# Switching Construction of Planar Functions on Finite Fields

@inproceedings{Pott2010SwitchingCO, title={Switching Construction of Planar Functions on Finite Fields}, author={Alexander Pott and Yue Zhou}, booktitle={WAIFI}, year={2010} }

A function f : Fpn → Fpn is planar, if f(x+a)-f(x) = b has precisely one solution for all a, b e Fpn, a ≠ 0. In this paper, we discuss possible extensions of the switching idea developed in [1] to the case of planar functions. We show that some of the known planar functions can be constructed from each other by switching.

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## References

SHOWING 1-10 OF 34 REFERENCES

### New semifields, PN and APN functions

- MathematicsDes. Codes Cryptogr.
- 2010

The main new contribution is the construction of a family of PN-functions and their corresponding commutative semifields of dimension 4s in arbitrary odd characteristic.

### Planar Functions and Planes of Lenz-Barlotti Class II

- MathematicsDes. Codes Cryptogr.
- 1997

Several classes of planar functions over a finite field are described, including a class whose associated affine planes are not translation planes or dual translation planes, and which cannot be obtained by derivation or lifting.

### Some Theorems on Planar Mappings

- MathematicsWAIFI
- 2008

It is proved that two planar functions are CCZ-equivalent exactly when they are EA-equ equivalent.

### Symplectic Spreads and Quadric Veronesean

- Mathematics
- 2009

In this paper we face with the problem of constructing semifield spreads in projective spaces of dimension larger than 3. To this aim we study the relationship between linear sets disjoint from the…

### A new almost perfect nonlinear function which is not quadratic

- Mathematics, Computer ScienceAdv. Math. Commun.
- 2009

It is shown that the approach shown can be used to construct a ''non-quadratic'' APN function, which is in remarkable contrast to all recently constructed functions which have all been quadratic.

### The permutation group of affine-invariant extended cyclic codes

- Computer Science, MathematicsIEEE Trans. Inf. Theory
- 1996

The formal expression of the permutation group of primitive BCH codes defined on any prime field is presented and it is proved, by studying some examples, that the tools are efficient.

### Nonlinear functions in abelian groups and relative difference sets

- MathematicsDiscret. Appl. Math.
- 2004

### Commutative semifields from projection mappings

- MathematicsDes. Codes Cryptogr.
- 2011

A general projection method to construct commutative semifields in odd characteristic yields a generalization of the Budaghyan–Helleseth family and also greatly simplifies the construction.