## CryptoDB

### Xiangyong Zeng

#### Affiliation: Hubei University

#### Publications

**Year**

**Venue**

**Title**

2010

EPRINT

Balanced Boolean Functions with (Almost) Optimal Algebraic Immunity and Very High Nonlinearity
Abstract

In this paper, we present a class of $2k$-variable balanced Boolean
functions and a class of $2k$-variable $1$-resilient Boolean functions for an integer $k\ge 2$, which both have the maximal algebraic degree and very high nonlinearity. Based on a newly proposed conjecture by Tu and Deng, it is shown that the proposed balanced Boolean functions have optimal algebraic immunity and the $1$-resilient Boolean functions have almost optimal algebraic immunity. Among all the known results of balanced Boolean
functions and $1$-resilient Boolean functions, our new functions possess the highest nonlinearity. Based on the fact that the conjecture has been verified for all $k\le 29$ by computer,
at least we have constructed a class of balanced Boolean functions and a class of $1$-resilient Boolean functions with the even number of variables $\le 58$, which are cryptographically optimal or almost
optimal in terms of balancedness, algebraic degree, nonlinearity, and algebraic immunity.

2007

EPRINT

FURTHER PROPERTIES OF SEVERAL CLASSES OF BOOLEAN FUNCTIONS WITH OPTIMUM ALGEBRAIC IMMUNITY
Abstract

Thanks to a method proposed by Carlet, several classes of balanced
Boolean functions with optimum algebraic immunity are obtained. By
choosing suitable parameters, for even $n\geq 8$, the balanced
$n$-variable functions can have nonlinearity
$2^{n-1}-{n-1\choose\frac{n}{2}-1}+2{n-2\choose\frac{n}{2}-2}/(n-2)$,
and for odd $n$, the functions can have nonlinearity
$2^{n-1}-{n-1\choose\frac{n-1}{2}}+\Delta(n)$, where the function
$\Delta(n)$ is describled in Theorem 4.4. The algebraic
degree of some constructed functions is also discussed.

2007

EPRINT

On The Inequivalence Of Ness-Helleseth APN Functions
Abstract

In this paper, the Ness-Helleseth functions over $F_{p^n}$ defined
by the form $f(x)=ux^{\frac{p^n-1}{2}-1}+x^{p^n-2}$ are proven to be
a new class of almost perfect nonlinear (APN) functions and they are
CCZ-inequivalent with all other known APN functions when $p\geq 7$.
The original method of Ness and Helleseth showing the functions are
APN for $p=3$ and odd $n\geq 3$ is also suitable for showing their
APN property for any prime $p\geq 7$ with $p\equiv 3\,({\rm
mod}\,4)$ and odd $n$.

#### Coauthors

- Claude Carlet (1)
- Lei Hu (3)
- Wenfeng Jiang (1)
- Chunlei Li (1)
- Xiaohu Tang (1)
- Deng Tang (1)
- Yang Yang (1)