Kim-type APN functions are affine equivalent to Gold functions

09/13/2020
βˆ™
by   Benjamin Chase, et al.
βˆ™
0
βˆ™

The problem of finding APN permutations of 𝔽_2^n where n is even and n>6 has been called the Big APN Problem. Li, Li, Helleseth and Qu recently characterized APN functions defined on 𝔽_q^2 of the form f(x)=x^3q+a_1x^2q+1+a_2x^q+2+a_3x^3, where q=2^m and mβ‰₯ 4. We will call functions of this form Kim-type functions because they generalize the form of the Kim function that was used to construct an APN permutation of 𝔽_2^6. We extend the result of Li, Li, Helleseth and Qu by proving that if a Kim-type function f is APN and mβ‰₯ 4, then f is affine equivalent to one of two Gold functions G_1(x)=x^3 or G_2(x)=x^2^m-1+1. Combined with the recent result of GΓΆloğlu and Langevin who proved that, for even n, Gold APN functions are never CCZ equivalent to permutations, it follows that for mβ‰₯ 4 Kim-type APN functions on 𝔽_2^2m are never CCZ equivalent to permutations.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
βˆ™ 08/19/2020

On CCZ-equivalence of the inverse function

The inverse function x ↦ x^-1 on 𝔽_2^n is one of the most studied functi...
research
βˆ™ 06/23/2023

On the Functions Which are CCZ-equivalent but not EA-equivalent to Quadratic Functions over 𝔽_p^n

For a given function F from 𝔽_p^n to itself, determining whether there e...
research
βˆ™ 11/24/2022

A doubly-infinite family of 0-APN monomials

We consider an infinite family of exponents e(l,k) with two parameters, ...
research
βˆ™ 09/06/2022

Towards non-linear quadrature formulae

Prompted by an observation about the integral of exponential functions o...
research
βˆ™ 03/31/2022

An Affine Type System with Hindley-Milner Style Type Inference

This article first provides an algorithm W based type inference algorith...
research
βˆ™ 01/25/2022

Gold Functions and Switched Cube Functions Are Not 0-Extendable in Dimension n > 5

In the independent works by Kalgin and Idrisova and by Beierle, Leander ...
research
βˆ™ 02/26/2021

Recovering or Testing Extended-Affine Equivalence

Extended Affine (EA) equivalence is the equivalence relation between two...

Please sign up or login with your details

Forgot password? Click here to reset