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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159 Citations
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Methods Citations
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On the Equivalence of Nonlinear Functions

  • Yves EdelA. Pott
  • Mathematics, Computer Science
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  • 2009
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References

SHOWING 1-10 OF 42 REFERENCES

Constructing new APN functions from known ones

A class of quadratic APN binomials inequivalent to power functions

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

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

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…

New cyclic difference sets with Singer parameters

On the classification of APN functions up to dimension five

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

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

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

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

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…