An Algorithm for Checking Normality of Boolean Functions

RuhrRuhrUniversityUniversityBochumBochumFaculty of MathematicsFaculty of MathematicsInformation-Security and CryptologyInformation-Security and Cryptology

An AlgorithmAn Algorithm

for Checking Normalityfor Checking Normality

of Boolean Functionsof Boolean Functions

Magnus DaumMagnus Daum Hans DobbertinHans Dobbertin Gregor LeanderGregor Leander


28.03.2003 Daum,Dobbertin,Leander: An Algorithm for Checking Normality of Boolean Functions

OverviewOverview

• Definitions and Motivation• General Idea and Algorithm• Details of the Algorithm• Complexity Evaluations



• 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

DefinitionsDefinitions



MotivationMotivation

• 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,…)



• # (subspaces of of dimension k):• # ( flats of of dimension k):

• Naive idea:– Enumerate all flats of dimension m and check

whether f is constant/affine or not

Motivation: Naive AlgorithmMotivation: Naive Algorithm

• # (flats of of dimension m)

m 4 5 6 7 8

log2(#flats) 21 31 44 57 73



General IdeaGeneral 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:

constant



) and are constant




• Idea for the algorithm:– find flats a+U and b+U, dim(U)=k

– combine them to

with :




Outline of the AlgorithmOutline of the Algorithm

Input: Boolean function f, starting dimension t0

For all subspaces do :

Determine all flats a+U, such that

a+U, b+U, dim U=k

! dim=k+1

a+U, b+U, dim U=k

! dim=k+1

m-dimensional flats on which f is affine

combine

repeat up to

dim=m-1



• Main problem:

can be split into

in many ways

• many ways to find some• try to avoid this redundant work

Outline of the AlgorithmOutline of the Algorithm



• need a unique representation of U to avoid looking at same U twice

• Basis of U is a Gauss Jordan Basis

Details: Representation of UDetails: Representation of U

(ui): index of the leftmost 1 in ui

• >: standard lexicographical ordering

DetailsDetails



• easy to enumerate all subspaces:

– loop over all

– fill corresponding to this scheme

– loop over all values in for

• Example:




• easy to enumerate all flats corresponding to U:

–

– a+U can be represented uniquely as a‘+U with a‘2

• Example:




• Main data structure:

List of all flats corresponding toon which f is constant equal to c

Details: Combining FlatsDetails: Combining Flats

if is GJB

otherwise



• store elements in sorted order


Lemma:

• compute once , check




Corollary:

• only need to consider if this is fulfilled

or



Details: AlgorithmDetails: Algorithm

Input: Boolean function f, starting dimension t0

Output: a list of all flats on which f is affine



ComplexityComplexity

• 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



• Check all flats of dimension t0:

about flats

• less than two comparisons per flat on average

n=14:

Complexity: Exhaustive SearchComplexity: Exhaustive Search

t0 1 2 3 4 5 6 7

log2(complexity) 28 38 46 52 56 58 58



• 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

Complexity: CombiningComplexity: Combining

t0 1 2 3 4 5 6 7

log2(complexity) 28 38 46 52 56 58 58

t0 1 2 3 4 5

log2(complexity) 43 43 41 30 1 naive

algorithm



ConclusionConclusion

• 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


Thank You !!!Thank You !!!

Questions ?!?Questions ?!?

An Algorithm for Checking Normality of Boolean Functions

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