arXiv subjects

Deng Tang

Publications and source records attributed to Deng Tang.

7 recordsLinked to original sources

arXiv2026

Three Infinite Classes of APN Permutations on $Z_n$

For any permutation of a nontrivial finite abelian group, the differential uniformity is at least two; permutations attaining this bound are called almost perfect nonlinear (APN). We construct three infinite classes of APN permutations on the cyclic group $\mathbb{Z}_n$ using Singer cycles, binomials inducing projective permutations, and completed reciprocals combined with parity and quadratic characters. The respective domain orders are $q+1$ for prime powers $q>2$, $(3^d-1)/2$ for integers $d\ge2$, and $2p$ for primes $p>5$ with $p\equiv5\pmod6$. Each class contains an infinite subclass of composite orders outside the standard forms $r-1$, $r-2$, $r-3$, and $r-4$, where $r$ is a prime power. These forms arise in the Welch--Costas, Panario--Sakzad--Stevens--Wang, and Golomb constructions. To the best of our knowledge, these are the first infinite APN constructions on $\mathbb{Z}_n$ reported since 2011 that yield infinitely many composite orders outside these standard forms.

math.CO↗

arXiv2026

Explicit determination of a class of permutation rational functions in any characteristic

In this paper, we make use of the classification results of low-degree permutation rational functions together with their geometric properties to investigate rational functions that induce permutations on the multiplicative subgroup mu_q+1, where q is a prime power. By carefully analyzing the structural conditions under which such rational functions permute muq+1, we obtain an explicit description of a broad class of permutation rational functions of small degree. As a direct application of these findings, we explicitly determine many permutation quadrinomials over Fq2 that are induced by degree-3 rational functions permuting muq+1. Our approach not only unifies and extends several existing results in the literature but also provides a concrete geometric perspective for characterizing permutation polynomials over Fq2.

math.NT↗

arXiv2025

A general secondary construction of Boolean functions including the indirect sum and its generalizations

We study a secondary construction of Boolean functions, which generalizes the direct sum and the indirect sum. We detail how these two classic secondary constructions are particular cases of this more general one, as well as two known generalizations of the indirect sum. This unifies the known secondary constructions of Boolean functions. We study very precisely the Walsh transform of the constructed functions. This leads us to an interesting observation on the Walsh transforms $W_g,W_{g'},W_{g''}$, and $W_{g\oplus g'\oplus g''}$ when $g,g',g''$ are Boolean functions such that $(g\oplus g')(g\oplus g'')$ equals the zero function.

cs.IT↗

arXiv2021

On Boolean Functions with Low Polynomial Degree and Higher Order Sensitivity

Boolean functions are important primitives in different domains of cryptology, complexity and coding theory. In this paper, we connect the tools from cryptology and complexity theory in the domain of Boolean functions with low polynomial degree and high sensitivity. It is well known that the polynomial degree of of a Boolean function and its resiliency are directly connected. Using this connection we analyze the polynomial degree-sensitivity values through the lens of resiliency, demonstrating existence and non-existence results of functions with low polynomial degree and high sensitivity on small number of variables (upto 10). In this process, borrowing an idea from complexity theory, we show that one can implement resilient Boolean functions on a large number of variables with linear size and logarithmic depth. Finally, we extend the notion of sensitivity to higher order and note that the existing construction idea of Nisan and Szegedy (1994) can provide only constant higher order sensitivity when aiming for polynomial degree of $n-\omega(1)$. In this direction, we present a construction with low ($n-\omega(1)$) polynomial degree and super-constant $\omega(1)$ order sensitivity exploiting Maiorana-McFarland constructions, that we borrow from construction of resilient functions. The questions we raise identify novel combinatorial problems in the domain of Boolean functions.

cs.CC↗

arXiv2021

Constructions of Binary Optimal Locally Repairable Codes via Intersection Subspaces

Locally repairable codes (LRCs), which can recover any symbol of a codeword by reading only a small number of other symbols, have been widely used in real-world distributed storage systems, such as Microsoft Azure Storage and Ceph Storage Cluster. Since binary linear LRCs can significantly reduce coding and decoding complexity, constructions of binary LRCs are of particular interest. The aim of this paper is to construct dimensional optimal binary locally repairable codes with disjoint local repair groups. We introduce how to connect intersection subspaces with binary locally repairable codes and construct dimensional optimal binary linear LRCs with locality $2^b$ ($b\geq 3$) and minimum distance $d\geq 6$ by employing intersection subspaces deduced from the direct sum. This method will sufficiently increase the number of possible repair groups of dimensional optimal LRCs, and thus efficiently expanding the range of the construction parameters while keeping the largest code rates compared with all known binary linear LRCs with minimum distance $d\geq 6$ and locality $2^b$ ($b\geq 3$).

cs.IT↗

arXiv2021

Constructing new APN functions through relative trace functions

In 2020, Budaghyan, Helleseth and Kaleyski [IEEE TIT 66(11): 7081-7087, 2020] considered an infinite family of quadrinomials over $\mathbb{F}_{2^{n}}$ of the form $x^3+a(x^{2^s+1})^{2^k}+bx^{3\cdot 2^m}+c(x^{2^{s+m}+2^m})^{2^k}$, where $n=2m$ with $m$ odd. They proved that such kind of quadrinomials can provide new almost perfect nonlinear (APN) functions when $\gcd(3,m)=1$, $ k=0 $, and $(s,a,b,c)=(m-2,\omega, \omega^2,1)$ or $((m-2)^{-1}~{\rm mod}~n,\omega, \omega^2,1)$ in which $\omega\in\mathbb{F}_4\setminus \mathbb{F}_2$. By taking $a=\omega$ and $b=c=\omega^2$, we observe that such kind of quadrinomials can be rewritten as $a {\rm Tr}^{n}_{m}(bx^3)+a^q{\rm Tr}^{n}_{m}(cx^{2^s+1})$, where $q=2^m$ and $ {\rm Tr}^n_{m}(x)=x+x^{2^m} $ for $ n=2m$. Inspired by the quadrinomials and our observation, in this paper we study a class of functions with the form $f(x)=a{\rm Tr}^{n}_{m}(F(x))+a^q{\rm Tr}^{n}_{m}(G(x))$ and determine the APN-ness of this new kind of functions, where $a \in \mathbb{F}_{2^n} $ such that $ a+a^q\neq 0$, and both $F$ and $G$ are quadratic functions over $\mathbb{F}_{2^n}$. We first obtain a characterization of the conditions for $f(x)$ such that $f(x) $ is an APN function. With the help of this characterization, we obtain an infinite family of APN functions for $ n=2m $ with $m$ being an odd positive integer: $ f(x)=a{\rm Tr}^{n}_{m}(bx^3)+a^q{\rm Tr}^{n}_{m}(b^3x^9) $, where $ a\in \mathbb{F}_{2^n}$ such that $ a+a^q\neq 0 $ and $ b $ is a non-cube in $ \mathbb{F}_{2^n} $.

cs.IT↗

arXiv2016

Binary Linear Codes From Vectorial Boolean Functions and Their Weight Distribution

Binary linear codes with good parameters have important applications in secret sharing schemes, authentication codes, association schemes, and consumer electronics and communications. In this paper, we construct several classes of binary linear codes from vectorial Boolean functions and determine their parameters, by further studying a generic construction developed by Ding \emph{et al.} recently. First, by employing perfect nonlinear functions and almost bent functions, we obtain several classes of six-weight linear codes which contains the all-one codeword. Second, we investigate a subcode of any linear code mentioned above and consider its parameters. When the vectorial Boolean function is a perfect nonlinear function or a Gold function in odd dimension, we can completely determine the weight distribution of this subcode. Besides, our linear codes have larger dimensions than the ones by Ding et al.'s generic construction.

cs.IT↗