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An Algorithm for Checking Normality of Boolean Functions

An Algorithm for Checking Normality of Boolean Functions. Magnus Daum Hans Dobbertin Gregor Leander. Overview. Definitions and Motivation General Idea and Algorithm Details of the Algorithm Complexity Evaluations. Definitions. Boolean function , n=2m

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An Algorithm for Checking Normality of Boolean Functions

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  1. An Algorithm for Checking Normality of Boolean Functions Magnus Daum Hans Dobbertin Gregor Leander

  2. Overview • Definitions and Motivation • General Idea and Algorithm • Details of the Algorithm • Complexity Evaluations Daum,Dobbertin,Leander: An Algorithm for Checking Normality of Boolean Functions

  3. Definitions • Boolean function , n=2m • f is normal :,9 m-dimensional flat a+U: is constant • f is weakly normal :,9 m-dimensional flat a+U: is affine Daum,Dobbertin,Leander: An Algorithm for Checking Normality of Boolean Functions

  4. Motivation • Open question: Are there non normal bent functions? • Dillon / Dobbertin found some new bent functions to be checked for normality (Kasami power functions) • Interesting cases: m ¸ 5 odd (i.e. n=10,14,…) Daum,Dobbertin,Leander: An Algorithm for Checking Normality of Boolean Functions

  5. # (flats of of dimension m) Motivation: Naive Algorithm • Naive idea: • Enumerate all flats of dimension m and check whether f is constant/affine or not • # ( flats of of dimension k): • # (subspaces of of dimension k): Daum,Dobbertin,Leander: An Algorithm for Checking Normality of Boolean Functions

  6. constant General Idea • flat a+U, u1,…,uk a basis of U • f is affine on a+U: ) f is constant on a+U or • f constant on a+U: Daum,Dobbertin,Leander: An Algorithm for Checking Normality of Boolean Functions

  7. ) and are constant General Idea Daum,Dobbertin,Leander: An Algorithm for Checking Normality of Boolean Functions

  8. General Idea • Idea for the algorithm: • find flats a+U and b+U, dim(U)=k • combine them to with : Daum,Dobbertin,Leander: An Algorithm for Checking Normality of Boolean Functions

  9. a+U, b+U, dim U=k ! dim=k+1 a+U, b+U, dim U=k ! dim=k+1 repeat up to dim=m-1 combine Outline of the Algorithm Input: Boolean function f, starting dimension t0 For all subspaces do : Determine all flats a+U, such that m-dimensional flats on which f is affine Daum,Dobbertin,Leander: An Algorithm for Checking Normality of Boolean Functions

  10. Outline of the Algorithm • Main problem: can be split into in many ways • many ways to find some • try to avoid this redundant work Daum,Dobbertin,Leander: An Algorithm for Checking Normality of Boolean Functions

  11. (ui): index of the leftmost 1 in ui Details: Representation of U Details • need a unique representation of U to avoid looking at same U twice • Basis of U is a Gauss Jordan Basis • >: standard lexicographical ordering Daum,Dobbertin,Leander: An Algorithm for Checking Normality of Boolean Functions

  12. Details: Representation of U • Example: • easy to enumerate all subspaces: • loop over all • fill corresponding to this scheme • loop over all values in for Daum,Dobbertin,Leander: An Algorithm for Checking Normality of Boolean Functions

  13. Details: Representation of U • Example: • easy to enumerate all flats corresponding to U: • a+U can be represented uniquely as a‘+U with a‘2 Daum,Dobbertin,Leander: An Algorithm for Checking Normality of Boolean Functions

  14. if is GJB otherwise Details: Combining Flats • Main data structure: List of all flats corresponding toon which f is constant equal to c Daum,Dobbertin,Leander: An Algorithm for Checking Normality of Boolean Functions

  15. compute once , check Details: Combining Flats • store elements in sorted order Lemma: Daum,Dobbertin,Leander: An Algorithm for Checking Normality of Boolean Functions

  16. only need to consider if this is fulfilled or Details: Combining Flats Corollary: Daum,Dobbertin,Leander: An Algorithm for Checking Normality of Boolean Functions

  17. Details: Algorithm Input: Boolean function f, starting dimension t0 Output: a list of all flats on which f is affine Daum,Dobbertin,Leander: An Algorithm for Checking Normality of Boolean Functions

  18. Complexity • How to choose starting dimension t0? • evaluate expected complexity under the assumption that f is a random Boolean function • Two parts: • „exhaustive search“ part • „combining“ part Daum,Dobbertin,Leander: An Algorithm for Checking Normality of Boolean Functions

  19. Complexity: Exhaustive Search • Check all flats of dimension t0: about flats • less than two comparisons per flat on average n=14: Daum,Dobbertin,Leander: An Algorithm for Checking Normality of Boolean Functions

  20. Complexity: Combining • complexity of combining part • n=14: • exh.search: • choose starting dimension t0=2 or t0=3 • implementation on Pentium IV, 1.5 GHz: about 50 hours for n=14 naive algorithm Daum,Dobbertin,Leander: An Algorithm for Checking Normality of Boolean Functions

  21. Conclusion • presented an algorithm that computes a list of all flats, on which a given Boolean function is affine/constant • can be used to check (weak) normality of Boolean functions much faster than with the naive algorithm • other applications also possible(like checking Maiorana-McFarland-property) • solved open question about existence of non (weakly) normal bent functions Daum,Dobbertin,Leander: An Algorithm for Checking Normality of Boolean Functions

  22. Thank You !!!Questions ?!?

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