Nonlinearity, Local and Global Avalanche Characteristics of Balanced Boolean Functions

Nonlinearity measures of random Boolean functions

The r-th order nonlinearity of a Boolean function is the minimum number of elements that have to be changed in its truth table to arrive at a Boolean function of degree at most r. It is shown that the (suitably normalised) r-th order nonlinearity of a random Boolean function converges strongly for all r\ge 1. This extends results by Rodier for r=1 and by Dib for r=2. The methods in the present paper are mostly of elementary combinatorial nature and also lead to simpler proofs in the cases that r=1 or 2.

Nonlinearity of quartic rotation symmetric Boolean functions

Nonlinearity of rotation symmetric Boolean functions is an important topic on cryptography algorithm. Let $e\ge 1$ be any given integer. In this paper, we investigate the following question: Is the nonlinearity of the quartic rotation symmetric Boolean function generated by the monomial $x_0x_ex_{2e}x_{3e}$ equal to its weight? We introduce some new simple sub-functions and develop new technique to get several recursive formulas. Then we use these recursive formulas to show that the nonlinearity of the quartic rotation symmetric Boolean function generated by the monomial $x_0x_ex_{2e}x_{3e}$ i…

Thin Sidon sets and the nonlinearity of vectorial Boolean functions

The vectorial nonlinearity of a vector valued function is its distance from the set of affine functions. In 2017, Liu, Mesnager and Chen conjectured a general upper bound for the vectorial linearity. Recently, Carlet proved a lower bound in terms of the differential uniformity. In this paper, we improve Carlet's lower bound. Our method is elementary, it relies on the fact that the level sets of an APN functions are Sidon sets. We give a survey on Sidon sets in elementary abelian 2-groups. We study the completeness problem of Sidon sets obtained from hyperbolas and ellipses of the finite affine…

关于布尔函数在仿射子空间上的代数次数与非线性度的最大最小值

On the max min of the algebraic degree and the nonlinearity of a Boolean function on an affine subspace

本文研究布尔函数在仿射子空间上代数次数与非线性度的最大最小值，给出上下界并提出特殊情形下的猜想。

Quasi-Random Influences of Boolean Functions

We examine a hierarchy of equivalence classes of quasi-random properties of Boolean Functions. In particular, we prove an equivalence between a number of properties including balanced influences, spectral discrepancy, local strong regularity, homomorphism enumerations of colored or weighted graphs and hypergraphs associated with Boolean functions as well as the $k$th-order strict avalanche criterion amongst others. We further construct families of quasi-random boolean functions which exhibit the properties of our equivalence theorem and separate the levels of our hierarchy.

Proof of a Conjecture about Rotation Symmetric Functions

Rotation symmetric Boolean functions have important applications in the design of cryptographic algorithms. In this paper, the Conjecture about rotation symmetric Boolean functions (RSBFs) of degree 3 proposed by Cusik and Stănică is proved. As a result, the nonlinearity of such kind of functions is determined.