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### Switching Construction of Planar Functions on Finite Fields [chapter]

Alexander Pott, Yue Zhou
2010 Lecture Notes in Computer Science
We show that some of the known planar functions can be constructed from each other by switching.  ...  has precisely one solution for all a, b ∈ Fpn , a = 0. In this paper, we discuss possible extensions of the switching idea developed in [1] to the case of planar functions.  ...  A finite field is a trivial example of a semifield. The first non-trivial examples were constructed by Dickson [2].  ...

### Another Class of Perfect Nonlinear Polynomial Functions

Menglong Su, Zhengbang Zha, Zhonghai Xu
2013 Mathematical Problems in Engineering
Perfect nonlinear (PN) functions have been an interesting subject of study for a long time and have applications in coding theory, cryptography, combinatorial designs, and so on.  ...  In this paper, the planarity of the trinomialsxpk+1+ux2+vx2pkover GF(p2k) are presented. This class of PN functions are all EA-equivalent tox2.  ...  This project was supported by NSFC-Union Science Foundation of Henan  ...

### Bundles, presemifields and nonlinear functions

K. J. Horadam, D. G. Farmer
2008 Designs, Codes and Cryptography
This technique produces functions with low differential uniformity, including PN functions ( p odd), and quadratic APN and differentially 4-uniform functions ( p = 2).  ...  We prove there are exactly p bundles of presemifields of order p 2 and give a representative of each.  ...  Acknowledgments Parts of this work form part of the Ph.D. thesis of the second author, taken under the supervision of the first author.  ...

### New Perfect Nonlinear Functions and Their Semifields [article]

Jinquan Luo, Junru Ma
2019 arXiv   pre-print
Furthermore, we investigate the nucleus of the corresponding semifields of these functions and show that the semifields are not isotopic to all the known semifields.  ...  In this paper, two new classes of perfect nonlinear functions over F_p^2m are proposed, where p is an odd prime.  ...  Note that CCZ-equivalent functions have equal differential uniformity and EA-equivalent is a particular case of CCZ-equivalence [9] .  ...

### New Perfect Nonlinear Functions over Finite Fields [article]

Jinquan Luo, Junru Ma, Min Tu
2019 arXiv   pre-print
In this paper we present a new class of perfect nonlinear polynomials over F_p^2k for any odd prime p.  ...  In addition, we show that the new perfect nonlinear functions are CCZ-inequivalent to all the previously known perfect nonlinear functions in general.  ...  Functions used as S-boxes are required to have low differential uniformity [4] , high nonlinearity [28] and high algebraic degree [22] .  ...

### Thermodynamic Semirings [article]

Matilde Marcolli, Ryan Thorngren
2012 arXiv   pre-print
The Witt construction describes a functor from the category of Rings to the category of characteristic 0 rings.  ...  Interestingly, they found that in characteristic one, the Witt construction depends critically on the Shannon entropy.  ...  This paper is based on the results of the second author's summer research project, supported by the Summer Undergraduate Research Fellowship program at Caltech.  ...

### Thermodynamic semirings

Matilde Marcolli, Ryan Thorngren
2014 Journal of Noncommutative Geometry
Besides the case of the Shannon entropy, which arises in the context of geometry over the field with one element and the Witt construction in characteristic one, there are other interesting thermodynamic  ...  A more general theory of thermodynamic semirings is then formulated in categorical terms, by encoding all partial associativity and commutativity constraints into an entropy operad and a corresponding  ...  This paper is based on the results of the second author's summer research project, supported by the Summer Undergraduate Research Fellowship program at Caltech.  ...

### Noncommutative Geometry, the spectral standpoint [article]

Alain Connes
2019 arXiv   pre-print
2) Advances on the Baum-Connes conjecture, on coarse geometry and on higher index theory, 3) The geometrization of the pseudo-differential calculi using smooth groupoids, 4) The development of Hopf cyclic  ...  Galois group, 7) The development of quantum field theory on noncommutative spaces, 8) The discovery of a simple equation whose irreducible representations correspond to 4-dimensional spin geometries with  ...  It is of characteristic 1, i.e. 1 ∨ 1 = 1 and contains the smallest semifield of characteristic 1, namely the Boolean semifield B = {0, 1}.  ...

### New families of perfect nonlinear polynomial functions

Zhengbang Zha, Xueli Wang
2009 Journal of Algebra
A new class of almost perfect nonlinear (APN) polynomial functions has been recently introduced.  ...  We give some generalizations of these functions and deduce new families of perfect nonlinear (PN) functions.  ...  Acknowledgments The authors would like to thank the anonymous referees for their helpful suggestions that much improved the organization and presentation of this paper.  ...

### Correspondence principle for idempotent calculus and some computer applications [article]

Grigori Litvinov, Victor Maslov
2001 arXiv   pre-print
construction of solutions.  ...  The theory is well advanced and includes, in particular, new integration theory, new linear algebra, spectral theory and functional analysis. It has a wide range of applications.  ...  Idempotent analogs for some basic ideas, constructions and results in Functional Analysis and Mathematical Physics are discussed from this point of view.  ...

### The Goldbach conjecture resulting from global-local cuspidal representations and deformations of Galois representations [article]

Christian Pierre
2008 arXiv   pre-print
In the basic general frame of the Langlands global program, a local p-adic elliptic semimodule corresponding to a local (left) cuspidal form is constructed from its global equivalent covered by p-th roots  ...  More particularly, the inverse quantum deformation of a closed curve responsible for its splitting directly leads to the Goldbach conjecture.  ...  ELLIP(2, x, K + p ) = ⊕ r λ 1 2 p (2, r, m r )(x) f (µ qr·r ) is constructed in one-to-one correspondence where x is a closed point of the finite Galois extension of the non archimedean p -adic left semifield  ...

### Skeleta in non-Archimedean and tropical geometry [article]

Andrew W. Macpherson
2017 arXiv   pre-print
The primary result of this paper is that the topological space underlying a non-Archimedean analytic space may locally be recovered from the sheaf of 'pointwise valuations' of its analytic functions.  ...  Skeleta are spaces equipped with a structure sheaf of topological semirings, and are locally modelled on the spectra of the same.  ...  The subset C 1 (X , R ∨ ) of differentiable functions is not a B-module, since the pointwise maximum f ∨ g of two differentiable functions f , g needn't be differentiable.  ...

### Skeleta in non-Archimedean and tropical geometry

Andrew W. Macpherson
2020 Annales de la Faculté des Sciences de Toulouse
I also thank the Cecil King foundation for funding my visit to Mark Gross in UCSD, where some of the aforementioned conversations, as well as part of the work writing this paper, occurred.  ...  The subset C 1 (X, R ∨ ) of differentiable functions is not a B-module, since the pointwise maximum f ∨ g of two differentiable functions f, g needn't be differentiable.  ...  The image of Trop(C/X/∆) in the Hausdorff quotient sk ∆ → ∆ ⊂ R 2 is the non-differentiability locus of the convex piecewise-affine function on ∆ defined by F .  ...

### Constructing APN functions through isotopic shifts [article]

Lilya Budaghyan, Marco Calderini, Claude Carlet, Robert S. Coulter, Irene Villa
2018 IACR Cryptology ePrint Archive
We apply the idea of isotopic equivalence to transform APN functions in characteristic 2 into other functions, some of which can be APN.  ...  In the case of quadratic planar functions, it is a particular case of isotopic equivalence.  ...  The existence of an involution in the additive group means such functions cannot exist in even characteristic; here, the best resistance belongs to functions that are differentially 2-uniform.  ...

### 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.  ...  Our final constructions demonstrate G-perfect nonlinear planar permutations (from Z a 2 to itself), thus providing an alternative implementation to current uses of almost perfect nonlinear functions.  ...  APN permutation when a is odd and differentially 4-uniform when a is even [27] .  ...
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