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| [[File:Boolean functions like 1000 nonlinearity.svg|thumb|2-ary bent functions with [[Hamming weight]] 1<br><br>Their nonlinearity is <br><math>2^{2-1} - 2^{\frac{2}{2}-1} = 2-1 = 1</math>]]
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| [[File:0001 0001 0001 1110 nonlinearity.svg|thumb|The nonlinearity of the bent function <math>x_1 x_2~+~x_3 x_4</math> is <br><math>2^{4-1} - 2^{\frac{4}{2}-1} = 8-2 = 6</math><br><br><math>~+~</math> stands for the [[exclusive or]]<br>(compare [[algebraic normal form]])]]
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| In the [[mathematics|mathematical]] field of [[combinatorics]], a '''bent function''' is a special type of [[Boolean function]]. Defined and named in the 1960s by [[Oscar Rothaus]] in research not published until 1976,<ref name=rothaus/> bent functions are so called because they are as different as possible from all [[linear map|linear]] and [[affine function]]s. They have been extensively studied for their applications in [[cryptography]], but have also been applied to [[spread spectrum]], [[coding theory]], and [[combinatorial design]]. The definition can be extended in several ways, leading to different classes of generalized bent functions that share many of the useful properties of the original.
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| == Walsh transform ==
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| Bent functions are defined in terms of the [[Walsh transform]]. The Walsh transform of a Boolean function {{nowrap|''ƒ'': '''Z'''{{sup sub|n|2}} → '''Z'''<sub>2</sub>}} is the function <math>\hat{f}:\Z_2^n \to \Z</math> given by
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| :<math> \hat{f}(a) = \sum_{\scriptstyle{x \in \Z_2^n}} (-1)^{f(x) + a \cdot x} </math>
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| where {{nowrap|1=''a'' · ''x'' = ''a''<sub>1</sub>''x''<sub>1</sub> + ''a''<sub>2</sub>''x''<sub>2</sub> + … + ''a''<sub>''n''</sub>''x''<sub>''n''</sub> (mod 2)}} is the [[dot product]] in '''Z'''{{sup sub|n|2}}.<ref name=bool/> Alternatively, let
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| {{nowrap|1=''S''<sub>0</sub>(''a'') = { ''x'' ∈ '''Z'''{{sup sub|n|2}} : ''ƒ''(''x'') = ''a'' · ''x'' } }} and
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| {{nowrap|1=''S''<sub>1</sub>(''a'') = { ''x'' ∈ '''Z'''{{sup sub|n|2}} : ''ƒ''(''x'') ≠ ''a'' · ''x'' } }}.
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| Then {{nowrap|1={{!}}''S''<sub>0</sub>(''a''){{!}} + {{!}}''S''<sub>1</sub>(''a''){{!}} = 2<sup>''n''</sup>}} and hence
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| :<math> \hat{f}(a) = |S_0(a)| - |S_1(a)| = 2 |S_0(a)| - 2^n. </math>
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| For any Boolean function ''ƒ'' and {{nowrap|''a'' ∈ '''Z'''{{sup sub|n|2}}}} the transform lies in the range
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| :<math> -2^n \leq \hat{f}(a) \leq 2^n. </math>
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| Moreover, the linear function
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| {{nowrap|1=''ƒ''<sub>0</sub>(''x'') = ''a'' · ''x''}}
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| and the affine function
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| {{nowrap|1=''ƒ''<sub>1</sub>(''x'') = ''a'' · ''x'' + 1}}
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| correspond to the two extreme cases, since
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| :<math>
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| \hat{f}_0(a) = 2^n,~
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| \hat{f}_1(a) = -2^n.
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| </math>
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| Thus, for each {{nowrap|''a'' ∈ '''Z'''{{sup sub|n|2}}}} the value of <math>\hat{f}(a)</math> characterizes where the function ''ƒ''(''x'') lies in the range from ''ƒ''<sub>0</sub>(''x'') to ''ƒ''<sub>1</sub>(''x'').
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| == Definition and properties ==
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| Rothaus defined a '''bent function''' as a Boolean function {{nowrap|''ƒ'': '''Z'''{{sup sub|n|2}} → '''Z'''<sub>2</sub>}} whose [[Walsh transform]] has constant [[absolute value]]. Bent functions are in a sense equidistant from all the affine functions, so they are equally hard to approximate with any affine function.
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| The simplest examples of bent functions, written in [[algebraic normal form]], are ''F''(''x''<sub>1</sub>,''x''<sub>2</sub>) =
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| ''x''<sub>1</sub>''x''<sub>2</sub> and ''G''(''x''<sub>1</sub>,''x''<sub>2</sub>,''x''<sub>3</sub>,''x''<sub>4</sub>) =
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| ''x''<sub>1</sub>''x''<sub>2</sub> + ''x''<sub>3</sub>''x''<sub>4</sub>. This pattern continues:
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| ''x''<sub>1</sub>''x''<sub>2</sub> + ''x''<sub>3</sub>''x''<sub>4</sub> + ... + ''x''<sub>''n'' − 1</sub>''x''<sub>''n''</sub> is a bent function {{nowrap|'''Z'''{{sup sub|n|2}} → '''Z'''<sub>2</sub>}} for every even ''n'', but there is a wide variety of different types of bent functions as ''n'' increases.<ref name=nonlin/> The sequence of values (−1)<sup>''ƒ''(''x'')</sup>, with {{nowrap|''x'' ∈ '''Z'''{{sup sub|n|2}}}} taken in [[lexicographical order]], is called a '''bent sequence'''; bent functions and bent sequences have equivalent
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| properties. In this ±1 form, the Walsh transform is easily computed as
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| :<math> \hat{f}(a) = W(2^n) (-1)^{f(a)}, </math>
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| where ''W''(2<sup>''n''</sup>) is the natural-ordered [[Walsh matrix]] and the sequence is treated as a [[column vector]].<ref name=dual/>
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| Rothaus proved that bent functions exist only for even ''n'', and that for a bent function ''ƒ'',
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| <math>|\hat{f}(a)|=2^{n/2}</math> for all {{nowrap|''a'' ∈ '''Z'''{{sup sub|n|2}}}}.<ref name=bool/> In fact, <math>\hat{f}(a)=2^{n/2}(-1)^{g(a)}</math>, where ''g'' is also bent. In this case,
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| <math>\hat{g}(a)=2^{n/2}(-1)^{f(a)}</math>, so ''ƒ'' and ''g'' are considered [[duality (mathematics)|dual]] functions.<ref name=dual/>
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| Every bent function has a [[Hamming weight]] (number of times it takes the value 1) of {{nowrap|2<sup>n − 1</sup> ± 2<sup>n/2 − 1</sup>}}, and in fact agrees with any affine function at one of those two numbers of points. So the ''nonlinearity'' of ''ƒ'' (minimum number of times it equals any affine function) is {{nowrap|2<sup>''n'' − 1</sup> − 2<sup>''n''/2 − 1</sup>}}, the maximum possible. Conversely, any Boolean function with nonlinearity {{nowrap|2<sup>''n'' − 1</sup> − 2<sup>''n''/2 − 1</sup>}} is bent.<ref name=bool/> The [[Degree of a polynomial|degree]] of ''ƒ'' in algebraic normal form (called the ''nonlinear order'' of ''ƒ'') is at most ''n''/2 (for ''n'' > 2).<ref name=nonlin/>
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| Although bent functions are vanishingly rare among Boolean functions of many variables, they come
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| in many different kinds. There has been detailed research into special classes of bent functions,
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| such as the [[homogeneous polynomial|homogeneous]] ones<ref name=homo/> or those arising from
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| a [[monomial]] over a [[finite field]],<ref name=mono/> but so far the bent functions have defied
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| all attempts at a complete enumeration or classification.
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| == Applications ==
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| As early as 1982 it was discovered that [[maximum length sequence]]s based on bent functions have [[cross-correlation]] and [[autocorrelation]] properties rivalling those of the [[Gold code]]s and [[Kasami code]]s for use in [[CDMA]].<ref name=seq/> These sequences have several applications in [[spread spectrum]] techniques.
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| The properties of bent functions are naturally of interest in modern digital [[cryptography]], which seeks to obscure relationships between input and output. By 1988 Forré recognized that the Walsh transform of a function can be used to show that it satisfies the [[Strict Avalanche Criterion]] (SAC) and higher-order generalizations, and recommended this tool to select candidates for good [[S-box]]es achieving near-perfect [[confusion and diffusion|diffusion]].<ref name=spectral/> Indeed, the functions satisfying the SAC to the highest possible order are always bent.<ref name=sac/> Furthermore, the bent functions are as far as possible from having what are called ''linear structures'', nonzero vectors a such that ''ƒ''(''x''+''a'') + ''ƒ''(''x'') is a constant. In the language of [[differential cryptanalysis]] (introduced after this property was discovered) the ''derivative'' of a bent function ''ƒ'' at every nonzero point ''a'' (that is, ''ƒ''<sub>''a''</sub>(''x'') = ''ƒ''(''x''+''a'') + ''ƒ''(''x'')) is a [[balanced boolean function|''balanced'']] Boolean function, taking on each value exactly half of the time. This property is called ''perfect nonlinearity''.<ref name=nonlin/>
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| Given such good diffusion properties, apparently perfect resistance to differential cryptanalysis, and resistance by definition to [[linear cryptanalysis]], bent functions might at first seem the ideal choice for secure cryptographic functions such as S-boxes. Their fatal flaw is that they fail to be balanced. In particular, an invertible S-box cannot be constructed directly from bent functions, and a [[stream cipher]] using a bent combining function is vulnerable to a [[correlation attack]]. Instead, one might start with a bent function and randomly complement appropriate values until the result is balanced. The modified function still has high nonlinearity, and as such functions are very rare the process should be much faster than a brute-force search.<ref name=nonlin/> But functions produced in this way may lose other desirable properties, even failing to satisfy the SAC—so careful testing is necessary.<ref name=sac/> A number of cryptographers have worked on techniques for generating balanced functions that preserve as many of the good cryptographic qualities of bent functions as possible.<ref name=nyberg/><ref name=highly/><ref name=cast/>
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| Some of this theoretical research has been incorporated into real cryptographic algorithms. The ''CAST'' design procedure, used by [[Carlisle Adams]] and [[Stafford Tavares]] to construct the S-boxes for the [[block ciphers]] [[CAST-128]] and [[CAST-256]], makes use of bent functions.<ref name=cast/> The [[cryptographic hash function]] [[HAVAL]] uses Boolean functions built from representatives of all four of the [[equivalence class]]es of bent functions on six variables.<ref name=haval/> The stream cipher [[Grain (cipher)|Grain]] uses an [[NLFSR]] whose nonlinear feedback polynomial is, by design, the sum of a bent function and a linear function.<ref name=grain/>
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| == Generalizations ==
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| The most common class of ''generalized bent functions'' is the [[modular arithmetic|mod m]] type,
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| <math>f:\mathbb{Z}_m^n \to \mathbb{Z}_m</math> such that
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| :<math> \hat{f}(a) = \sum_{x \in \mathbb{Z}_m^n} e^{\frac{2 \pi i}{m} (f(x) - a \cdot x)} </math>
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| has constant absolute value ''m''<sup>''n''/2</sup>. Perfect nonlinear functions <math>f:\mathbb{Z}_m^n \to \mathbb{Z}_m</math>, those such that for all nonzero ''a'', ''ƒ''(''x''+''a'') − ''ƒ''(''a'') takes on each value {{nowrap|''m''<sup>''n'' − 1</sup>}} times, are generalized bent. If ''m'' is [[prime number|prime]], the converse is true. In most cases only prime ''m'' are considered. For odd prime ''m'', there are generalized bent functions for every positive ''n'', even and odd. They have many of the same good cryptographic properties as the binary bent functions.<ref name=nyberg2/>
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| '''Semi-bent functions''' are an odd-order counterpart to bent functions. A semi-bent function is | |
| <math>f:\mathbb{Z}_m^n \to \mathbb{Z}_m</math> with ''n'' odd, such that <math>|\hat{f}|</math> takes only the values 0 and ''m''<sup>(''n''+1)/2</sup>. They also have good cryptographic characteristics, and some of them are balanced, taking on all possible values equally often.<ref name=semi/>
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| The '''partially bent functions''' form a large class defined by a condition on the Walsh transform and autocorrelation functions. All affine and bent functions are partially bent. This is in turn a proper subclass of the ''plateaued functions''.<ref name=plat/>
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| The idea behind the '''hyper-bent functions''' is to maximize the minimum distance to ''all'' Boolean
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| functions coming from [[bijection|bijective]] monomials on the finite field ''GF''(2<sup>''n''</sup>), not just the affine functions. For these functions this distance is constant, which may make them resistant to an [[interpolation attack]].
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| Other related names have been given to cryptographically important classes of functions {{nowrap|'''Z'''{{sup sub|n|2}} → '''Z'''{{sup sub|n|2}}}}, such as '''almost bent functions''' and '''crooked functions'''. While not Boolean functions themselves, these are closely related to the bent functions and have good nonlinearity properties.
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| == References ==
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| {{Reflist|2|refs=
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| <ref name=rothaus>{{Cite journal | author = O. S. Rothaus | title = On "Bent" Functions | journal = [[Journal of Combinatorial Theory]], Series A | issn = 0097-3165 | volume = 20 | issue = 3 | pages = 300–305 |date=May 1976 | url = https://www.sciencedirect.com/science/article/pii/0097316576900248 | accessdate = 16 December 2013}}</ref>
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| <ref name=bool>{{Cite journal | author = C. Qu | coauthors = [[Jennifer Seberry|J. Seberry]], T. Xia
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| | date=29 December 2001| title = Boolean Functions in Cryptography | url = http://citeseer.ist.psu.edu/old/700097.html | accessdate = 14 September 2009}}</ref>
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| <ref name=nonlin>{{cite conference | author = W. Meier | coauthors = O. Staffelbach
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| | title = Nonlinearity Criteria for Cryptographic Functions | conference = [[Eurocrypt]] '89
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| |date=April 1989 | pages = 549–562}}</ref>
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| <ref name=dual>{{cite conference | author = C. Carlet | coauthors = L.E. Danielsen, M.G. Parker, P. Solé | title = Self Dual Bent Functions
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| | conference = Fourth International Workshop on Boolean Functions: Cryptography and Applications (BFCA '08)
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| | url = http://www.ii.uib.no/~matthew/bfcasdb.pdf | date = 19 May 2008
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| | accessdate = 21 September 2009}}</ref>
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| <ref name=homo>{{cite journal | author = T. Xia | coauthors = J. Seberry, [[Josef Pieprzyk|J. Pieprzyk]], C. Charnes | title = Homogeneous bent functions of degree n in 2n variables do not exist for n > 3 | journal = Discrete Applied Mathematics | issn = 0166-218X | volume = 142
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| | issue = 1–3 | pages = 127–132 |date=June 2004 | url = http://ro.uow.edu.au/infopapers/291/ | accessdate = 21 September 2009 | doi = 10.1016/j.dam.2004.02.006}}</ref>
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| <ref name=mono>{{cite journal | author = A. Canteaut| coauthors = P. Charpin, G. Kyureghyan | |
| | title = A new class of monomial bent functions | journal = Finite Fields and Their Applications
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| | issn = 1071-5797 | volume = 14 | issue = 1 | pages = 221–241 |date=January 2008 | url = http://www-roc.inria.fr/secret/Anne.Canteaut/Publications/CanChaKuy07.pdf
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| | format = PDF | accessdate = 21 September 2009 | doi = 10.1016/j.ffa.2007.02.004}}</ref>
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| <ref name=seq>{{cite journal | author = J. Olsen | coauthors = R. Scholtz, L. Welch
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| | title = Bent-Function Sequences | journal = [[IEEE Transactions on Information Theory]]
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| | issn = 0018-9448 | volume = IT-28 | issue = 6 | pages = 858–864 |date=November 1982 | url = http://www.costasarrays.org/costasrefs/b2hd-olsen82bent-function.html
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| | accessdate = 24 September 2009}}</ref>
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| <ref name=spectral>{{cite conference | author = R. Forré | title = The Strict Avalanche Criterion: Spectral Properties of Boolean Functions and an Extended Definition | conference = [[CRYPTO]] '88
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| |date=August 1988 | pages = 450–468}}</ref>
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| <ref name=sac>{{Cite journal | author = [[Carlisle Adams|C. Adams]] | coauthors = [[Stafford Tavares|S. Tavares]] | title = The Use of Bent Sequences to Achieve Higher-Order Strict Avalanche Criterion in S-Box Design | version = Technical Report TR 90-013
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| | publisher = [[Queen's University]] |date=January 1990
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| | id = {{citeseerx|10.1.1.41.8374}}
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| | accessdate = 23 September 2009}}</ref>
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| <ref name=nyberg>{{cite conference | author = [[Kaisa Nyberg|K. Nyberg]] | title = Perfect nonlinear S-boxes | conference = Eurocrypt '91 |date=April 1991
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| | pages = 378–386}}</ref>
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| <ref name=highly>{{cite conference | author = J. Seberry | coauthors = X. Zhang | title = Highly Nonlinear 0-1 Balanced Boolean Functions Satisfying Strict Avalanche Criterion
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| | conference = [[AUSCRYPT]] '92 |date=December 1992 | pages = 143–155
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| | id = {{citeseerx|10.1.1.57.4992}}
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| | accessdate = 24 September 2009}}</ref>
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| <ref name=cast>{{cite journal | author = C. Adams | title = Constructing Symmetric Ciphers Using the CAST Design Procedure | journal = Designs, Codes, and Cryptography | issn = 0925-1022
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| | volume = 12 | issue = 3 | pages = 283–316 |date=November 1997
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| | url = http://jya.com/cast.html | accessdate = 20 September 2009 | doi = 10.1023/A:1008229029587}}</ref>
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| <ref name=haval>{{cite conference | author = [[Yuliang Zheng|Y. Zheng]] | coauthors = J. Pieprzyk,
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| J. Seberry | title = HAVAL—a one-way hashing algorithm with variable length of output
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| | conference = AUSCRYPT '92 |date=December 1992 | pages = 83–104
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| | url = http://labs.calyptix.com/files/haval-paper.pdf | format = PDF
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| | accessdate = 24 September 2009}}</ref>
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| <ref name=grain>{{Cite journal | author = M. Hell | coauthors = T. Johansson, A. Maximov, W. Meier
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| | title = A Stream Cipher Proposal: Grain-128
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| | url = http://www.ecrypt.eu.org/stream/p2ciphers/grain/Grain128_p2.pdf | format = PDF
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| | accessdate = 24 September 2009}}</ref>
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| <ref name=nyberg2>{{cite conference | author = K. Nyberg | title = Constructions of bent functions and difference sets | conference = Eurocrypt '90 |date=May 1990
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| | pages = 151–160}}
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| </ref>
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| <ref name=semi>{{cite journal | author = K. Khoo | coauthors = G. Gong, [[Doug Stinson|D. Stinson]] | title = A new characterization of bent and semi-bent functions on finite fields
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| | journal = Designs, Codes, and Cryptography | issn = 0925-1022 | volume = 38 | issue = 2
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| | pages = 279–295 |date=February 2006
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| | url = http://www.cacr.math.uwaterloo.ca/~dstinson/papers/dcc-final.ps | format = [[PostScript]]
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| | accessdate = 24 September 2009 | doi = 10.1007/s10623-005-6345-x}}</ref>
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| <ref name=plat>{{cite conference | author = Y. Zheng | coauthors = X. Zhang | title = Plateaued Functions | conference = Second International Conference on Information and Communication
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| Security (ICICS '99) |date=November 1999 | pages = 284–300
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| | url = http://citeseer.ist.psu.edu/old/291018.html | accessdate = 24 September 2009}}</ref>
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| }}
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| == Further reading ==
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| * {{cite conference | author = C. Carlet | title = Two New Classes of Bent Functions
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| | conference = Eurocrypt '93 |date=May 1993 | pages = 77–101}}
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| * {{cite journal | author = J. Seberry | coauthors = X. Zhang | title = Constructions of Bent Functions from Two Known Bent Functions | journal = Australasian Journal of Combinatorics
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| | issn = 1034-4942 | volume = 9 | pages = 21–35 |date=March 1994
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| | id = {{citeseerx|10.1.1.55.531}}
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| | accessdate = 17 September 2009}}
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| * {{Cite journal | author = T. Neumann | coauthors = advisor: U. Dempwolff | title = Bent Functions
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| |date=May 2006 | id = {{citeseerx|10.1.1.85.8731}}
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| | accessdate = 23 September 2009}}
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| * {{cite book | first1 = Charles J. | last1 = Colbourn | author1-link = Charles Colbourn | first2 = Jeffrey H. | last2 = Dinitz | author2-link = Jeff Dinitz | title = Handbook of Combinatorial Designs | edition = 2nd | publisher = [[CRC Press]] | year = 2006 | isbn = 978-1-58488-506-1 | pages = 337–339}}
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| {{DEFAULTSORT:Bent Function}}
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| [[Category:Boolean algebra]]
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| [[Category:Combinatorics]]
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| [[Category:Symmetric-key cryptography]]
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| [[Category:Theory of cryptography]]
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