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G-Perfect nonlinear functions

James A. Davis, Laurent Poinsot
2007 Designs, Codes and Cryptography  
We construct several examples of 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.  ... 
doi:10.1007/s10623-007-9137-7 fatcat:iwtdsmek5nb3ppwcrbcyrl3u64

NON-BOOLEAN ALMOST PERFECT NONLINEAR FUNCTIONS ON NON-ABELIAN GROUPS

LAURENT POINSOT, ALEXANDER POTT
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 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.  ... 
doi:10.1142/s0129054111008751 fatcat:gdshihbsezh5zggjdgovwlapta

Generalized Boolean Bent Functions [chapter]

Laurent Poinsot, Sami Harari
2004 Lecture Notes in Computer Science  
In this paper we largely develop this concept to define 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 .  ... 
doi:10.1007/978-3-540-30556-9_10 fatcat:naj3lza7vfdztelg6gmglmcz3y

Constructions of vectorial Boolean functions with good cryptographic properties

Luyang Li, Weiguo Zhang
2016 Science China Information Sciences  
The upper bound on the 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.  ... 
doi:10.1007/s11432-015-0863-3 fatcat:uylw4di2vfeejmrowpso2kryoq

Nonlinear functions in abelian groups and relative difference sets

Alexander Pott
2004 Discrete Applied Mathematics  
During the past decade, 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.  ... 
doi:10.1016/s0166-218x(03)00293-2 fatcat:neu65fwbcjfrhlptviyekfvkgm

Doubly Perfect Nonlinear Boolean Permutations [article]

Laurent Poinsot
2010 arXiv   pre-print
We call them doubly 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) (  ... 
arXiv:1003.4919v1 fatcat:bqtrswsadzhydayfqh6ongod7a

Nonlinear functions and difference sets on group actions [article]

Yun Fan, Bangteng Xu
2016 arXiv   pre-print
They have more interesting properties than 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) .  ... 
arXiv:1603.01016v1 fatcat:em3ygfwm6beq3ge2yydxnnkp4m

Doubly perfect nonlinear boolean permutations

Laurent Poinsot
2010 Journal of Discrete Mathematical Sciences and Cryptography  
We call them doubly 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) (  ... 
doi:10.1080/09720529.2010.10698315 fatcat:ncbamitxfjb2dg43edtrs7w3sa

A new characterization of group action-based perfect nonlinearity

Laurent Poinsot
2009 Discrete Applied Mathematics  
In this paper we show that this generalized concept of 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.  ... 
doi:10.1016/j.dam.2009.02.001 fatcat:yz6nvtmohbbovbxidixfj7sjb4

C-differential bent functions and perfect nonlinearity [article]

Pantelimon Stanica, Sugata Gangopadhyay, Aaron Geary, Constanza Riera, Anton Tkachenko
2020 arXiv   pre-print
We further extend the notion of 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.  ... 
arXiv:2006.12535v1 fatcat:eqau2dnew5cbfbz3sno3dvygaa

Highly nonlinear mappings

Claude Carlet, Cunsheng Ding
2004 Journal of Complexity  
Ding / Journal of Complexity 20 (2004) 205-244 206 Known 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.  ... 
doi:10.1016/j.jco.2003.08.008 fatcat:rtvmmkb3krg5vbjk6fuimswoqu

Cartesian authentication codes from functions with optimal nonlinearity

Samuel Chanson, Cunsheng Ding, Arto Salomaa
2003 Theoretical Computer Science  
In this paper, we present several classes of authentication codes using 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] .  ... 
doi:10.1016/s0304-3975(02)00077-4 fatcat:uppzjxq2trciza7x2yorltzaui

Non Abelian Bent Functions [article]

Laurent Poinsot
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  ... 
arXiv:1012.4079v1 fatcat:fsztbq4jmff6rdle2le7hvgx4m

A family of skew Hadamard difference sets

Cunsheng Ding, Jin Yuan
2006 Journal of combinatorial theory. Series A  
In this paper, we present a family of new 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.  ... 
doi:10.1016/j.jcta.2005.10.006 fatcat:gsmhemrfungo7azlkeqthwxlxi

Two characterizations of crooked functions [article]

Aidan Roy, Chris Godsil
2007 arXiv   pre-print
We give two characterizations of crooked 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.  ... 
arXiv:0704.1293v1 fatcat:gqaa32dck5al3iucdw4tufspcm
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