Kangquan Li

Haode Yan, Kangquan Li

Niho exponents have found important applications in sequence design, coding theory, and cryptography. Determining the differential spectrum of a power function with Niho exponent is a topic of considerable interest. In this paper, we investigate the power function $F(x) = x^{3q - 2}$ over $\mathbb{F}_{q^2}$, where $q = 2^m$ and $m\geq 4$ is an even integer. Notably, the exponent $3q - 2$ is a Niho exponent. By analyzing the properties of certain polynomials over $\mathbb{F}_{q^2}$, we determine the differential spectrum of $F$. Our results show that $F$ is locally differentially $4$-uniform, which complements existing results on the differential spectra of power functions with Niho exponents.

Tailin Niu, Kangquan Li, Longjiang Qu et al.

$n$-to-$1$ mappings have wide applications in many areas, especially in cryptography, finite geometry, coding theory and combinatorial design. In this paper, many classes of $n$-to-$1$ mappings over finite fields are studied. First, we provide a characterization of general $n$-to-$1$ mappings over $\mathbb{F}_{p^m}$ by means of the Walsh transform. Then, we completely determine $3$-to-$1$ polynomials with degree no more than $4$ over $\mathbb{F}_{p^{m}}$. Furthermore, we obtain an AGW-like criterion for characterizing an equivalent relationship between the $n$-to-$1$ property of a mapping over finite set $A$ and that of another mapping over a subset of $A$. Finally, we apply the AGW-like criterion into several forms of polynomials and obtain some explicit $n$-to-$1$ mappings. Especially, three explicit constructions of the form $x^rh\left( x^s \right) $ from the cyclotomic perspective, and several classes of $n$-to-$1$ mappings of the form $ g\left( x^{q^k} -x +δ\right) +cx$ are provided.

Kangquan Li, Longjiang Qu, Bing Sun et al.

In EUROCRYPT 2018, Cid et al. \cite{BCT2018} introduced a new concept on the cryptographic property of S-boxes: Boomerang Connectivity Table (BCT for short) for evaluating the subtleties of boomerang-style attacks. Very recently, BCT and the boomerang uniformity, the maximum value in BCT, were further studied by Boura and Canteaut \cite{BC2018}. Aiming at providing new insights, we show some new results about BCT and the boomerang uniformity of permutations in terms of theory and experiment in this paper. Firstly, we present an equivalent technique to compute BCT and the boomerang uniformity, which seems to be much simpler than the original definition from \cite{BCT2018}. Secondly, thanks to Carlet's idea \cite{Carlet2018}, we give a characterization of functions $f$ from $\mathbb{F}_{2}^n$ to itself with boomerang uniformity $δ_{f}$ by means of the Walsh transform. Thirdly, by our method, we consider boomerang uniformities of some specific permutations, mainly the ones with low differential uniformity. Finally, we obtain another class of $4$-uniform BCT permutation polynomials over $\mathbb{F}_{2^n}$, which is the first binomial.