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...ts7sBalanced t-th order correlation immune functions are called t-resilient functions.sFrom this point on, we will consider a balanced Boolean function as 0-resilient (thissconvention is accepted in =-=[CamCarCS1991]-=-, [SarMai1999], [PasJoh1999],s[Tara2000]) and an arbitrary Boolean function as (–1)-resilient.sThe importance of correlation immunity is that, it provides zero mutual informationsbetween any t-tuple s...

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...he nonlinearity of f can be obtained from the Walsh-Hadamard transformsassω ω)(max 2 12 1 f n f Wnl −= −sThis measure of nonlinearity is important for linear cryptanalysis [Heys2000].sCarlet and Ding =-=[CarDin2004]-=- gives a more general definition for nonlinearity andsinvestigates perfect nonlinear functions.sA very well known theorem says that for any Boolean function f the sum of squaredsvalues of the Walsh-Ha...

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...n H W gfd −−= −sTherefore, the nonlinearity of f can be obtained from the Walsh-Hadamard transformsassω ω)(max 2 12 1 f n f Wnl −= −sThis measure of nonlinearity is important for linear cryptanalysis =-=[Heys2000]-=-.sCarlet and Ding [CarDin2004] gives a more general definition for nonlinearity andsinvestigates perfect nonlinear functions.sA very well known theorem says that for any Boolean function f the sum of ...

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...( = ∈ω ωsDefinition 2.7. For a perfectly nonlinear Boolean function, also called bents[Roth1976], the squared spectrum is flat andsn fW 2)(2 =ω , for all ∈ GF(2)n.sDefinition 2.8. Xiao and Massey =-=[GouMas1988]-=- provided a spectralscharacterization of correlation immune functions. Here we state that as a definitionsof correlation immunity:sA Boolean function f is t-th order correlation immune if and only ifs...

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...iorana-McFarland like technique is one of the construction techniques usedsto construct resilient Boolean functions and has been investigated previously in thesliterature ([CamCarCS1991], [Carl1997], =-=[CheLeeLS1996]-=-, [SarMai2000-3],s[SebZhaZ1994]). This technique of construction basically uses the following idea.sLet be a map from { }r1,0sto { }k1,0 , where for any x ∈ { }r1,0 , wt((x)) t + 1. Let fsbe a Bo...

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...Mai2000-1] have found an upper bound for the nonlinearitysof t-resilient Boolean functions and inspired a number of researchers ([SarMai20001], [SarMai2000-2], [FedTar2001], [JohPas2003], [Tara2001], =-=[PasJohMS2001]-=-,s[Mait2000]) for discovering functions satisfying that upper bound, which iss11 22 +− −≤ tnfnlsfor 22 −> ntsand 1121 222 +−− −−≤ t n n fnlsfor 22 −≤ nt . Theyshave based their results to the fact tha...

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...lation immune functions are called t-resilient functions.sFrom this point on, we will consider a balanced Boolean function as 0-resilient (thissconvention is accepted in [CamCarCS1991], [SarMai1999], =-=[PasJoh1999]-=-,s[Tara2000]) and an arbitrary Boolean function as (–1)-resilient.sThe importance of correlation immunity is that, it provides zero mutual informationsbetween any t-tuple selected from the input vecto...

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...–1.sSarkar and Maitra [SarMai2000-1] have found an upper bound for the nonlinearitysof t-resilient Boolean functions and inspired a number of researchers ([SarMai20001], [SarMai2000-2], [FedTar2001], =-=[JohPas2003]-=-, [Tara2001], [PasJohMS2001],s[Mait2000]) for discovering functions satisfying that upper bound, which iss11 22 +− −≤ tnfnlsfor 22 −> ntsand 1121 222 +−− −−≤ t n n fnlsfor 22 −≤ nt . Theyshave based t...

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...f the document and also includes thesdiscussion on the nonlinearity bound of Sarkar and Maitra.sThe construction method described in Chapter 3 is a previously introduced methodsby Maity and Johansson =-=[MaiJoh2002]-=- to construct t-resilient Boolean functions ofsnonlinearity 1121 ' 222 +−− −−= t n n fnl , for t 2 ns- 2. In their method, Maity andsJohansson start with a bent function and complement some values o...

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...niquesThe Maiorana-McFarland like technique is one of the construction techniques usedsto construct resilient Boolean functions and has been investigated previously in thesliterature ([CamCarCS1991], =-=[Carl1997]-=-, [CheLeeLS1996], [SarMai2000-3],s[SebZhaZ1994]). This technique of construction basically uses the following idea.sLet be a map from { }r1,0sto { }k1,0 , where for any x ∈ { }r1,0 , wt((x)) t + ...

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...then t + d n–1.sSarkar and Maitra [SarMai2000-1] have found an upper bound for the nonlinearitysof t-resilient Boolean functions and inspired a number of researchers ([SarMai20001], [SarMai2000-2], =-=[FedTar2001]-=-, [JohPas2003], [Tara2001], [PasJohMS2001],s[Mait2000]) for discovering functions satisfying that upper bound, which iss11 22 +− −≤ tnfnlsfor 22 −> ntsand 1121 222 +−− −−≤ t n n fnlsfor 22 −≤ nt . The...

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... the Constructed FunctionssIn this section, we make a comparison of the functions constructed in the previousschapters of this work, together with some other functions provided by some previoussworks =-=[KavYüc2003]-=-. Table 5-1 gives the comparison of functions in terms ofsnonlinearity, order of resiliency, absolute indicator and sum-of-squared-errors.sTable 5-1. Comparison of 8-Variable FunctionssFunctionsf nlfs...

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... and is given by PiotrsPorwik [Porw2003].ssGupta and Sarkar have introduced an alternative computation method for computingsWalsh-Hadamard values of a Boolean function from its algebraic normal forms=-=[GupSar2003]-=-. Although their method is good for calculating the Walsh-Hadamardstransform at a specific point, the algorithm does not find all values of the spectrum,swhich is necessary for finding the nonlinearit...