Pair of Binary Sequences with Ideal Two-Level Crosscorrelation - PowerPoint PPT Presentation

Pair of Binary Sequences with Ideal Two-Level Crosscorrelation Seok-Yong Jin and Hong-Yeop Song {sy.jin, hysong}@yonsei.ac.kr Coding and Crypto Lab Yonsei University, Seoul, KOREA 2008 IEEE International Symposium on Information Theory Toronto, Canada July 6-11, 2008 Jin, Song (Yonsei Univ) Ideal 2-level crosscorrelation ISIT 2008 1 / 22

Outline Introduction 1 Structure and Property of Associated Cyclic Difference 2 Pair Ideal Cyclic Difference Pair with k − λ = 1 : 3 Parameterizations and Construction Exhaustive Search for Short Lengths 4 Jin, Song (Yonsei Univ) Ideal 2-level crosscorrelation ISIT 2008 2 / 22

Definition of Correlation a = ( a 0 , ··· , a v − 1 ) and b = ( b 0 , ··· , b v − 1 ): binary (0,1)-sequences of length v Periodic correlation function v − 1 � ( − 1) a i + b i + τ θ a , b ( τ ) = i = 0 Jin, Song (Yonsei Univ) Ideal 2-level crosscorrelation ISIT 2008 3 / 22

Ideal 2-level Correlation: Single Sequence 2-level (auto)-correlation of a sequence ( ⇔ cyclic difference set ) � v , τ = 0 θ a , a ( τ ) = γ ( �= v ) , otherwise . Ideal 2-level (auto)-correlation � Small | γ | is desirable for various applications � γ = 0: currently no such example found, except for v = 4 � γ = − 1: called ideal 2-level autocorrelation (m-sequences, GMW sequences, 3-term and 5-term sequences, etc.) Jin, Song (Yonsei Univ) Ideal 2-level crosscorrelation ISIT 2008 4 / 22

Ideal 2-level Correlation: Sequence Pair Generalization to pair of binary sequences Binary sequence pair ( a , b ) has 2-level correlation if � Γ 1 , τ = 0 θ a , b ( τ ) = Γ 2 ( �= Γ 1 ) , τ �= 0 (mod v ), Γ 2 = 0: Ideal 2-level correlation � Γ ( �= 0) , τ = 0 θ a , b ( τ ) = 0 , else. Jin, Song (Yonsei Univ) Ideal 2-level crosscorrelation ISIT 2008 5 / 22

Notations s = ( s 0 , s 1 , ··· , s v − 1 ): binary sequence of period v Support set and characteristic sequence � Support set: supp ( s ) = { i | s i = 1,0 ≤ i ≤ v − 1} ⊂ Z v ( s is called the characteristic sequence) � � � Weight: wt ( s ) = | { i | s i = 1,0 ≤ i ≤ v − 1} | = � supp ( s ) � Operations on binary sequences � Cyclic shift: ρ i ( s ) = ( s i , s i + 1 , ··· , s i + v − 1 ) � � � Decimation: s ( d ) = s d · 0 , s d · 1 , ··· , s d · ( v − 1) � � � Negation: s ′ = s ′ 0 , ··· , s ′ , where s ′ i = 1 if s i = 0 and s ′ i = 0 if s i = 1 v − 1 � � s ′ 0 , s 1 , s ′ � Alternation at even positions: s E = 2 , s 3 , ··· Jin, Song (Yonsei Univ) Ideal 2-level crosscorrelation ISIT 2008 6 / 22

Notations s = ( s 0 , s 1 , ··· , s v − 1 ): binary sequence of period v Support set and characteristic sequence Operations on binary sequences Hall polynomial: h s ( z ) = s 0 + s 1 z 1 +···+ s v − 1 z v − 1 (mod z v − 1) Canonical form of circulant matrix associated with s :   s 0 s v − 1 s v − 2 ... s 1 s 1 s 0 s v − 1 ... s 2     s 2 s 1 s 0 ... s 3   M s =   . . . . ...   . . . . . . . .   s v − 1 s v − 2 s v − 3 ... s 0 The sequence s is called the defining array of M s . Jin, Song (Yonsei Univ) Ideal 2-level crosscorrelation ISIT 2008 7 / 22

Correlation Coefficients by Set Notation ( a , b ): binary sequence pair of length v A : = supp ( a ), B : = supp ( b ), k a : = wt ( a ), k b : = wt ( b ) k : = | A ∩ B | , d A , B ( τ ) = | A ∩ ( τ + B ) | Calculation of correlation coefficients of binary sequences a : 1 ··· 1 1 ··· 1 0 ··· 0 0 ··· 0 ρ τ ( b ) : 1 ··· 1 0 ··· 0 1 ··· 1 0 ··· 0 � �� � � �� � � �� � � �� � # of times : d τ k a − d τ k b − d τ v − ( k a + k b ) + d τ θ a , b ( τ ) = v − 2( k a + k b ) + 4 d A , B ( τ ) For a sequence pair ( a , b ) with ideal 2-level correlation: d A , B (0) = k ⇒ Γ = v − 2( k a + k b ) + 4 k d A , B ( τ ) = λ , ∀ τ �= 0 0 = v − 2( k a + k b ) + 4 λ ⇒ Jin, Song (Yonsei Univ) Ideal 2-level crosscorrelation ISIT 2008 8 / 22

Cyclic Difference Pair (CDP) Binary sequence with 2-level correlation ⇔ cyclic difference set Binary sequence pair with 2-level correlation ⇔ ? Definition (Cyclic Difference Pair) X and Y : k x -subset and k y -subset of Z v with | X ∩ Y | = k ( X , Y ) is a ( v , k x , k y , k , λ )-cyclic difference pair (CDP) if For every nonzero w ∈ Z v , w is expressed in exactly λ ways in the form w = x − y (mod v ) where x ∈ X and y ∈ Y . Especially when v = 2( k 1 + k 2 ) − 4 λ and k �= λ , it is called an ideal cyclic difference pair. Jin, Song (Yonsei Univ) Ideal 2-level crosscorrelation ISIT 2008 9 / 22

Relation: CDP and Binary Sequence Pair Theorem (Existence and Relation) ( a , b ) : binary sequence pair of period v with 2-level correlation such that � In-phase correlation coefficient: Γ � Out-of-phase correlation coefficients: γ � wt ( a ) = k a and wt ( b ) = k b Their support set pair ( A , B ) forms a ( v , k a , k b , k , λ ) -cyclic difference pair, where � k = | A ∩ B | satisfies Γ = v − 2( k a + k b ) + 4 k � λ is such that γ = v − 2( k a + k b ) + 4 λ . Moreover, any cyclic difference pair arises in this way. Jin, Song (Yonsei Univ) Ideal 2-level crosscorrelation ISIT 2008 10 / 22

Characterization: Three Equations Inphase and out-of-phase correlation coefficient: 1 v − 2( k a + k b ) + 4 k = Γ (e-I) v − 2( k a + k b ) + 4 λ = 0 (e-II) Counting the number of elements of A × B : 2 k a k b = λ v + ( k − λ ) (e-III) If there exists a binary sequence pair of period v having ideal 2-level correlation, then v is even. Γ = 4( k − λ ) Jin, Song (Yonsei Univ) Ideal 2-level crosscorrelation ISIT 2008 11 / 22

Characterization: using Hall Polynomial A , B : k a -subset and k b -subset of Z v with | A ∩ B | = k a , b : the characteristic binary sequences of A and B of period v Theorem Let h a ( z ) and h b ( z ) denote the associated hall polynomial of a and b , respectively. Then ( A , B ) is a ( v , k a , k b , k , λ ) -cyclic difference pair if and only if h a ( z ) h b ( z − 1 ) = ( k − λ ) + λ (1 + z +···+ z v − 1 ) Jin, Song (Yonsei Univ) Ideal 2-level crosscorrelation ISIT 2008 12 / 22

Characterization: using Circulant Matrix Under the same notations: A , B , ( k a and k b -subset), k = | A ∩ B | , and a and b Theorem M a , M b : canonical form of the circulant matrix associated with a and b ( A , B ) is a ( v , k a , k b , k , λ ) -cyclic difference pair, if and only if T = ( k − λ ) I + λ J M a M b Matrices are viewed over the integers or over the reals. Jin, Song (Yonsei Univ) Ideal 2-level crosscorrelation ISIT 2008 13 / 22

Necessary Condition: Determinants ( A , B ): ( v , k a , k b , k , λ )-cyclic difference pair ( a , b ): the corresponding characteristic binary sequence pair Theorem Let M a and M b be the canonical form of circulant matrices associated with a and b , respectively. Then det ( M a ) · det ( M b ) = k a k b ( k − λ ) v − 1 Jin, Song (Yonsei Univ) Ideal 2-level crosscorrelation ISIT 2008 14 / 22

Property Preserving Transformations If ( A , B ) is an ideal ( v , k a , k b , k , λ )-cyclic difference pair: Cyclic Difference Pair Parameters ( τ + A , τ + B ), τ = 0,1,... ( v , k a , k b , k , λ ) � A ( d ) , B ( d ) � , gcd ( d , v ) = 1 ( v , k a , k b , k , λ ) ( B , A ) ( v , k b , k a , k , λ ) ( A , B C ) ( v , k a , v − k b , k a − k , k a − λ ) ( A C , B ) ( v , v − k a , k b , k b − k , k b − λ ) ( A C , B C ) ( v , v − k a , v − k b , k ′ , λ ′ ), k ′ = v − ( k a + k b ) + k , λ ′ = v − ( k a + k b ) + λ ( v , k ′′ a , k ′′ b , k ′′ , λ ′′ ), ( A E , B E ) k ′′ a = k a + ( v /2 − 2 e a ), k ′′ b = k b + ( v /2 − 2 e b ), k ′′ = k + ( v /2 − ( e a + e b )), λ ′′ = λ + ( v /2 − ( e a + e b )) Jin, Song (Yonsei Univ) Ideal 2-level crosscorrelation ISIT 2008 15 / 22

Parameterizations For any ( v , k a , k b , k , λ )-cyclic difference pair, we assume without loss of generality: v /2 ≥ k a ≥ k b ≥ k > λ , and λ > 0 for v > 4 4( k − λ ) = ( v − 2 k a )( v − 2 k b ) Γ = 4( k − λ ) �= 0 ⇒ k � λ . If λ = 0: k a = k b = k = 1, a = b = (1000). Jin, Song (Yonsei Univ) Ideal 2-level crosscorrelation ISIT 2008 16 / 22

Ideal CDP with k − λ = 1 : Parameterizations Theorem If an ideal ( v , k a , k b , k , λ ) -cyclic difference pair with k − λ = 1 exists, then ( v , k a , k b , k , λ ) = (4 t , 2 t − 1, 2 t − 1, t , t − 1) Note: ( v , k , λ ) = (4 t − 1,2 t − 1, t − 1): cyclic difference set with Hadamard parameters ( v , k a , k b , k , λ ) = (4 t ,2 t − 1,2 t − 1, t , t − 1): cyclic difference pair with “Hadamard" parameters Jin, Song (Yonsei Univ) Ideal 2-level crosscorrelation ISIT 2008 17 / 22

Ideal CDP with k − λ = 1 : Construction det ( M a ) · det ( M b ) = k a k b ( k − λ ) v − 1 k − λ = 1 : det ( M a ) · det ( M b ) = k a · k b Q: a and b with det ( M a ) = k a and det ( M b ) = k b ?? One part: If the sequence a is such that 2 t − 1 2 t + 1 � �� � � �� � , a = ( 11 ··· 1 00 ··· 0 ) � �� � 4 t then det ( M a ) = 2 t − 1 = wt ( a ). The other part: even position negation and shift of a Jin, Song (Yonsei Univ) Ideal 2-level crosscorrelation ISIT 2008 18 / 22