x ) . Ferruh ¨ Ozbudak (joint work with Alexander Pott) Non-extendable F q -quadratic Perfect Nonlinear Maps 16/57