## CryptoDB

### Lilya Budaghyan

#### Publications

**Year**

**Venue**

**Title**

2010

EPRINT

On isotopisms of commutative presemifields and CCZ-equivalence of functions
Abstract

A function $F$ from \textbf{F}$_{p^n}$ to itself is planar if for any $a\in$\textbf{F}$_{p^n}^*$ the function $F(x+a)-F(x)$ is a permutation. CCZ-equivalence is the most general known equivalence relation of functions preserving planar property. This paper considers two possible extensions of CCZ-equivalence for functions over fields of odd characteristics, one proposed by Coulter and Henderson and the other by Budaghyan and Carlet, and we show that they in fact coincide with CCZ-equivalence. We prove that two finite commutative presemifields of odd order are isotopic if and only if they are strongly isotopic. This result implies that two isotopic commutative presemifields always define CCZ-equivalent planar functions (this was unknown for the general case). Further we prove that, for any odd prime $p$ and any positive integers $n$ and $m$, the indicators of the graphs of functions $F$ and $F'$ from \textbf{F}$_{p^n}$ to \textbf{F}$_{p^m}$ are CCZ-equivalent if and only if $F$ and $F'$ are CCZ-equivalent.
We also prove that, for any odd prime $p$, CCZ-equivalence of functions from \textbf{F}$_{p^n}$ to \textbf{F}$_{p^m}$, is strictly more general than EA-equivalence when $n\ge3$ and $m$ is greater or equal to the smallest positive divisor of $n$ different from 1.

2009

EPRINT

CCZ-equivalence and Boolean functions
Abstract

We study further CCZ-equivalence of $(n,m)$-functions. We prove that
for Boolean functions (that is, for $m=1$), CCZ-equivalence coincides
with EA-equivalence. On the contrary, we show that for $(n,m)$-
functions, CCZ-equivalence is strictly more general than EA-equivalence when $n\ge5$ and $m$ is greater or equal to the smallest
positive divisor of $n$ different from 1.
Our result on Boolean functions allows us to study the natural
generalization of CCZ-equivalence corresponding to the CCZ-equivalence
of the indicators of the graphs of the functions. We show that it
coincides with CCZ-equivalence.

2009

EPRINT

On CCZ-equivalence and its use in secondary constructions of bent functions
Abstract

We prove that, for bent vectorial functions, CCZ-equivalence coincides with EA-equivalence.
However, we show that CCZ-equivalence can be used for constructing bent functions which are new up to CCZ-equivalence. Using this approach we construct classes of nonquadratic bent Boolean and bent vectorial functions.

2009

EPRINT

New commutative semifields defined by PN multinomials
Abstract

We introduce infinite families of perfect nonlinear Dembowski-Ostrom multinomials over $F_{p^{2k}}$ where $p$ is any odd prime. We prove that for $k$ odd and $p\ne3$ these PN functions define new commutative semifields (in part by studying the nuclei of these semifields). This implies that these functions are CCZ-inequivalent to all previously known PN mappings.

2007

EPRINT

The simplest method for constructing APN polynomials EA-inequivalent to power functions
Abstract

The first APN polynomials EA-inequivalent to power functions have been constructed in [1,2] by applying CCZ-equivalence to the Gold APN functions. It is a natural question whether it is possible to construct APN polynomials EA-inequivalent to power functions by using only EA-equivalence and inverse transformation on a power APN function: this would be the simplest method to construct APN polynomials EA-inequivalent to power functions. In the present paper we prove that the answer to this question is positive. By this method we construct a class of APN polynomials EA-inequivalent to power functions. On the other hand it is shown that the APN polynomials from [1,2] cannot be obtained by the introduced method.
[1] L. Budaghyan, C. Carlet, A. Pott. New Classes of Almost Bent and Almost Perfect Nonlinear Functions. IEEE Trans. Inform. Theory, vol. 52, no. 3, pp. 1141-1152, March 2006.
[2] L. Budaghyan, C. Carlet, A. Pott. New Constructions of Almost Bent and Almost Perfect Nonlinear Functions. Proceedings of the Workshop on Coding and Cryptography 2005, pp. 306-315, 2005.

2007

EPRINT

Constructing new APN functions from known ones
Abstract

We present a method for constructing new quadratic APN functions from known ones. Applying this method to the Gold power functions we construct an APN function $x^3+\tr(x^9)$ over $\F_{2^n}$. It is proven that in general this function is CCZ-inequivalent to the Gold functions (and therefore EA-inequivalent to power functions), to the inverse and Dobbertin mappings, and in the case $n=7$ it is CCZ-inequivalent to all power mappings.

2007

EPRINT

Classes of Quadratic APN Trinomials and Hexanomials and Related Structures
Abstract

A method for constructing differentially 4-uniform quadratic hexanomials has been recently introduced by J. Dillon. We give various generalizations of this method and we deduce the constructions of new infinite classes of almost perfect nonlinear quadratic trinomials and hexanomials from $\mathbb{F}_{2^{2m}}$ to $\mathbb{F}_{2^{2m}}$. We check for $m=3$ that some of these functions are CCZ-inequivalent to power functions.

2006

EPRINT

Another class of quadratic APN binomials over $\F_{2^n}$: the case $n$ divisible by 4
Abstract

We exhibit an infinite class of almost perfect nonlinear quadratic binomials from $\mathbb{F}_{2^{n}}$ to $\mathbb{F}_{2^{n}}$ with $n=4k$ and $k$ odd. We prove that these functions are CCZ-inequivalent to known APN power functions when $k\ne 1$. In particular it means that for $n=12,20,28$, they are CCZ-inequivalent to any power function.

2006

EPRINT

A class of quadratic APN binomials inequivalent to power functions
Abstract

We exhibit an infinite class of almost perfect nonlinear quadratic binomials from $\mathbb{F}_{2^n}$ to $\mathbb{F}_{2^n}$ ($n\geq 12$, $n$ divisible by 3 but not by 9). We prove that these functions are EA-inequivalent to any power function and that they are CCZ-inequivalent to any Gold function and to any Kasami function. It means that for $n$ even they are CCZ-inequivalent to any known APN function, and in particular for $n=12,24$, they are therefore CCZ-inequivalent to any power function.
It is also proven that, except in particular cases, the Gold mappings are CCZ-inequivalent to the Kasami and Welch functions.

2005

EPRINT

An infinite class of quadratic APN functions which are not equivalent to power mappings
Abstract

We exhibit an infinite class of almost
perfect nonlinear quadratic polynomials from $\mathbb{F}_{2^n}$ to
$\mathbb{F}_{2^n}$ ($n\geq 12$, $n$ divisible by 3 but not by 9).
We prove that these functions are EA-inequivalent to any power
function. In the forthcoming version of the present paper we will
proof that these functions are CCZ-inequivalent to any Gold
function and to any Kasami function, in particular for $n=12$,
they are therefore CCZ-inequivalent to power functions.

#### Coauthors

- Claude Carlet (7)
- Patrick Felke (1)
- Tor Helleseth (2)
- Gregor Leander (4)