Constructing Generalized Bent Functions from Trace Forms over Galois - PowerPoint PPT Presentation

Background Bent functions and generalized Bent functions Galois rings Constructions of generalized Bent functions Constructing Generalized Bent Functions from Trace Forms over Galois Rings Xiaoming Zhang Key Laboratory of Mathematics Mechanization, CAS Oct. 26, 2012, Beijing Joint work with Zhuojun Liu, Baofeng Wu and Qingfang Jin Xiaoming Zhang KLMM, AMSS, CAS

Background Bent functions and generalized Bent functions Galois rings Constructions of generalized Bent functions Outline of this talk Background 1 Bent functions and generalized Bent functions 2 Galois rings 3 Constructions of generalized Bent functions 4 Xiaoming Zhang KLMM, AMSS, CAS

Background Bent functions and generalized Bent functions Galois rings Constructions of generalized Bent functions Outline of this talk Background 1 Bent functions and generalized Bent functions 2 Galois rings 3 Constructions of generalized Bent functions 4 Xiaoming Zhang KLMM, AMSS, CAS

Background Bent functions and generalized Bent functions Galois rings Constructions of generalized Bent functions A constant-amplitude code is a code that reduces the peak-to-average power ratio (PAPR) in multicode code-division multiple access (MC-CDMA) systems to the favorable value 1. Kai-Uwe Schmidt showed the conncetion between codes with PAPR equal to 1 and functions from the binary m -tuples to Z 4 having the bent property. Kai-Uwe Schmidt proposed a technique to consturct generalized bent functions using trace form over Galois rings. Xiaoming Zhang KLMM, AMSS, CAS

Background Bent functions and generalized Bent functions Galois rings Constructions of generalized Bent functions Outline of this talk Background 1 Bent functions and generalized Bent functions 2 Galois rings 3 Constructions of generalized Bent functions 4 Xiaoming Zhang KLMM, AMSS, CAS

Background Bent functions and generalized Bent functions Galois rings Constructions of generalized Bent functions Boolean function Let f : F m 2 − → F 2 , then f is called a Boolean function with m variables. f can be represented as a polynomial in ( x 2 1 + x 1 , x 2 2 + x 2 , · · · , x 2 � F 2 [ x 1 , x 2 , · · · , x m ] m + x m ) . Xiaoming Zhang KLMM, AMSS, CAS

Background Bent functions and generalized Bent functions Galois rings Constructions of generalized Bent functions Walsh Transform The Walsh transform of a Boolean function f at u is defined by � ( − 1 ) f ( x )+ x · u W f ( u ) = x ∈ F m 2 where x · u = � 1 ≤ i ≤ m x i u i for x = ( x 1 , x 2 , · · · , x m ) , u = ( u 1 , u 2 , · · · , u m ) ∈ F m 2 . Xiaoming Zhang KLMM, AMSS, CAS

Background Bent functions and generalized Bent functions Galois rings Constructions of generalized Bent functions Walsh Transform The Walsh transform of a Boolean function f at u is defined by � ( − 1 ) f ( x )+ x · u W f ( u ) = x ∈ F m 2 where x · u = � 1 ≤ i ≤ m x i u i for x = ( x 1 , x 2 , · · · , x m ) , u = ( u 1 , u 2 , · · · , u m ) ∈ F m 2 . Bent function → F 2 is called a Bent function if | W f ( u ) | = 2 m / 2 for all f : F m 2 − u = ( u 1 , u 2 , · · · , u m ) ∈ F m 2 . The number of variables m must be even. Xiaoming Zhang KLMM, AMSS, CAS

Background Bent functions and generalized Bent functions Galois rings Constructions of generalized Bent functions Generalized Boolean function A generalized Boolean function is defined as a map f : F m 2 − → Z 2 h , where h is a positive integer. Write k = ( k 1 , k 1 , ..., k m ) for k ∈ { 0 , 1 } m , every such function can be uniquely expressed in the polynomial form m k j � � f ( x ) = f ( x 1 , ..., x m ) = c k x j , c k ∈ Z 2 h k ∈{ 0 , 1 } m j = 1 Xiaoming Zhang KLMM, AMSS, CAS

Background Bent functions and generalized Bent functions Galois rings Constructions of generalized Bent functions Generalized Walsh Transform For f : F m 2 − → Z 2 h , the generalized Walsh transform of f is given by ˆ f : F m 2 − → C with ˆ � ω f ( x ) ( − 1 ) x · u f ( u ) = x ∈ F m 2 where " · " denotes the scalar product in F m 2 and ω is a primitive 2 h -th root of unity in C . Xiaoming Zhang KLMM, AMSS, CAS

Background Bent functions and generalized Bent functions Galois rings Constructions of generalized Bent functions Generalized Bent function A function f : F m 2 − → Z 2 h is called a generalized Bent function if f ( u ) | = 2 m / 2 for all u ∈ F m | ˆ 2 . The number of variables m can be even or odd. Xiaoming Zhang KLMM, AMSS, CAS

Background Bent functions and generalized Bent functions Galois rings Constructions of generalized Bent functions Outline of this talk Background 1 Bent functions and generalized Bent functions 2 Galois rings 3 Constructions of generalized Bent functions 4 Xiaoming Zhang KLMM, AMSS, CAS

Background Bent functions and generalized Bent functions Galois rings Constructions of generalized Bent functions Notations: Define µ : Z 2 h − → F 2 , � h − 1 i = 0 a i 2 i �− → a 0 µ : Z 2 h [ x ] − → F 2 [ x ] � m � m i = 0 b i x i i = 0 µ ( b i ) x i �− → A polynomial p ( x ) ∈ Z 2 h [ x ] is called monic basic irreducible if p ( x ) is monic and its projection µ ( p ( x )) is irreducible over F 2 . Xiaoming Zhang KLMM, AMSS, CAS

Background Bent functions and generalized Bent functions Galois rings Constructions of generalized Bent functions Galois ring The Galois ring R h , m is defined by R h , m ∼ = Z 2 h [ x ] / ( p ( x )) , where p ( x ) is a basic irreducible polynomial over Z 2 h of degree m . Let ξ ∈ R h , m be a root of p ( x ) , then R h , m ∼ = Z 2 h [ x ] / ( p ( x )) ∼ = Z 2 h [ ξ ] . The map µ can be extended to R h , m . Xiaoming Zhang KLMM, AMSS, CAS

Background Bent functions and generalized Bent functions Galois rings Constructions of generalized Bent functions Teichemüler set The set T h , m := { 0 } ∪ T ∗ h , m is called the Teichmüller set of R h , m , where T ∗ h , m is the cyclic group generated by ξ . µ ( ξ ) is a primitive element of F 2 m , so µ ( T h , m ) = F 2 m . Xiaoming Zhang KLMM, AMSS, CAS

Background Bent functions and generalized Bent functions Galois rings Constructions of generalized Bent functions Every element z ∈ R h , m can be uniquely expressed as: Additive representation m − 1 � z i ξ i , z i ∈ Z 2 h z = i = 0 2-adic Representation h − 1 � z i 2 i , z i ∈ T h , m z = i = 0 Xiaoming Zhang KLMM, AMSS, CAS

Background Bent functions and generalized Bent functions Galois rings Constructions of generalized Bent functions Frobenius automorphism For any z = � h − 1 i = 0 z i 2 i , z i ∈ T h , m , the map σ : R h , m − → R h , m defined by h − 1 � z 2 i 2 i σ ( z ) = i = 0 is called the Frobenius automorphism of R h , m with respect to the ground ring Z 2 h . Xiaoming Zhang KLMM, AMSS, CAS

Background Bent functions and generalized Bent functions Galois rings Constructions of generalized Bent functions Trace function The trace function Tr : R h , m − → Z 2 h is defined to be m − 1 � σ i ( z ) . Tr ( z ) = i = 0 Tr ( 2 r ) = 2 tr ( µ ( r )) for any r ∈ R h , m , where " tr " is the trace function over F 2 m . Xiaoming Zhang KLMM, AMSS, CAS

Background Bent functions and generalized Bent functions Galois rings Constructions of generalized Bent functions Outline of this talk Background 1 Bent functions and generalized Bent functions 2 Galois rings 3 Constructions of generalized Bent functions 4 Xiaoming Zhang KLMM, AMSS, CAS

Background Bent functions and generalized Bent functions Galois rings Constructions of generalized Bent functions Schmidt’s construction Theorem (K.-U. Schmidt) → Z 4 be given by Suppose m ≥ 3 and let f : T 2 , m − f ( x ) = ε + Tr ( ax + 2 bx 3 ) , ε ∈ Z 4 , a ∈ R 2 , m , b ∈ T ∗ 2 , m . Then f ( x ) is a generalized Bent function if either of the following conditions holds: µ ( a ) = 0 and x 3 + 1 µ ( b ) = 0 has no solution in F 2 m ; 1 µ ( a ) � = 0 and x 3 + x + µ ( b ) 2 µ ( a ) 6 = 0 has no solution in F 2 m . 2 Here, µ is the modulo 2 reduction map on R 2 , m . Xiaoming Zhang KLMM, AMSS, CAS

Background Bent functions and generalized Bent functions Galois rings Constructions of generalized Bent functions Question: 1 Can we generalize Schmidt’s construction? Can we say something more about the conditions to be satisfied? 2 Xiaoming Zhang KLMM, AMSS, CAS

Background Bent functions and generalized Bent functions Galois rings Constructions of generalized Bent functions Our construction Theorem Suppose m ≥ 5 and let f ( x ) = ε + Tr ( ax + 2 bx 1 + 2 k ) , where ε ∈ Z 4 , a ∈ R 2 , m , b ∈ T ∗ 2 , m . Then f ( x ) is a generalized Bent function if either of the following conditions holds: µ ( a ) = 0 and x 2 2 k − 1 + 1 µ ( b ) 2 k − 1 = 0 has no solution in F 2 m ; 1 µ ( a ) � = 0 and µ ( b ) 2 k x 2 2 k − 1 + µ ( a ) 2 k + 1 x 2 k − 1 + µ ( b ) = 0 2 has no solution in F 2 m . Schmidt’s construction is the special case k = 1 of ours. Xiaoming Zhang KLMM, AMSS, CAS

Background Bent functions and generalized Bent functions Galois rings Constructions of generalized Bent functions Remark For any positive integer k , there always exist a ∈ R 2 , m and b ∈ T ∗ 2 , m such that the function we construct is a generalized Bent function. Hence our construction greatly generalize Schmidt’s. Xiaoming Zhang KLMM, AMSS, CAS