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G-Perfect nonlinear functions

2007
*
Designs, Codes and Cryptography
*

We construct several examples of

doi:10.1007/s10623-007-9137-7
fatcat:iwtdsmek5nb3ppwcrbcyrl3u64
*G*-*perfect**nonlinear**functions*, both Z 2 -valued and Z a 2 -valued. ... The new cryptosystems, when combined with*G*-*perfect**nonlinear**functions*(similar to classical*perfect**nonlinear**functions*with one XOR replaced by a general group action), allow us to construct systems ... Moreover, note that |H| must divide |X| in order to have a*G*-*perfect**nonlinear**function*. We now consider the connection between*G*-*perfect**nonlinear**functions*and difference sets. ...##
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NON-BOOLEAN ALMOST PERFECT NONLINEAR FUNCTIONS ON NON-ABELIAN GROUPS

2011
*
International Journal of Foundations of Computer Science
*

The purpose of this paper is to present the extended definitions and characterizations of the classical notions of APN and maximum

doi:10.1142/s0129054111008751
fatcat:gdshihbsezh5zggjdgovwlapta
*nonlinear*Boolean*functions*to deal with the case of mappings from a finite ...*Perfect**Nonlinear**Functions*on Non-Abelian Groups 15Table 1. ... Moreover, if (K, N ) = (S 3 , Z 6 ) each almost*perfect**nonlinear**functions*is also maximal*nonlinear*. ...##
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Generalized Boolean Bent Functions
[chapter]

2004
*
Lecture Notes in Computer Science
*

In this paper we largely develop this concept to define

doi:10.1007/978-3-540-30556-9_10
fatcat:naj3lza7vfdztelg6gmglmcz3y
*G*-*perfect**nonlinearity*and*G*-bent*functions*, where*G*is an Abelian group of involutions, and to show their equivalence as in the classical case. ... The notions of*perfect**nonlinearity*and bent*functions*are closely dependent on the action of the group of translations over IF m 2 . ... A*function*f : IF m 2 −→ IF n 2 is*G*-*perfect**nonlinear*if ∆ f = 2 m−n . ...##
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Constructions of vectorial Boolean functions with good cryptographic properties

2016
*
Science China Information Sciences
*

The upper bound on the

doi:10.1007/s11432-015-0863-3
fatcat:uylw4di2vfeejmrowpso2kryoq
*nonlinearity*of (n, m)*functions*is 2 n−1 − 2 n/2−1 , and*functions*achieving this maximum*nonlinearity*are called*perfect**nonlinear**functions*[2]. ... Unfortunately,*perfect**nonlinear**functions*cannot be used directly because they are not balanced or correlation immune. ... Let*G*= (*g*1 ,*g*2 , . . . ,*g*m ) be an (n/2, m)*perfect**nonlinear**function*with*g*i (0 k ) = 0. Let C = {C 0 , C 1 , . . . , C 2 k } be a set of [n, k] disjoint linear codes. ...##
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Nonlinear functions in abelian groups and relative difference sets

2004
*
Discrete Applied Mathematics
*

During the past decade,

doi:10.1016/s0166-218x(03)00293-2
fatcat:neu65fwbcjfrhlptviyekfvkgm
*perfect*, almost*perfect*and maximum*nonlinear**functions*on ÿnite ÿelds have been thoroughly investigated. ... It is the purpose of this paper to show that the main results on*nonlinear**functions*can be easily generalized to the case of arbitrary abelian groups if the Walsh-Hadamard transform is replaced by the ... Therefore, we call a*function*f : K → N almost*perfect**nonlinear*if a;b [ f (a; b)] 2 6 a;b [*g*(a; b)] 2 ∀*g*: K → N (2) but f is not*perfect**nonlinear*. ...##
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Doubly Perfect Nonlinear Boolean Permutations
[article]

2010
*
arXiv
*
pre-print

We call them doubly

arXiv:1003.4919v1
fatcat:bqtrswsadzhydayfqh6ongod7a
*perfect**nonlinear*permutations. ... In this contribution we study the*functions*that offer the best resistance against a differential attack based on a finite field multiplication. ... A*function*f : X → H is called*perfect**nonlinear*(by respect to the action of*G*on X) or*G*-*perfect**nonlinear*if for each α ∈*G** , the derivative of f in direction α d α f : X → H x → f (α.x) − f (x) ( ...##
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Nonlinear functions and difference sets on group actions
[article]

2016
*
arXiv
*
pre-print

They have more interesting properties than

arXiv:1603.01016v1
fatcat:em3ygfwm6beq3ge2yydxnnkp4m
*perfect**nonlinear**functions*from*G*itself to H. ... Let*G*, H be finite groups and let X be a finite*G*-set.*G*-*perfect**nonlinear**functions*from X to H have been studied in several papers. ... of*G*-*perfect**nonlinear**functions*and*G*-bent*functions*on X(see Theorems 4.7, 5.4, 5.8, and 6.2) . ...##
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Doubly perfect nonlinear boolean permutations

2010
*
Journal of Discrete Mathematical Sciences and Cryptography
*

We call them doubly

doi:10.1080/09720529.2010.10698315
fatcat:ncbamitxfjb2dg43edtrs7w3sa
*perfect**nonlinear*permutations. ... In this contribution we study the*functions*that offer the best resistance against a differential attack based on a finite field multiplication. ... A*function*f : X → H is called*perfect**nonlinear*(by respect to the action of*G*on X) or*G*-*perfect**nonlinear*if for each α ∈*G** , the derivative of f in direction α d α f : X → H x → f (α.x) − f (x) ( ...##
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A new characterization of group action-based perfect nonlinearity

2009
*
Discrete Applied Mathematics
*

In this paper we show that this generalized concept of

doi:10.1016/j.dam.2009.02.001
fatcat:yz6nvtmohbbovbxidixfj7sjb4
*nonlinearity*is actually equivalent to a new bentness notion that deals with*functions*defined on a finite abelian group*G*that acts on a finite set ... The left-regular multiplication is explicitly embedded in the notion of*perfect**nonlinearity*. ... Note that in (3; 8), we have proved the existence of a*G*-*perfect**nonlinear**function*f : X → H such that it exists at least one x 0 ∈ X for which f (x 0 ) :*G*→ H is not classical*perfect**nonlinear*. ...##
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C-differential bent functions and perfect nonlinearity
[article]

2020
*
arXiv
*
pre-print

We further extend the notion of

arXiv:2006.12535v1
fatcat:eqau2dnew5cbfbz3sno3dvygaa
*perfect*c-*nonlinear*introduced in , also in two different ways, and show that, in both cases, the concepts of c-differential bent and*perfect*c-*nonlinear*are equivalent ... Drawing inspiration from Nyberg's paper on*perfect**nonlinearity*and the c-differential notion we defined in , in this paper we introduce the concept of c-differential bent*functions*in two different ways ... If m = n and δ = 1, then F is called a*perfect**nonlinear*(PN)*function*, or planar*function*. If m = n and δ = 2, then F is called an almost*perfect**nonlinear*(APN)*function*. ...##
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Highly nonlinear mappings

2004
*
Journal of Complexity
*

Ding / Journal of Complexity 20 (2004) 205-244 206 Known

doi:10.1016/j.jco.2003.08.008
fatcat:rtvmmkb3krg5vbjk6fuimswoqu
*perfect**nonlinear*power*functions*x s from GF ðp m Þ to GF ðp m Þ; where p42; are the following [23,45]: is odd, and gcdðm; kÞ ¼ 1: * s ¼ p m À ... We also present open problems about*functions*with high*nonlinearity*. r C. Carlet, C. ... We consider only the case B ¼ Z 2 Â Z 2 : For any affine*function*lðxÞ; gðxÞ ¼ f ðxÞ À lðxÞ must have*perfect**nonlinearity*P*g*¼ 1 4 as f ðxÞ has*perfect**nonlinearity*. ...##
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Cartesian authentication codes from functions with optimal nonlinearity

2003
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Theoretical Computer Science
*

In this paper, we present several classes of authentication codes using

doi:10.1016/s0304-3975(02)00077-4
fatcat:uppzjxq2trciza7x2yorltzaui
*functions*with*perfect**nonlinearity*and optimum*nonlinearity*. Some of the authentication codes are optimal. ... In many cases, almost*perfect**nonlinear**functions*have optimal*nonlinearity*as 2=|B| is the minimum possible value a*function*from A to B can take on. ... There are several classes of*functions*from GF(q) 2t to GF(q) with*perfect**nonlinearity*. When q = 2, they are in fact the bent*functions*[4, 23] . ...##
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Non Abelian Bent Functions
[article]

2010
*
arXiv
*
pre-print

*Perfect*

*nonlinear*

*functions*from a finite group

*G*to another one H are those

*functions*f:

*G*→ H such that for all nonzero α∈

*G*, the derivative d_αf: x f(α x) f(x)^-1 is balanced. ... In the case where both

*G*and H are Abelian groups, f:

*G*→ H is

*perfect*

*nonlinear*if and only if f is bent i.e for all nonprincipal character χ of H, the (discrete) Fourier transform of χ∘ f has a constant ... Accord- ing to lemma 4, we deduce that φ(ρ V ) = 0 End (V ) for all ρ V ∈

*G** . ⊓ ⊔ 4 On

*perfect*

*nonlinear*

*functions*Some basic definitions

*Perfect*

*nonlinearity*must be seen as the fundamental notion ...

##
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A family of skew Hadamard difference sets

2006
*
Journal of combinatorial theory. Series A
*

In this paper, we present a family of new

doi:10.1016/j.jcta.2005.10.006
fatcat:gsmhemrfungo7azlkeqthwxlxi
*perfect**nonlinear*(also called planar)*functions*, and construct a family of skew Hadamard difference sets using these*perfect**nonlinear**functions*. ... The class of new*perfect**nonlinear**functions*has applications in cryptography, coding theory, and combinatorics. ... program for proving the inequivalence of Image(*g*1 ) \ {0} and Q for the case m = 5. ...##
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Two characterizations of crooked functions
[article]

2007
*
arXiv
*
pre-print

We give two characterizations of crooked

arXiv:0704.1293v1
fatcat:gqaa32dck5al3iucdw4tufspcm
*functions*: one based on the minimum distance of a Preparata-like code, and the other based on the distance-regularity of a crooked graph. ... Almost*Perfect**Nonlinear**Functions*Before considering crooked*functions*we need to characterize a more general class, namely almost*perfect**nonlinear**functions*. ... However*functions*do exist in several lesser categories of*nonlinearity*, such as almost*perfect**nonlinear*, almost bent, and crooked. ...
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