## Bounds on the degree of APN polynomials: the case of x −1 + g(x)

Publication: Research - peer-reviewJournal article – Annual report year: 2011

### Standard

Bounds on the degree of APN polynomials: the case of x −1 + g(x). / Leander, Gregor; Rodier, François.

In: Designs, Codes and Cryptography, Vol. 59, No. 1-3, 2011, p. 207-222.

Publication: Research - peer-reviewJournal article – Annual report year: 2011

### Author

Leander, Gregor; Rodier, François / Bounds on the degree of APN polynomials: the case of x −1 + g(x).

In: Designs, Codes and Cryptography, Vol. 59, No. 1-3, 2011, p. 207-222.

Publication: Research - peer-reviewJournal article – Annual report year: 2011

### Bibtex

title = "Bounds on the degree of APN polynomials: the case of x −1 + g(x)",
abstract = "In this paper we consider APN functions $${f:\mathcal{F}_{2^m}\to \mathcal{F}_{2^m}}$$ of the form f(x) = x −1 + g(x) where g is any non $${\mathcal{F}_{2}}$$-affine polynomial. We prove a lower bound on the degree of the polynomial g. This bound in particular implies that such a function f is APN on at most a finite number of fields $${\mathcal{F}_{2^m}}$$. Furthermore we prove that when the degree of g is less than 7 such functions are APN only if m ≤ 3 where these functions are equivalent to x 3.",
author = "Gregor Leander and François Rodier",
year = "2011",
doi = "10.1007/s10623-010-9456-y",
volume = "59",
pages = "207--222",
journal = "Designs, Codes and Cryptography",
issn = "0925-1022",
publisher = "Springer New York LLC",
number = "1-3",

}

### RIS

TY - JOUR

T1 - Bounds on the degree of APN polynomials: the case of x −1 + g(x)

AU - Leander,Gregor

AU - Rodier,François

PY - 2011

Y1 - 2011

N2 - In this paper we consider APN functions $${f:\mathcal{F}_{2^m}\to \mathcal{F}_{2^m}}$$ of the form f(x) = x −1 + g(x) where g is any non $${\mathcal{F}_{2}}$$-affine polynomial. We prove a lower bound on the degree of the polynomial g. This bound in particular implies that such a function f is APN on at most a finite number of fields $${\mathcal{F}_{2^m}}$$. Furthermore we prove that when the degree of g is less than 7 such functions are APN only if m ≤ 3 where these functions are equivalent to x 3.

AB - In this paper we consider APN functions $${f:\mathcal{F}_{2^m}\to \mathcal{F}_{2^m}}$$ of the form f(x) = x −1 + g(x) where g is any non $${\mathcal{F}_{2}}$$-affine polynomial. We prove a lower bound on the degree of the polynomial g. This bound in particular implies that such a function f is APN on at most a finite number of fields $${\mathcal{F}_{2^m}}$$. Furthermore we prove that when the degree of g is less than 7 such functions are APN only if m ≤ 3 where these functions are equivalent to x 3.

U2 - 10.1007/s10623-010-9456-y

DO - 10.1007/s10623-010-9456-y

M3 - Journal article

VL - 59

SP - 207

EP - 222

JO - Designs, Codes and Cryptography

T2 - Designs, Codes and Cryptography

JF - Designs, Codes and Cryptography

SN - 0925-1022

IS - 1-3

ER -