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Xiang-dong Hou

Publications and source records attributed to Xiang-dong Hou.

At least 37 records · Page 2Linked to original sources
arXiv2020

On a Type of Permutation Rational Functions over Finite Fields

Let $p$ be a prime and $n$ be a positive integer. Let $f_b(X)=X+(X^p-X+b)^{-1}$, where $b\in\Bbb F_{p^n}$ is such that $\text{Tr}_{p^n/p}(b)\ne 0$. In 2008, Yuan et al. \cite{Yuan-Ding-Wang-Pieprzyk-FFA-2008} showed that for $p=2,3$, $f_b$ permutes $\Bbb F_{p^n}$ for all $n\ge 1$. Using the Hasse-Weil bound, we show that when $p>3$ and $n\ge 5$, $f$ does not permute $\Bbb F_{p^n}$. For $p>3$ and $n=2$, we prove that $f_b$ permutes $\Bbb F_{p^2}$ if and only if $\text{Tr}_{p^2/p}(b)=\pm 1$. We conjecture that for $p>3$ and $n=3,4$, $f_b$ does not permute $\Bbb F_{p^n}$.

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arXiv2019

The Fifth International Students' Olympiad in Cryptography -- NSUCRYPTO: problems and their solutions

Problems and their solutions of the Fifth International Students' Olympiad in cryptography NSUCRYPTO'2018 are presented. We consider problems related to attacks on ciphers and hash functions, Boolean functions, quantum circuits, Enigma, etc. We discuss several open problems on orthogonal arrays, Sylvester matrices and disjunct matrices. The problem of existing an invertible Sylvester matrix whose inverse is again a Sylvester matrix was completely solved during the Olympiad.

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arXiv2019

An Application of the Hasse-Weil Bound to Rational Functions over Finite Fields

We use the Aubry-Perret bound for singular curves, a generalization of the Hasse-Weil bound, to prove the following curious result about rational functions over finite fields: Let $f(X),g(X)\in\Bbb F_q(X)\setminus\{0\}$ be such that $q$ is sufficiently large relative to $\text{deg}\, f$ and $\text{deg}\, g$, $f(\Bbb F_q)\subset g(\Bbb F_q\cup\{\infty\})$, and for ``most'' $a\in\Bbb F_q\cup\{\infty\}$, $|\{x\in \Bbb F_q:g(x)=g(a)\}|>(\text{deg}\, g)/2$. Then there exists $h(X)\in\Bbb F_q(X)$ such that $f(X)=g(h(X))$. A generalization to multivariate rational functions is also included.

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arXiv2019

On the Tu-Zeng Permutation Trinomial of Type $(1/4,3/4)$

Let $q$ be a power of $2$. Recently, Tu and Zeng considered trinomials of the form $f(X)=X+aX^{(1/4)q^2(q-1)}+bX^{(3/4)q^2(q-1)}$, where $a,b\in\Bbb F_{q^2}^*$. They proved that $f$ is a permutation polynomial of $\Bbb F_{q^2}$ if $b=a^{2-q}$ and $X^3+X+a^{-1-q}$ has no root in $\Bbb F_q$. In this paper, we show that the above sufficient condition is also necessary.

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arXiv2019

Determination of a Class of Permutation Trinomials in Characteristic Three

Let $f(X)=X(1+aX^{q(q-1)}+bX^{2(q-1)})\in\Bbb F_{q^2}[X]$, where $a,b\in\Bbb F_{q^2}^*$. In a series of recent papers by several authors, sufficient conditions on $a$ and $b$ were found for $f$ to be a permutation polynomial (PP) of $\Bbb F_{q^2}$ and, in characteristic $2$, the sufficient conditions were shown to be necessary. In the present paper, we confirm that in characteristic 3, the sufficient conditions are also necessary. More precisely, we show that when $\text{char}\,\Bbb F_q=3$, $f$ is a PP of $\Bbb F_{q^2}$ if and only if $(ab)^q=a(b^{q+1}-a^{q+1})$ and $1-(b/a)^{q+1}$ is a square in $\Bbb F_q^*$.

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arXiv2018

On a conjecture on permutation polynomials over finite fields

Let $\Bbb F_q$ be the finite field with $q$ elements and let $p=\text{char}\,\Bbb F_q$. It was conjectured that for integers $e\ge 2$ and $1\le a\le pe-2$, the polynomial $X^{q-2}+X^{q^2-2}+\cdots+X^{q^a-2}$ is a permutation polynomial of $\Bbb F_{q^e}$ if and only if (i) $a=2$ and $q=2$, or (ii) $a=1$ and $\text{gcd}(q-2,q^e-1)=1$. In the present paper we confirm this conjecture.

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arXiv2018

On a Class of Permutation Trinomials in Characteristic 2

Recently, Tu, Zeng, Li, and Helleseth considered trinomials of the form $f(X)=X+aX^{q(q-1)+1}+bX^{2(q-1)+1}\in\Bbb F_{q^2}[X]$, where $q$ is even and $a,b\in\Bbb F_{q^2}^*$. They found sufficient conditions on $a,b$ for $f$ to be a permutation polynomial (PP) of $\Bbb F_{q^2}$ and they conjectured that the sufficient conditions are also necessary. The conjecture has been confirmed by Bartoli using the Hasse-Weil bound. In this paper, we give an alternative solution to the question. We also use the Hasse-Weil bound, but in a different way. Moreover, the necessity and sufficiency of the conditions are proved by the same approach.

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arXiv2017

Optimal Binary Constant Weight Codes and Affine Linear Groups over Finite Fields

Let $\text{AGL}(1,\Bbb F_q)$ be the affine linear group of dimension $1$ over a finite field $\Bbb F_q$. $\text{AGL}(1,\Bbb F_q)$ acts sharply 2-transitively on $\Bbb F_q$. Given $S<\text{AGL}(1,\Bbb F_q)$ and an integer $k$ with $1\le k\le q$, does there exist a subset $B\subset\Bbb F_q$ with $|B|=k$ such that $S=\text{AGL}(1,\Bbb F_q)_B$? ($\text{AGL}(1,\Bbb F_q)_B=\{σ\in\text{AGL}(1,\Bbb F_q):σ(B)=B\}$ is the stabilizer of $B$ in $\text{AGL}(1,\Bbb F_q)$.) We derive a sum that holds the answer to this question. This result determines all possible parameters of binary constant weight codes that are constructed from the action of $\text{AGL}(1,\Bbb F_q)$ on $\Bbb F_q$ to meet the Johnson bound. Consequently, the values of the function $A_2(n,d,w)$ are determined for many parameters, where $A_2(n,d,w)$ is the maximum number of codewords in a binary constant weight code of length $n$, weight $w$ and minimum distance $\ge d$.

math.CO↗
arXiv2017

On the DLW Conjectures

In 2007, Dmytrenko, Lazebnik and Williford posed two related conjectures about polynomials over finite fields. Conjecture~1 is a claim about the uniqueness of certain monomial graphs. Conjecture~2, which implies Conjecture~1, deals with certain permutation polynomials of finite fields. Two natural strengthenings of Conjecture~2, referred to as Conjectures~A and B in the present paper, were also insinuated. In a recent development, Conjecture~2 and hence Conjecture~1 have been confirmed. The present paper gives a proof of Conjecture~A.

math.CO↗
arXiv2016

Permutation Polynomials of the form ${\tt X}^r(a+{\tt X}^{2(q-1)})$ --- A Nonexistence Result

Let $f={\tt X}^r(a+{\tt X}^{2(q-1)})\in{\Bbb F}_{q^2}[{\tt X}]$, where $a\in{\Bbb F}_{q^2}^*$ and $r\ge 1$. The parameters $(q,r,a)$ for which $f$ is a permutation polynomial (PP) of ${\Bbb F}_{q^2}$ have been determined in the following cases: (i) $a^{q+1}=1$; (ii) $r=1$; (iii) $r=3$. These parameters together form three infinite families. For $r>3$ (there is a good reason not to consider $r=2$) and $a^{q+1}\ne 1$, computer search suggested that $f$ is not a PP of ${\Bbb F}_{q^2}$ when $q$ is not too small relative to $r$. In the present paper, we prove that this claim is true. In particular, for each $r>3$, there are only finitely many $(q,a)$, where $a^{q+1}\ne 1$, for which $f$ is a PP of ${\Bbb F}_{q^2}$.

math.CO↗
arXiv2016

On Gauss Periods

Let $q$ be a prime power, and let $r=nk+1$ be a prime such that $r\nmid q$, where $n$ and $k$ are positive integers. Under a simple condition on $q$, $r$ and $k$, a Gauss period of type $(n,k)$ is a normal element of $\Bbb F_{q^n}$ over $\Bbb F_q$; the complexity of the resulting normal basis of $\Bbb F_{q^n}$ over $\Bbb F_q$ is denoted by $C(n,k;q)$. Recent works determined $C(n,k;q)$ for $k\le 7$ and all qualified $n$ and $q$. In this paper, we show that for any given $k>0$, $C(n,k;q)$ is given by an explicit formula except for finitely many primes $r=nk+1$ and the exceptional primes are easily determined. Moreover, we describe an algorithm that allows one to compute $C(n,k;q)$ for the exceptional primes $r=nk+1$. The numerical results of the paper cover $C(n,k;q)$ for $k\le 20$ and all qualified $n$ and $q$.

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arXiv2015

Polynomials Meeting Ax's Bound

Let $f\in\Bbb F_q[X_1,\dots,X_n]$ with $°f=d>0$ and let $Z(f)=\{(x_1,\dots,x_n)\in \Bbb F_q^n: f(x_1,\dots,x_n)=0\}$. Ax's theorem states that $|Z(f)|\equiv 0\pmod {q^{\lceil n/d\rceil-1}}$, that is, $ν_p(|Z(f)|)\ge m(\lceil n/d\rceil-1)$, where $p=\text{char}\,\Bbb F_q$, $q=p^m$, and $ν_p$ is the $p$-adic valuation. In this paper, we determine a condition on the coefficients of $f$ that is necessary and sufficient for $f$ to meet Ax's bound, that is, $ν_p(|Z(f)|)=m(\lceil n/d\rceil-1)$. Let $R_q(d,n)$ denote the $q$-ary Reed-Muller code $\{f\in\Bbb F_q[X_1,\dots,X_n]: °f\le d,\ °_{X_j}f\le q-1,\ 1\le j\le n\}$, and let $N_q(d,n;t)$ be the number of codewords of $R_q(d,n)$ with weight divisible by $p^t$. As applications of the aforementioned result, we find explicit formulas for $N_q(d,n;t)$ in the following cases: (i) $q=2^m$, $n$ even, $d=n/2$, $t=m+1$; (ii) $q=2$, $n/2\le d\le n-2$, $t=2$; (iii) $q=3^m$, $d=n$, $t=1$; (iv) $q=3$, $n\le d\le 2n$, $t=1$.

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arXiv2015

Permutation Polynomials of $\Bbb F_{q^2}$ of the form $a{\tt X}+{\tt X}^{r(q-1)+1}$

Let $q$ be a prime power, $2\le r\le q$, and $f=a{\tt X}+{\tt X}^{r(q-1)+1}\in\Bbb F_{q^2}[{\tt X}]$, where $a\ne 0$. The conditions on $r,q,a$ that are necessary and sufficient for $f$ to be a permutation polynomial (PP) of ${\Bbb F}_{q^2}$ are not known. (Such conditions are known under an additional assumption that $a^{q+1}=1$.) In this paper, we prove the following: (i) If $f$ is a PP of ${\Bbb F}_{q^2}$, then $\text{gcd}(r,q+1)>1$ and $(-a)^{(q+1)/\text{gcd}(r,q+1)}\ne 1$. (ii) For a fixed $r>2$ and subject to the conditions that $q+1\equiv 0\pmod r$ and $a^{q+1}\ne 1$, there are only finitely many $(q,a)$ for which $f$ is a PP of ${\Bbb F}_{q^2}$. Combining (i) and (ii) confirms a recent conjecture regarding the type of permutation binomial considered here.

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arXiv2015

Proof of a conjecture on monomial graphs

Let $e$ be a positive integer, $p$ be an odd prime, $q=p^{e}$, and $\Bbb F_q$ be the finite field of $q$ elements. Let $f,g \in \Bbb F_q [X,Y]$. The graph $G=G_q(f,g)$ is a bipartite graph with vertex partitions $P=\Bbb F_q^3$ and $L=\Bbb F_q^3$, and edges defined as follows: a vertex $(p)=(p_1,p_2,p_3)\in P$ is adjacent to a vertex $[l] = [l_1,l_2,l_3]\in L$ if and only if $p_2 + l_2 = f(p_1,l_1)$ and $p_3 + l_3 = g(p_1,l_1)$. Motivated by some questions in finite geometry and extremal graph theory, Dmytrenko, Lazebnik and Williford conjectured in 2007 that if $f$ and $g$ are both monomials and $G$ has no cycle of length less than eight, then $G$ is isomorphic to the graph $G_q(XY,XY^2)$. They proved several instances of the conjecture by reducing it to the property of polynomials $A_k= X^k[(X+1)^k - X^k]$ and $B_k= [(X+1)^{2k} - 1] X^{q-1-k} - 2X^{q-1}$ being permutation polynomials of $\Bbb F_q$. In this paper we prove the conjecture by obtaining new results on the polynomials $A_k$ and $B_k$, which are also of interest on their own.

math.CO↗
arXiv2015

From $r$-Linearized Polynomial Equations to $r^m$-Linearized Polynomial Equations

Let $r$ be a prime power and $q=r^m$. For $0\le i\le m-1$, let $f_i\in \mathbb{F}_r[x]$ be $q$-linearized and $a_i\in \mathbb{F}_q$. Assume that $z\in \mathbb{\bar{F}}_r$ satisfies the equation $\sum_{i=0}^{m-1}a_if_i(z)^{r^i}=0$, where $\sum_{i=0}^{m-1}a_if_i^{r_i}\in \mathbb{F}_q[x]$ is an $r$-linearized polynomial. It is shown that $z$ satisfies a $q$-linearized polynomial equation with coefficients in $\mathbb{F}_r$. This result provides an explanation for numerous permutation polynomials previously obtained through computer search.

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arXiv2014

Switchings of semifield multiplications

Let $B(X,Y)$ be a polynomial over $\mathbb{F}_{q^n}$ which defines an $\mathbb{F}_q$-bilinear form on the vector space $\mathbb{F}_{q^n}$, and let $ξ$ be a nonzero element in $\mathbb{F}_{q^n}$. In this paper, we consider for which $B(X,Y)$, the binary operation $xy+B(x,y)ξ$ defines a (pre)semifield multiplication on $\mathbb{F}_{q^n}$. We prove that this question is equivalent to finding $q$-linearized polynomials $L(X)\in\mathbb{F}_{q^n}[X]$ such that $Tr_{q^n/q}(L(x)/x)\neq 0$ for all $x\in\mathbb{F}_{q^n}^*$. For $n\le 4$, we present several families of $L(X)$ and we investigate the derived (pre)semifields. When $q$ equals a prime $p$, we show that if $n>\frac{1}{2}(p-1)(p^2-p+4)$, $L(X)$ must be $a_0 X$ for some $a_0\in\mathbb{F}_{p^n}$ satisfying $Tr_{q^n/q}(a_0)\neq 0$. Finally, we include a natural connection with certain cyclic codes over finite fields, and we apply the Hasse-Weil-Serre bound for algebraic curves to prove several necessary conditions for such kind of $L(X)$.

math.CO↗
arXiv2014

On Global $\mathcal P$-Forms

Let $\Bbb F_q$ be a finite field with $\text{char}\,\Bbb F_q=p$ and $n>0$ an integer with $\text{gcd}(n, \log_pq)=1$. Let $(\ )^*:\Bbb F_q({\tt x}_0,\dots,{\tt x}_{n-1})\to\Bbb F_q({\tt x}_0,\dots,{\tt x}_{n-1})$ be the $\Bbb F_q$-monomorphism defined by ${\tt x}_i^*={\tt x}_{i+1}$ for $0\le i< n-1$ and ${\tt x}_{n-1}^*={\tt x}_0^q$. For $f,g\in\Bbb F_q({\tt x}_0,\dots,{\tt x}_{n-1})\setminus\Bbb F_q$, define $f\circ g=f(g,g^*,\dots,g^{(n-1)*})$. Then $(\Bbb F_q({\tt x}_0,\dots,{\tt x}_{n-1})\setminus\Bbb F_q,\,\circ)$ is a monoid whose invertible elements are called global $\mathcal P$-forms. Global $\mathcal P$-forms were first introduced by H. Dobbertin in 2001 with $q=2$ to study certain type of permutation polynomials of $\Bbb F_{2^m}$ with $\text{gcd}(m,n)=1$; global $\mathcal P$-forms with $q=p$ for an arbitrary prime $p$ were considered by W. More in 2005. In this paper, we discuss some fundamental questions about global $\mathcal P$-forms, some of which are answered and others remain open.

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