American Institute of Mathematical Sciences

Advanced
February  2009, 3(1): 59-81. doi: 10.3934/amc.2009.3.59

A new almost perfect nonlinear function which is not quadratic

Yves Edel 1, and  Alexander Pott 2,

1. 

Department of Pure Mathematics and Computer Algebra, Ghent University, Krijgslaan 281, S22, B-9000 Ghent, Belgium
2. 

Faculty of Mathematics, Otto-von-Guericke-University Magdeburg, D-39016 Magdeburg, Germany

Received  October 2008 Revised  January 2009 Published  January 2009

Following an example in [12], we show how to change one coordinate function of an almost perfect nonlinear (APN) function in order to obtain new examples. It turns out that this is a very powerful method to construct new APN functions. In particular, we show that our approach can be used to construct a ''non-quadratic'' APN function. This new example is in remarkable contrast to all recently constructed functions which have all been quadratic. An equivalent function has been found independently by Brinkmann and Leander [8]. However, they claimed that their function is CCZ equivalent to a quadratic one. In this paper we give several reasons why this new function is not equivalent to a quadratic one.
Keywords: equivalence of functions, almost bent., Walsh spectrum, Almost perfect nonlinear.
Mathematics Subject Classification: Primary: 11T71; Secondary: 05E20, 94A6.
Citation: Yves Edel, Alexander Pott. A new almost perfect nonlinear function which is not quadratic. Advances in Mathematics of Communications, 2009, 3 (1) : 59-81. doi: 10.3934/amc.2009.3.59
[1]

Yang Yang, Xiaohu Tang, Guang Gong. New almost perfect, odd perfect, and perfect sequences from difference balanced functions with d-form property. Advances in Mathematics of Communications, 2017, 11 (1) : 67-76. doi: 10.3934/amc.2017002
[2]

Markku Lehtinen, Baylie Damtie, Petteri Piiroinen, Mikko Orispää. Perfect and almost perfect pulse compression codes for range spread radar targets. Inverse Problems & Imaging, 2009, 3 (3) : 465-486. doi: 10.3934/ipi.2009.3.465
[3]

Sihem Mesnager, Fengrong Zhang. On constructions of bent, semi-bent and five valued spectrum functions from old bent functions. Advances in Mathematics of Communications, 2017, 11 (2) : 339-345. doi: 10.3934/amc.2017026
[4]

Marko Kostić. Almost periodic type functions and densities. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021008
[5]

Álvaro Castañeda, Gonzalo Robledo. Dichotomy spectrum and almost topological conjugacy on nonautonomus unbounded difference systems. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2287-2304. doi: 10.3934/dcds.2018094
[6]

Felipe García-Ramos, Brian Marcus. Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 729-746. doi: 10.3934/dcds.2019030
[7]

Ahmed Y. Abdallah. Attractors for first order lattice systems with almost periodic nonlinear part. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1241-1255. doi: 10.3934/dcdsb.2019218
[8]

Amira M. Boughoufala, Ahmed Y. Abdallah. Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1549-1563. doi: 10.3934/dcdsb.2020172
[9]

Benjamin Dodson. Improved almost Morawetz estimates for the cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2011, 10 (1) : 127-140. doi: 10.3934/cpaa.2011.10.127
[10]

Ying Zhang, Changjun Yu, Yingtao Xu, Yanqin Bai. Minimizing almost smooth control variation in nonlinear optimal control problems. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1663-1683. doi: 10.3934/jimo.2019023
[11]

Alp Eden, Elİf Kuz. Almost cubic nonlinear Schrödinger equation: Existence, uniqueness and scattering. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1803-1823. doi: 10.3934/cpaa.2009.8.1803
[12]

Jacques Wolfmann. Special bent and near-bent functions. Advances in Mathematics of Communications, 2014, 8 (1) : 21-33. doi: 10.3934/amc.2014.8.21
[13]

Tingting Pang, Nian Li, Li Zhang, Xiangyong Zeng. Several new classes of (balanced) Boolean functions with few Walsh transform values. Advances in Mathematics of Communications, 2021, 15 (4) : 757-775. doi: 10.3934/amc.2020095
[14]

Chao Wang, Ravi P Agarwal. Almost automorphic functions on semigroups induced by complete-closed time scales and application to dynamic equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 781-798. doi: 10.3934/dcdsb.2019267
[15]

Michel Cahen, Simone Gutt, John Rawnsley. On twistor almost complex structures. Journal of Geometric Mechanics, 2021, 13 (3) : 313-331. doi: 10.3934/jgm.2021006
[16]

Bimal Mandal, Aditi Kar Gangopadhyay. A note on generalization of bent boolean functions. Advances in Mathematics of Communications, 2021, 15 (2) : 329-346. doi: 10.3934/amc.2020069
[17]

Claude Carlet, Fengrong Zhang, Yupu Hu. Secondary constructions of bent functions and their enforcement. Advances in Mathematics of Communications, 2012, 6 (3) : 305-314. doi: 10.3934/amc.2012.6.305
[18]

Sihem Mesnager, Fengrong Zhang, Yong Zhou. On construction of bent functions involving symmetric functions and their duals. Advances in Mathematics of Communications, 2017, 11 (2) : 347-352. doi: 10.3934/amc.2017027
[19]

Ayça Çeşmelioğlu, Wilfried Meidl. Bent and vectorial bent functions, partial difference sets, and strongly regular graphs. Advances in Mathematics of Communications, 2018, 12 (4) : 691-705. doi: 10.3934/amc.2018041
[20]

Nikolaos Bournaveas. Local well-posedness for a nonlinear dirac equation in spaces of almost critical dimension. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 605-616. doi: 10.3934/dcds.2008.20.605

2020 Impact Factor: 0.935

Tools

Metrics

  • PDF downloads (279)
  • HTML views (0)
  • Cited by (68)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences