Semifields, Relative Difference Sets, and Bent Functions

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1 Semifields, Relative Difference Sets, and Bent Functions Alexander Pott Otto-von-Guericke-University Magdeburg December 09, / 34

2 Outline, or: 2 / 34

3 Outline, or: Why I am nervous 2 / 34

4 Outline, or: Why I am nervous bent functions... CLAUDE CARLET, TOR HELLESETH 2 / 34

5 Outline, or: Why I am nervous bent functions... CLAUDE CARLET, TOR HELLESETH relative difference sets (uninteresting generalization?) 2 / 34

6 Outline, or: Why I am nervous bent functions... CLAUDE CARLET, TOR HELLESETH relative difference sets (uninteresting generalization?) Z 4 (very old?) 2 / 34

7 Describe connection between... relative difference sets semifields projections of relative difference sets KNUTH operation on semifields bent functions Z 4 -valued bent functions 3 / 34

8 The team YUE ZHOU KAI-UWE SCHMIDT ALEX. P. 4 / 34

9 The team YUE ZHOU KAI-UWE SCHMIDT ALEX. P.... all this is also related to KATHY HORADAM s work, but less general and more concrete... 4 / 34

10 Bent functions, even characteristic Bent functions f : F 2 n F 2 such that f (x + a) f (x) is balanced for all a 0. Example f (x) = Trace(βx 3 ) on F 2 n: f (x + a) f (x) = Trace(β(x 2 a + a 2 x + a 2 )) hence 1 + a 3 β 0 for all a 0. Necessary condition n = 2m is even. = Trace(x 2 β(a + βa 4 ) + βa 2 ) 5 / 34

11 Bent functions, odd characteristic Bent functions f : F p n F p such that f (x + a) f (x) is balanced for all a 0. Example f (x) = Trace(βx 2 ): f (x + a) f (x) = Trace(2xβa) + βa 2. Any n. 6 / 34

12 Vectorial versions, even characteristic Bent functions F : F n 2 Fk 2 such that F(x + a) F(x) is balanced for all a 0. Component functions Trace(βF(x)) are bent. Hence: Vector space of bent functions. Necessary condition n = 2m is even and k m. Example F(x, y) = xy (x, y F 2 m) is vectorial bent F 2 m F 2 m F 2 m 7 / 34

13 Vectorial versions, odd characteristic Bent functions F : F n p F k p such that F(x + a) F(x) is balanced for all a 0. Component functions Trace(βF(x)) are bent. Hence: Vector space of bent functions. Necessary condition k n. Example F (x) = x 2 : F(x + a) F(x) = 2xa + a 2. 8 / 34

14 F : F n p F k p bent p = 2 p odd n even and k n 2 k n k = n: planar functions 9 / 34

15 Bent functions and relative difference sets If F : F n p F k p is bent, the set G F := {(x, F(x)) : x F n p} F n p F k p is a relative difference set: 10 / 34

16 Bent functions and relative difference sets If F : F n p F k p is bent, the set G F := {(x, F(x)) : x F n p} F n p F k p is a relative difference set: Every element outside {0} F k p has the same number of difference representations g = d d with d, d G F : is equivalent to x y = a, F(x) F(y) = b F(y + a) F(y) = b 10 / 34

17 Other groups? group G subgroup N (forbidden subgroup) subset D g G \ N has constant number of representations g = d d with d, d D, no element in N. Example D = {1, 2, 4} Z 8, forbidden subgroup {0, 4}. In this talk: D = G N, hence from each coset of N exactly one element. 11 / 34

18 The projection construction If U < N is a normal subgroup of G and D relative difference set, then is a relative difference set in D/U G/U with forbidden subgroup The size is N/U. D/U = D. One relative difference set produces a chain of relative difference sets. 12 / 34

19 Planar functions: n = k Definition A function F : F p n F p n is planar if F(x + a) F(x) is a permutation for all a 0. We obtain (vectorial) bent functions via projection. p must be odd. 13 / 34

20 Planar functions: n = k Definition A function F : F p n F p n is planar if F(x + a) F(x) is a permutation for all a 0. We obtain (vectorial) bent functions via projection. p must be odd. If p = 2, then generalize to?? 13 / 34

21 Two generalizations to characteristic 2 almost perfect nonlinear functions (APN): F(x + a) F(x) = b has at most 2 solutions. relative difference sets in other groups (not elementary abelian), related to semifields. 14 / 34

22 Two generalizations to characteristic 2 almost perfect nonlinear functions (APN): F(x + a) F(x) = b has at most 2 solutions. relative difference sets in other groups (not elementary abelian), related to semifields. Dream: Use the many semifields with p = 2 to construct many APN. 14 / 34

23 Semifields (S, +, ) is a finite (pre)semifield (field without associativity) if (S, +) is a finite abelian group. x a = b has a unique solution x if a 0. a y = b has a unique solution y if a 0. x (y + z) = x y + x z and (x + y) z = x z + y z. Example: Finite field! 15 / 34

24 Semifields (S, +, ) is a finite (pre)semifield (field without associativity) if (S, +) is a finite abelian group. x a = b has a unique solution x if a 0. a y = b has a unique solution y if a 0. x (y + z) = x y + x z and (x + y) z = x z + y z. Example: Finite field! T a : S S with T a (x) := x a is an isomorphism. T a + T a = T a+a. Vector space of invertible linear mappings. S is elementary abelian (additive group of a field F p n), multiplication not always commutative. 15 / 34

25 Why? Construct projective plane from a semifield: 16 / 34

26 Why? Construct projective plane from a semifield: Points: S S Lines: {x, m x + y : x S}. 16 / 34

27 How many? p odd: quite a few, but not many. LAVRAUW, POLVERINO (2012). p = 2: very many commutative KANTOR (2003). Question Number is not bounded by a polynomial. 17 / 34

28 Semifields and relative difference sets Any semifield gives rise to a relative difference set in a group of order p 2n with forbidden subgroup of order p n : 18 / 34

29 Semifields and relative difference sets Any semifield gives rise to a relative difference set in a group of order p 2n with forbidden subgroup of order p n : Consider the set S S with addition (a, b) (a, b ) = (a + a, b + b + a a ). 18 / 34

30 Semifields and relative difference sets Any semifield gives rise to a relative difference set in a group of order p 2n with forbidden subgroup of order p n : Consider the set S S with addition (a, b) (a, b ) = (a + a, b + b + a a ). Difference set: {(a, 0) : a S} 18 / 34

31 What are these strange groups? Let S be commutative: F 2n p if p is odd, Z n 4 if p = 2. Forbidden subgroup: 2Zn 4. In the p odd case: Planar function. 19 / 34

32 The KNUTH cube Basis e i of S e i e j = k a i,j,k e k with a i,j,k F p. Linear mappings are described by matrices (a i,k ). Permuting the indices gives six semifields (KNUTH). 20 / 34

33 The KNUTH cube Basis e i of S e i e j = k a i,j,k e k with a i,j,k F p. Linear mappings are described by matrices (a i,k ). Permuting the indices gives six semifields (KNUTH). If S is commutative, the linear mappings associated with one of the 6 semifields are symmetric: Vector space of symmetric invertible matrices. 20 / 34

34 What are the projections of RDS, p odd? One semifield in Knuth orbit of a commutative semifield is vector space of symmetric invertible matrices, another one can be described by planar function (RDS). The invertible matrices associated with F are not symmetric. x F(x + a) F(x) F(a) + F(0). 21 / 34

35 What are the projections of RDS, p odd? One semifield in Knuth orbit of a commutative semifield is vector space of symmetric invertible matrices, another one can be described by planar function (RDS). The invertible matrices associated with F are not symmetric. x F(x + a) F(x) F(a) + F(0). p odd: Symmetric invertible matrix (a i,j ) gives a bent function f (x 1,..., x n ) = i,j a i,j x i x j. These are the projections of the planar function F! 21 / 34

36 What are the projections of RDS, p odd? One semifield in Knuth orbit of a commutative semifield is vector space of symmetric invertible matrices, another one can be described by planar function (RDS). The invertible matrices associated with F are not symmetric. x F(x + a) F(x) F(a) + F(0). p odd: Symmetric invertible matrix (a i,j ) gives a bent function f (x 1,..., x n ) = i,j a i,j x i x j. These are the projections of the planar function F! Nice representation of KNUTH operation in terms of relative difference sets. 21 / 34

37 p = 2: Difference set in Z n 4 KNUTH gives invertible symmetric matrices (a i,j ). Projections are relative difference set in the group Z 4 Z n / 34

38 p = 2: Difference set in Z n 4 KNUTH gives invertible symmetric matrices (a i,j ). Projections are relative difference set in the group Z 4 Z n 1 2. The group defined on the set Z n 2 Z 2 with addition (v, x) (w, y) = (v + w, x + y + i a i,i v i w i ) is Z 4 Z n / 34

39 p = 2: Difference set in Z n 4 KNUTH gives invertible symmetric matrices (a i,j ). Projections are relative difference set in the group Z 4 Z n 1 2. The group defined on the set Z n 2 Z 2 with addition (v, x) (w, y) = (v + w, x + y + i a i,i v i w i ) is Z 4 Z n 1 2. The set {(x, i<j a i,j x i x j ) : x Z n 2 } is a projection RDS in that group. 22 / 34

40 The group Z 4 Z n 1 2 Consider the set Z n 2 Z 2 and define addition (v, x) (w, y) = (v + w, x + y + v, w ) 23 / 34

41 The group Z 4 Z n 1 2 Consider the set Z n 2 Z 2 and define addition If f : Z n 2 Z 2, then (v, x) (w, y) = (v + w, x + y + v, w ) G f := {(x, f (x)) : x Z n 2 } is a relative difference set in Z 4 Z n 1 2 if and only if is balanced for all a 0. f (x + a) + f (x) + x, a 23 / 34

42 The group Z 4 Z n 1 2 Consider the set Z n 2 Z 2 and define addition If f : Z n 2 Z 2, then (v, x) (w, y) = (v + w, x + y + v, w ) G f := {(x, f (x)) : x Z n 2 } is a relative difference set in Z 4 Z n 1 2 if and only if f (x + a) + f (x) + x, a is balanced for all a 0. Character theoretic characterization: negabent. 2 ( 1) x,a +f (x) i w(x) = 2 n x F n 2 23 / 34

43 Construction of difference sets in Z 4 Z n 1 2 Theorem ( ) D, E difference sets in G. Then {0} D {1} E {2} (G \ D) {3} (G \ E) is a relative difference set in Z 4 G relative to 2Z 4 {0}. Start with bent function difference sets. If G = Z n 1 2 : negabent (equivalently Z 4 -valued bent). 24 / 34

44 Construction of difference sets in Z 4 Z n 1 2 Theorem (Arasu, Jungnickel, Ma, P. ) D, E difference sets in G. Then {0} D {1} E {2} (G \ D) {3} (G \ E) is a relative difference set in Z 4 G relative to 2Z 4 {0}. Start with bent function difference sets. If G = Z n 1 2 : negabent (equivalently Z 4 -valued bent). 24 / 34

45 Construction of difference sets in Z 4 Z n 1 2 Theorem (Arasu, Jungnickel, Ma, P. (1990)) D, E difference sets in G. Then {0} D {1} E {2} (G \ D) {3} (G \ E) is a relative difference set in Z 4 G relative to 2Z 4 {0}. Start with bent function difference sets. If G = Z n 1 2 : negabent (equivalently Z 4 -valued bent). 24 / 34

46 If you want to find new objects related to RDS s, look at Bernhard Schmidt s thesis Davis/Jedwab 25 / 34

47 Another look at RDS in Z n 4 Consider the set F 2 n F 2 n with addition (x, y) (x, y ) = (x + x, y + y + x x ). Group: Z n 4, and {(0, y) : y F 2n} is elementary abelian subgroup. 26 / 34

48 Another look at RDS in Z n 4 Consider the set F 2 n F 2 n with addition (x, y) (x, y ) = (x + x, y + y + x x ). Group: Z n 4, and {(0, y) : y F 2n} is elementary abelian subgroup. Question Which subsets are relative difference sets. {(x, F(x)) : x F 2 n} 26 / 34

49 Another look at RDS in Z n 4 Consider the set F 2 n F 2 n with addition (x, y) (x, y ) = (x + x, y + y + x x ). Group: Z n 4, and {(0, y) : y F 2n} is elementary abelian subgroup. Question Which subsets are relative difference sets. {(x, F(x)) : x F 2 n} Theorem (ZHOU (2012)) Relative difference set if and only if is a permutation for all a 0. Such an F is called planar. F(x + a) F(x) + a x 26 / 34

50 ridiculous example F(x) = 0 27 / 34

51 ridiculous example F(x) = 0 is a permutation for all a 0. F(x + a) + F(x) + a x = a x 27 / 34

52 Many semifields Thanks to KANTOR (2003): There are many semifields! 28 / 34

53 Many semifields Thanks to KANTOR (2003): There are many semifields! Theorem K = K 0 K 1 K n of characteristic 2 with [K : K n ] odd. Let tr i be the relative trace from K to K i. Then, for all nonzero ζ 1,..., ζ n K, the mapping F : K K given by F (x) = ( x n tr i (ζ i x) i=1 is planar. Examples are inequivalent. ) 2 28 / 34

54 Construction of new commutative semifields New symplectic spreads (symmetric invertible matrices) (Italian school) New planar functions (German-Chinese-Norwegian-Armenian school) 29 / 34

55 The COULTER-MATTHEWS (1998) example The only known planar function not corresponding to semifields: Theorem The function F(x) = x d with d = 3a is PN in F 3 n iff gcd(a, n) = 1 and a odd. 30 / 34

56 The COULTER-MATTHEWS (1998) example The only known planar function not corresponding to semifields: Theorem The function F(x) = x d with d = 3a is PN in F 3 n iff gcd(a, n) = 1 and a odd. p even??? 30 / 34

57 Switching Planar function such that image set of F has size 2? Theorem (ZHOU 2012) 31 / 34

58 Switching Planar function such that image set of F has size 2? Theorem (ZHOU 2012) No 31 / 34

59 Power mappings F (x) = α x d F(x + a) F(x) + a x permutation. 32 / 34

60 Power mappings F (x) = α x d F(x + a) F(x) + a x permutation. Known power mappings αx d which are planar: d condition 2 k no folklore 2 k + 1 n = 2k SCHMIDT, ZHOU 4 k (4 k + 1) n = 6k SCHERR, ZIEVE 32 / 34

61 Power mappings F (x) = α x d F(x + a) F(x) + a x permutation. Known power mappings αx d which are planar: d condition 2 k no folklore 2 k + 1 n = 2k SCHMIDT, ZHOU 4 k (4 k + 1) n = 6k SCHERR, ZIEVE Theorem (MÜLLER, ZIEVE (2013)) Let d be a positive integer such that d 4 2 m and let c F 2 m be nonzero. Then the function x αx d is planar on F 2 m if and only if d is a power of / 34

62 Conclusion Semifields give RDS (old result) In the commutative case, projections give KNUTH operation (new to me) Z 4 valued bent functions have been studied before (new) Non-commutative case? (work in progress) p = 2 Planar functions of low weight (ZHOU) p = 2 Power planar? (ZHOU, SCHMIDT, ZIEVE, MÜLLER, SCHERR) p = 2 RDS not semifield (Open) 33 / 34

63 Epilogue: My dream Use KANTOR to construct many APN. 34 / 34

64 Epilogue: My dream Use KANTOR to construct many APN. Almost true, but the generalization lives on the wrong face of KNUTH cube (DEMPWOLFF, KANTOR (2013)). 34 / 34

Nonlinear Functions A topic in Designs, Codes and Cryptography

Nonlinear Functions A topic in Designs, Codes and Cryptography Alexander Pott Otto-von-Guericke-Universität Magdeburg September 21, 2007 Alexander Pott (Magdeburg) Nonlinear Functions September 21, 2007