Skew Hadamard Difference Sets Alexander Pott (with Cunsheng Ding and Qi Wang) Otto-von-Guericke-University Magdeburg December 06, 2013 1 / 22
Difference set Subset D of a group G such that every g ∈ G , g � = 0, has the same number of difference representations d − d ′ with d , d ′ ∈ D . Example { 1 , 2 , 4 } ⊆ Z 7 . 2 / 22
Construction of difference sets ◮ Use trivial additive sub-structures, interprete multiplicatively. ◮ Use trivial multiplicative sub-structures, interprete additively. 3 / 22
Construction of difference sets ◮ Use trivial additive sub-structures, interprete multiplicatively. ◮ Use trivial multiplicative sub-structures, interprete additively. Example ◮ trace ( x ) = 0 in F ∗ 2 n ◮ squares in F q 3 / 22
How can we generalize trace ( x ) = 0? ◮ G ORDON -M ILLS -W ELCH (1962): Modify trace 4 / 22
How can we generalize trace ( x ) = 0? ◮ G ORDON -M ILLS -W ELCH (1962): Modify trace Breakthrough: M ASCHIETTI (1998) 2 n : trace ( x ) = 0 } = { y 2 + y : y ∈ F ∗ { x ∈ F ∗ 2 n , y � = 1 } Difference set is the image set of y 2 + y in F ∗ 2 n . 4 / 22
How can we generalize trace ( x ) = 0? ◮ G ORDON -M ILLS -W ELCH (1962): Modify trace Breakthrough: M ASCHIETTI (1998) 2 n : trace ( x ) = 0 } = { y 2 + y : y ∈ F ∗ { x ∈ F ∗ 2 n , y � = 1 } Difference set is the image set of y 2 + y in F ∗ 2 n . Generalize this description: Use 2-to-1 mappings. 4 / 22
Hyperovals Maschietti used monomial hyperovals: 1 0 0 : x ∈ F 2 n } ∪ { , { x 1 0 } x d 0 1 is a hyperoval in PG ( 2 , 2 n ) if and only if y d + y is 2-to-1. 5 / 22
S IDELNIKOV { x 2 − 1 : x ∈ F q } ⊆ F ∗ q “almost” difference set in F ∗ q , yields sequences with optimal autocorrelation properties. 6 / 22
Generalizing Squares I Cyclotomy: Unions of cosets of multiplicative subgroup. T AO F ENG , K OJI M OMIHARA , Q ING X IANG use small subgroups. 7 / 22
Generalizing Squares II Squares are image set of a 2-to-1 mapping f : F q → F q ! But in the additive group. 8 / 22
Generalizing Squares II Squares are image set of a 2-to-1 mapping f : F q → F q ! But in the additive group. Consider the graph G f = { ( x , f ( x )) : x ∈ F q } If G f has “nice” properties with respect to addition, then perhaps also the image set. 8 / 22
Planar functions f : F q → F q is planar if f ( x + a ) − f ( x ) is a permutation for all a � = 0. Example f ( x ) = x 2 : ( x + a ) 2 − x 2 = 2 xa + a 2 is a permutation on F q if q odd. Hence: Squares are image sets of a class of planar functions! 9 / 22
Squares in F q are nice The set of squares are a difference set: d − d ′ = x has q − 3 4 solutions with d , d ′ ∈ D for all x , 10 / 22
Squares in F q are nice The set of squares are a difference set: d − d ′ = x has q − 3 4 solutions with d , d ′ ∈ D for all x , and D ∪ ( − D ) ∪ { 0 } = F q ( ∗ ) 10 / 22
Squares in F q are nice The set of squares are a difference set: d − d ′ = x has q − 3 4 solutions with d , d ′ ∈ D for all x , and D ∪ ( − D ) ∪ { 0 } = F q ( ∗ ) Example ( q = 7) { 1 , 2 , 4 } ∪ { 3 , 5 , 6 } ∪ { 0 } = F 7 skew Hadamard difference sets Hadamard difference set: without ( ∗ ) . 10 / 22
Are there others? Brilliant idea due to D ING and Y UAN (2006): Try other planar functions! 11 / 22
Are there others? Brilliant idea due to D ING and Y UAN (2006): Try other planar functions! Exactly one gives new example: f ( x ) = x 10 + x 6 − x 2 in F 3 n C OULTER , M ATTHEWS (1998). 11 / 22
Are there others? Brilliant idea due to D ING and Y UAN (2006): Try other planar functions! Exactly one gives new example: f ( x ) = x 10 + x 6 − x 2 in F 3 n C OULTER , M ATTHEWS (1998). ... still no theoretical proof that it is “new” in general 11 / 22
... rekindled interest in planar functions... D ING and Y UAN also proved: f ( x ) = x 10 − x 6 − x 2 is planar and also gives skew Hadamard difference set. 12 / 22
Another look at Ding-Yuan composition of a permutation polynomial and x 2 : ( x 5 ± x 3 − x ) ◦ x 2 D ICKSON of order 5. 13 / 22
D ING , W ANG , X IANG (2007) q = 3 2 h + 1 , α = 3 h + 1 , u ∈ F q Use permutation polynomial f ( x ) = x 2 α + 3 + ( ux ) α − u 2 x (which is not planar): 14 / 22
D ING , W ANG , X IANG (2007) q = 3 2 h + 1 , α = 3 h + 1 , u ∈ F q Use permutation polynomial f ( x ) = x 2 α + 3 + ( ux ) α − u 2 x (which is not planar): Image set of f ◦ x 2 is skew Hadamard. Inequivalence only in small cases proved. 14 / 22
D ING , P., W ANG (2013) q = 3 m , m �≡ 0 mod 3, u ∈ F q Use D ICKSON of order 7: f ( x ) = x 7 − ux 5 − u 2 x 3 − u 3 x . (which is not planar). Inequivalence only in small cases proved. 15 / 22
Proof I Proof resembles Ding, Wang, Xiang. Have to show | Ψ( D ) | 2 = 3 m + 1 for additive characters Ψ . 4 Thanks to C HEN , S EHGAL , X IANG (1994), it is sufficient to show: Ψ( D ) ≡ 3 ( m − 1 ) / 2 − 1 mod 3 ( m − 1 ) / 2 . 2 16 / 22
Proof II Show � Ψ β ( f ( z )) χ ( z ) ≡ 0 mod 3 ( m − 1 ) / 2 S β = z ∈ F ∗ q where χ is the quadratic character and Ψ β ( z ) = ζ Trace ( β z ) . 3 This reduces to ζ Trace ( z 7 + η z 5 + γ z ) � χ ( z ) 3 z ∈ F ∗ q for some η and γ . 17 / 22
Proof III ζ Trace ( z 7 + η z 5 + γ z ) � χ ( z ) 3 z ∈ F ∗ q Use q − 2 1 ζ Trace ( z ) � g ( ω − b ) ω b ( z ) = 3 q − 1 b = 0 where g ( ω − b ) is Gauss sum with respect to multiplicative character ω − b , where ω has order q − 1. 18 / 22
Proof IV If γ = 0, we obtain q − 2 1 g ( ω − b ) g ( ω − q − 1 � 2 + 5 − 1 7 b ) × root of unity S β = ± q − 1 b = 0 Then use S TICKELBERGER and combinatorial arguments. Case γ � = 0 is similar. 19 / 22
... use polynomials ... ◮ to construct more Hadamard difference sets; ◮ to construct Sidelnikov sequences x 2 − 1; ◮ to construct more skew Hadamard difference sets. Problem: Show inequivalence! 20 / 22
M UZYCHUK (2010) Mikhail Muzychuk has another construction in F q 3 using orbits of vectors in F 3 q under the action of GL ( 3 , q ) . 21 / 22
M UZYCHUK (2010) Mikhail Muzychuk has another construction in F q 3 using orbits of vectors in F 3 q under the action of GL ( 3 , q ) . He can show inequivalence. 21 / 22
M UZYCHUK (2010) Mikhail Muzychuk has another construction in F q 3 using orbits of vectors in F 3 q under the action of GL ( 3 , q ) . He can show inequivalence. Inequivalence of some cyclotomic examples and squares has been shown by K OJI M OMIHARA . 21 / 22
Inequivalence Difference set corresponds to a design! ◮ triple intersection numbers; ◮ rank of incidence matrix; ◮ automorphism groups. 22 / 22
Inequivalence Difference set corresponds to a design! ◮ triple intersection numbers; M OMIHARA , computer ◮ rank of incidence matrix; ◮ automorphism groups. 22 / 22
Inequivalence Difference set corresponds to a design! ◮ triple intersection numbers; M OMIHARA , computer ◮ rank of incidence matrix; always the same for skew H.d.s ◮ automorphism groups. 22 / 22
Inequivalence Difference set corresponds to a design! ◮ triple intersection numbers; M OMIHARA , computer ◮ rank of incidence matrix; always the same for skew H.d.s ◮ automorphism groups. M UZYCHUK 22 / 22
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