# 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…

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## References

SHOWING 1-10 OF 42 REFERENCES

### Constructing new APN functions from known ones

- Mathematics, Computer ScienceFinite Fields Their Appl.
- 2007

### A class of quadratic APN binomials inequivalent to power functions

- MathematicsIACR 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.

### Two Classes of Quadratic APN Binomials Inequivalent to Power Functions

- MathematicsIEEE 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…

### 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…

### On the classification of APN functions up to dimension five

- Computer Science, MathematicsDes. 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.

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

- Computer ScienceIEEE 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/.

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

- Mathematics, Computer ScienceInf. 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…

### New Perfect Nonlinear Multinomials over Ffor Any Odd Prime p

- Mathematics, Computer ScienceSETA
- 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.

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

- MathematicsIEEE 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…