# A new almost perfect nonlinear function which is not quadratic

@article{Edel2009ANA, title={A new almost perfect nonlinear function which is not quadratic}, author={Yves Edel and Alexander Pott}, journal={Adv. Math. Commun.}, year={2009}, volume={3}, pages={59-81} }

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… Expand

#### 130 Citations

On the Equivalence of Nonlinear Functions

- Mathematics, Computer Science
- Enhancing Cryptographic Primitives with Techniques from Error Correcting Codes
- 2009

Different concepts of equivalence between almost perfect nonlinear (APN) and almost bent (AB) functions are summarized, and it is shown that CCZ equivalence is the same as extended affine equivalence if F is a vectorial bent function. Expand

A matrix approach for constructing quadratic APN functions

- Mathematics, Computer Science
- Des. Codes Cryptogr.
- 2014

This paper has found 471 new CCZ-inequivalent quadratic APN functions, which is 20 times more than the number of the previously known ones, and this number is still increasing. Expand

On Quadratic Almost Perfect Nonlinear Functions and Their Related Algebraic Object

- 2013

It is well known that almost perfect nonlinear (APN) functions achieve the lowest possible differential uniformity for functions defined on fields with even characteristic, and hence, from this point… Expand

Antiderivative Functions over F 2 n

- Mathematics
- 2015

In this paper, we use a linear algebra point of view to describe the derivatives and higher order derivatives over F2n. On one hand, this new approach enables us to prove several properties of these… Expand

Constructing new APN functions from known ones

- Computer Science, Mathematics
- Finite Fields Their Appl.
- 2007

Using this method to the Gold power functions, an APN function x^3+tr(x^9) over F"2"^"n" is constructed and it is proven that for n>=7 this function is CCZ-inequivalent to the gold functions. Expand

The classification of quadratic APN functions in 7 variables

- Computer Science
- IACR Cryptol. ePrint Arch.
- 2020

It is proved that the updated list of quadratic APN functions in dimension 6 is complete up to CCZ-equivalence and the first approach exploited a secondary construction idea and allowed us to find a new APN function in 7 variables. Expand

On Some Properties of Quadratic APN Functions of a Special Form

- Mathematics, Computer Science
- ArXiv
- 2017

In a recent paper, it is shown that functions of the form $L_1(x^3)+L_2(x^9)$, where $L_1$ and $L_2$ are linear, are a good source for construction of new infinite families of APN functions. In the… Expand

Antiderivative functions over $$\mathbb {F}_{2^n}$$F2n

- Mathematics, Computer Science
- Des. Codes Cryptogr.
- 2017

A linear algebra point of view is used to describe the derivatives and higher order derivatives over F2n, and a new equivalence of functions is defined, which links functions that share the same derivatives in directions given by some subspace. Expand

A divisibility criterion for exceptional APN functions

- 2017

We are interested in the functions from F2m to itself which are Almost Perfectly Nonlinear over infinitely many extensions of F2, namely, the exceptional APN functions. In particular, we study the… Expand

Quadratic almost bent functions - their partial characterization and design in the spectral domain

- Computer Science
- IACR Cryptol. ePrint Arch.
- 2021

This article could for the first time provide the design of quadratic AB functions in the spectral domain by identifying (using computer simulations) suitable sets of bent dual functions which give rise to possibly new quadrato-AB functions in a generic manner. Expand

#### References

SHOWING 1-10 OF 57 REFERENCES

Constructing new APN functions from known ones

- Computer Science, Mathematics
- Finite Fields Their Appl.
- 2007

Using this method to the Gold power functions, an APN function x^3+tr(x^9) over F"2"^"n" is constructed and it is proven that for n>=7 this function is CCZ-inequivalent to the gold functions. Expand

A class of quadratic APN binomials inequivalent to power functions

- Mathematics, Computer Science
- IACR Cryptol. ePrint Arch.
- 2006

It is proven that for n even they are CCZ-inequivalent to any known APN function, and in particular for n = 12, 24, they are therefore CCZ to any power function. Expand

Two Classes of Quadratic APN Binomials Inequivalent to Power Functions

- Mathematics, Computer Science
- IEEE Transactions on Information Theory
- 2008

This paper introduces the first found infinite classes of almost perfect nonlinear (APN) polynomials which are not Carlet-Charpin-Zinoviev (CCZ)-equivalent to power functions (at least for some… Expand

Almost Perfect Nonlinear Power Functions on GF(2n): A New Case for n Divisible by 5

- Mathematics
- 2001

We prove that for d = 24s + 23s + 22s + 2 s − 1 the power function x d is almost perfect nonlinear (APN) on L = GF(25s ), i.e. for each a ∈ L the equation (x + 1) d + x d = a has either no or… Expand

New cyclic difference sets with Singer parameters

- Computer Science, Mathematics
- Finite Fields Their Appl.
- 2004

There are today no sporadic examples of difference sets with Singer parameters; i.e. every known such difference set belongs to a series given by a constructive theorem. Expand

On the classification of APN functions up to dimension five

- Mathematics, Computer Science
- Des. Codes Cryptogr.
- 2008

It is demonstrated that up to dimension 5 any APN function is CCZ equivalent to a power function, while it is well known that in dimensions 4 and 5 there exist APN functions which are not extended affine (EA) equivalent to any power function. Expand

Almost Perfect Nonlinear Power Functions on GF(2n): The Welch Case

- Mathematics, Computer Science
- IEEE Trans. Inf. Theory
- 1999

The first case supports a well-known conjecture of Welch stating that for odd n=2m+1, the power function x/sup 2m+3/ is even maximally nonlinear or, in other terms, that the crosscorrelation function between a binary maximum-length linear shift register sequence and a decimation of that sequence by 2/sup m/+3 takes on precisely the three values -1, -1/spl plusmn/2/Sup m+1/. Expand

Almost Perfect Nonlinear Power Functions on GF(2n): The Niho Case

- Computer Science, Mathematics
- Inf. Comput.
- 1999

Almost perfect nonlinear (APN) mappings are of interest for applications in cryptography We prove for odd n and the exponent d=22r+2r?1, where 4r+1?0modn, that the power functions xd on GF(2n) is… Expand

New Perfect Nonlinear Multinomials over Ffor Any Odd Prime p

- Mathematics, Computer Science
- SETA
- 2008

Two infinite families of perfect nonlinear Dembowski-Ostrom multinomials over p where pis any odd prime are introduced and it is proved that in general these functions are CCZ-inequivalent to previously known PN mappings. Expand

A new APN function which is not equivalent to a power mapping

- Mathematics, Computer Science
- IEEE Transactions on Information Theory
- 2006

A new almost-perfect nonlinear function (APN) on F(2/sup 10/) which is not equivalent to any of the previously known APN mappings is constructed. This is the first example of an APN mapping which is… Expand