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}
}
  • Y. Edel, A. Pott
  • Published 2009
  • Mathematics, Computer Science
  • Adv. Math. Commun.
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
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References

SHOWING 1-10 OF 57 REFERENCES
Constructing new APN functions from known ones
TLDR
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
TLDR
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
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 someExpand
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 orExpand
New cyclic difference sets with Singer parameters
TLDR
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
TLDR
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
  • H. Dobbertin
  • Mathematics, Computer Science
  • IEEE Trans. Inf. Theory
  • 1999
TLDR
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
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) isExpand
New Perfect Nonlinear Multinomials over Ffor Any Odd Prime p
TLDR
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
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 isExpand
...
1
2
3
4
5
...