# On the Ambiguity of Differentially Uniform Functions

@article{Fu2017OnTA, title={On the Ambiguity of Differentially Uniform Functions}, author={Shihui Fu and Xiutao Feng and Qiang Wang}, journal={ArXiv}, year={2017}, volume={abs/1710.07765} }

Recently, the ambiguity and deficiency of a given bijective mapping $F$ over a finite abelian group $G$ were introduced by Panario et al. to measure the balancedness of the derivatives $D_a F(x)=F(x+a)-F(x)$ for all $a\in G\setminus\{0\}$. In this paper, we extend the study of the ambiguity and deficiency to functions between any two finite abelian groups $G_1$, $G_2$ with possible different orders. Many functions in cryptography are of this type. We investigate the optimum lower bound of…

## References

SHOWING 1-10 OF 43 REFERENCES

### Ambiguity and Deficiency of Permutations Over Finite Fields With Linearized Difference Map

- MathematicsIEEE Transactions on Information Theory
- 2013

This paper provides an explicit formula in terms of the ranks of matrices on the ambiguity and deficiency of a Dembowski-Ostrom (DO) polynomial, and derives exact values for the ambiguities and deficiencies of DO permutations obtained from trace functions.

### Two New Measures for Permutations: Ambiguity and Deficiency

- Mathematics, Computer ScienceIEEE Transactions on Information Theory
- 2011

It is concluded that a twisted permutation polynomial of a finite field is again closer to being optimal in ambiguity than the APN function employed in the SAFER cryptosystem.

### Constructing Differentially 4-Uniform Permutations Over ${\BBF}_{2^{2k}}$ via the Switching Method

- Mathematics, Computer ScienceIEEE Transactions on Information Theory
- 2013

The powerful switching method is applied to discover many CCZ-inequivalent infinite families of such functions on F(22k ) with optimal algebraic degree, where k is an arbitrary positive integer, and implies that some infinite families have high nonlinearity.

### Symplectic spreads, planar functions, and mutually unbiased bases

- MathematicsArXiv
- 2013

By using the new notion of pseudo-planar functions over fields of characteristic two, new explicit constructions of complete sets of MUBs and orthogonal decompositions of special Lie algebras are given.

### A New Approach to Constructing Quadratic Pseudo-Planar Functions over $\gf_{2^n}$

- MathematicsArXiv
- 2016

A new approach to constructing quadratic pseudo-planar functions is given and these functions not only lead to projective planes, relative difference sets and presemifields, but also give optimal codebooks meeting the Levenstein bound and compressed sensing matrices with low coherence.

### Constructing Differentially 4-uniform Permutations over GF(22k) from the Inverse Function Revisited

- Mathematics, Computer ScienceIACR Cryptol. ePrint Arch.
- 2013

It is shown that numerous differentially 4-uniform permutations over F22k can be constructed by composing the inverse function and cycles over F 22k by giving sufficient conditions to ensure that the differential uniformity of the corresponding compositions equals 4.

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

- MathematicsDiscret. Appl. Math.
- 2004

### (2^n,2^n,2^n,1)-relative difference sets and their representations

- Mathematics
- 2012

We show that every $(2^n,2^n,2^n,1)$-relative difference set $D$ in $\Z_4^n$ relative to $\Z_2^n$ can be represented by a polynomial $f(x)\in \F_{2^n}[x]$, where $f(x+a)+f(x)+xa$ is a permutation for…

### New Construction of Differentially 4-Uniform Bijections

- Mathematics, Computer ScienceInscrypt
- 2013

A method is proposed for constructing a large family of differentially 4-uniform bijections in even dimensions and this method exhibits a subclass of functions having high nonlinearity and being CCZ-inequivalent to all known differentially 3- uniform power bijection and to quadratic functions.

### A highly nonlinear differentially 4 uniform power mapping that permutes fields of even degree

- Mathematics, Computer ScienceFinite Fields Their Appl.
- 2010