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Hongfeng Wu

Publications and source records attributed to Hongfeng Wu.

18 recordsLinked to original sources
arXiv2026

Singular Cholesky Fibers over Finite Fields

For a finite field $\F_q$, consider the triangular Cholesky map $\Gamma_{n,q}(U)=U^TU$ from upper triangular matrices to symmetric matrices. Generalized Cholesky theory describes the regular locus on which all leading principal minors are nonzero, but it does not determine the multiplicities or root ranks in a fiber over a singular target. We develop a fixed-target fiber theory for this singular boundary and answer three questions posed by Cooper and Whitlatch. Our principal results concern the zero fiber. Over $\F_2$ we construct an explicit, invertible, rank-preserving recursive bijection between square-zero upper triangular matrices and upper triangular matrices satisfying $U^TU=0$. Over every finite field we determine the entire rank distribution of this zero fiber: in even characteristic the square-zero recurrence, and hence the rank-refined equinumerosity, persists over every $\F_{2^e}$, whereas in odd characteristic its failure is measured by an explicit quadratic-character correction. We place these results in a uniform framework by proving an exact first-pivot recursion for the fiber cardinality over an arbitrary symmetric target. Its regular specialization gives the constant fiber sizes on leading-principal-minor cones, while its zero-pivot branch explains the new singular behavior. For binary diagonal targets we further compress the rank-refined count to $O(n^2)$ integer-arithmetic transitions and show that it depends on the order, not merely the number, of the diagonal entries. Executable implementations and exhaustive low-order checks are collected in an appendix.

math.RA↗
arXiv2026

Peirce Stability of Null Polynomials and a Sharp Radical Bound for Finite Rings

Let $R$ be a finite associative ring with identity, and let $\Nul(R)$ denote the set of polynomials in a central indeterminate that vanish identically on $R$ under right evaluation. We establish two results concerning the failure of $\Nul(R)$ to be a right ideal. First, if $e$ is an idempotent, $f=1-e$, and either $eRf=0$ or $fRe=0$, then right multiplication by $e$ preserves $\Nul(R)$. Thus both off-diagonal Peirce components must be nonzero whenever $e$ witnesses a failure of right stability. Second, if $J$ is a finite nilpotent ideal whose additive group is a $2$-group and $e,f$ are complementary idempotents such that \[ eJf\neq0,\qquad fJe\neq0,\qquad J^3\neq0, \] then $|J|\geq32$. Werner's theorem that $\Jac(R)^3=0$ implies the two-sidedness of $\Nul(R)$ therefore yields $|R|\geq128$ whenever $\Nul(R)$ is not two-sided. We also construct a tiled matrix ring of characteristic $4$ and order $128$ whose null ideal is not two-sided.

math.RA↗
arXiv2026

A New Characteristic-Uniform Model for Elliptic Curves -- Theory, Arithmetic, and Applications

We develop a characteristic-uniform arithmetic theory for \[ \mathcal C_d:\quad (u^2+u)(v^2+v)=d. \] For \(d(1-16d)\ne0\), its smooth \((2,2)\)-completion has four rational boundary points forming \(\mathbb Z/4\mathbb Z\), an intrinsic \(D_8\)-action, the inverse \(-(u,v)=(u,-v-1)\), and the native Kummer map \(\kappa_d(u,v)=(u+1)\). Working natively, we derive complete full-point and differential laws, Kummer ladders and recovery, halving, tripling, \(2P+Q\), division polynomials, isogenies, CM endomorphisms, and pairings, with dedicated formulas in characteristics two and three. Classical models provide proof and optimization dictionaries while all endpoints remain native. For Cd25519, the Kummer line is exactly the X25519 line, and a native Segre recoding realizes the optimized complete \(a=-1\) Edwards full-point dependency graph. We further study \[ \begin{gathered} \mathcal C_{a,b,d}:(u^2+u+a)(v^2+v+b)=d,\quad \mathcal T_{a,d}:(u^2+u+a)(v^2+v)=d,\\ \mathcal R_{\tau,\sigma,\kappa}:(x^2-\tau)(y^2-\sigma)=\kappa xy,\quad \mathcal Q_{\alpha,\beta,\gamma}:x^2y^2+\alpha(x^2+y^2)+\beta xy+\gamma=0. \end{gathered} \] For these product, one-sided twisted, reciprocal, and QRT families, we determine their genus-one geometry, finite-field forms, arithmetic, and isogenies. On each smooth QRT fibre, the Vieta--McMillan map is a fixed elliptic translation. Marking \(D\) gives the state \(P\mapsto(\kappa(P),\kappa(P+D))\), with maps realizing \(n\mapsto mn+r\). This yields logarithmic ladders and an elliptic Lucas calculus with nonlinear addition, fast-index doubling, state-division polynomials, and bridges to elliptic divisibility sequences, sigma functions, and elliptic nets. In characteristic two, every ordinary pointed elliptic curve over a perfect field admits the binary state model. We further develop the \(\mathcal C_d\) platform for isogeny-based cryptography.

math.NT↗
arXiv2026

The Optimal Coefficients-Based Criterion for Primitive Quadratic Polynomials over Finite Fields

Let $\mathbb F_q$ be a finite field and consider quadratic polynomials $f(X)=X^2+bX+c\in\mathbb F_q[X]$ with primitive constant term $c$. We construct an optimal coefficients-based determining polynomial for primitive quadratic polynomials over every finite field. More precisely, for every primitive $c\in\mathbb F_q$, its specialization is the unique monic square-free polynomial whose roots are exactly the coefficients $b$ for which $X^2+bX+c$ is primitive. The construction is based on Lucas polynomials and Lucas atoms over finite fields. In odd characteristic, the optimal determining polynomial is the $(q+1)$-st Lucas atom. In characteristic $2$, this Lucas atom has multiplicity $2$ in the variable $B$, and the optimal polynomial is obtained by removing this multiplicity through the inverse Frobenius over $A=\mathbb F_q[C]/(\Phi_{q-1}(C))$. We prove that the determining polynomial is unique for each fixed primitive constant term and, globally, unique as an element of $A[B]$. We also give equivalent criteria involving only recursively computable Lucas polynomials and, as an application, a first-zero coefficient description of the binomial order and order of an irreducible quadratic polynomial.

math.NT↗
arXiv2026

Primitive Quadratic Polynomials in Additive Coset Families

Let \(q\) be an odd prime power. For \(\mu\in\mathbb F_q^\times\) and an additive coset \(\bar\alpha\in\mathbb F_{q^2}/\mathbb F_q\), consider the family \[ \{x^2+\mu x-\alpha:\alpha\in\bar\alpha\}, \] where each polynomial is regarded over its coefficient field \(\mathbb F_q(\alpha)\). We prove that, for \[ q\notin{7,11,13,19,29,31,41,43}, \] every such family contains a primitive polynomial. For the zero coset, the result follows from Cohen's prescribed-trace theorem. For a nonzero coset, after a natural normalization the roots are parameterized by two smooth projective conics arising from the two \(q^2\)-Frobenius eigenspaces in \(\mathbb F_{q^4}\). Their affine \(\mathbb F_q\)-point counts, \(q-1\) and \(q+1\), correspond respectively to the reducible and irreducible members of the family. On the irreducible root conic, \(q^2\)-Frobenius induces a fixed-point-free involution on rational points. Passing to the quotient conic allows the relative norm of the root function to descend to a rational function. Tensor induction then yields order-sensitive character-sum bounds with constants \(6,8\) on the root conic and the sharper constants \(2,4\) after norm descent. Combining these estimates with a double-core prime sieve and an exact residue-cover verification completes the finite range. As consequences, Gow and McGuire's Conjecture~1 holds for every odd prime power \(q\ne13\), while their Conjectures~2 and~3 hold for every odd prime power \(q>43\); the threshold \(43\) is sharp.

math.NT↗
arXiv2026

Consecutive non-square non-primitive tuples in finite fields

Let $q$ be an odd prime power and put \[ \theta_q=\frac{\varphi(q-1)}{q-1}. \] An element of $\Fq$ is called non-square non-primitive, or \emph{NSNP}, if it is both a non-square and a non-primitive element. Let $\ell$ be an odd prime divisor of $q-1$. We obtain a character-sum estimate for the number of translates of an arbitrary finite set lying in the set of non-square $\ell$th powers. Combining this estimate with a finite computation, we prove that $\theta_q<4/15$ guarantees three consecutive NSNP elements. On the boundary $\theta_q=4/15$, the only exceptions are $q\in\{31,61,121\}$. As a further application, we show that for $\operatorname{char}\Fq>3$, the inequality $\theta_q<8/35$ guarantees four consecutive NSNP elements, with $q=211$ as the unique exception on the boundary $\theta_q=8/35$. Thus both constants are best possible.

math.NT↗
arXiv2026

BCH and LCD cyclic codes of length $n=\lambda(q^m+1)$ over finite fields

BCH and LCD cyclic codes of length $n=\lambda(q^m+1)$ with $\lambda\mid q-1$ are studied. A complete characterization of $q$-cyclotomic cosets modulo $n$ is given: Theorem \ref{th4} provides a necessary and sufficient condition for any $0\le \gamma<n$ to be a coset leader, and for odd $m$, the two largest coset leaders are explicitly determined (Theorem \ref{th9} and Theorem \ref{th14}). Based on these results, the dimensions of several families of BCH codes are determined, and the lower bound on the minimal distance of $\mathcal{C}_{(q,n,2\delta+1,n-\delta+1)}$ is raised to $2(\delta+1)$ (Theorem \ref{th15}--\ref{th5}). Notably, several of these codes are optimal. When $m$ is odd, the necessary and sufficient condition for the BCH code $\mathcal{C}_{(q,n,\delta,0)}$ to be dually-BCH is proved (Theorem \ref{th11}). Finally, an exact enumeration of all LCD cyclic codes of this length is derived (Theorem \ref{th3}). All of the above results extend previous results that were limited to $\lambda=1$.

cs.IT↗
arXiv2026

The Arithmetic Singleton Bound on the Hamming Distances of Simple-rooted Constacyclic Codes over Finite Fields

In this work, We introduce a new upper bound on the Hamming distance of simple-root constacyclic codes over finite fields, which we call the arithmetic Singleton bound. The main technical tool is the notion of a multiple equal-difference (MED) representation. Via the MED representations of the defining set of the generator polynomial of a simple-root constacyclic code, we obtain a family of upper bounds on its Hamming distance, among which the weakest one coincides with the Singleton bound, while the strongest one is defined to be the arithmetic Singleton bound for this code. Consequently, the arithmetic Singleton bound is always at least as strong as the classical Singleton bound, and is in fact strictly stronger in numerous nontrivial cases. The arithmetic Singleton bound partially measures the restriction on the Hamming distance of a simple-root constacyclic code imposed by its arithmetic structure. In particular, for an irreducible constacyclic code, the MED representations of the defining set of its generator polynomial are completely determined, via which the arithmetic Singleton bound is computed concretely. Finally for any simple-root cyclic code the arithmetic Singleton bound and the BCH bound are compared.

cs.IT↗
arXiv2025

Finite fields whose members are the sum of a potent and a 5-potent

We show that there are only finitely many finite fields whose members are the sum of an $n$-potent element and a $5$-potent element. Combining this with the algorithmic results provided by S.D. Cohen {\it et al.}, we confirm in the affirmative the conjecture in \cite{Cohen} concerning all finite fields satisfying this condition. Furthermore, we obtain several elementary results for General problem, proving that the number of finite fields satisfying general condition is also finite.

math.NT↗
arXiv2025

The Minimal Binomial Multiples of Polynomials over Finite Fields

Let $f(X)$ be a nonconstant polynomial over $\mathbb{F}_{q}$, with a nonzero constant term. The order of $f(X)$ is a classical notion in the theory of polynomials over finite fields, and recently the definition of freeness of binomials of $f(X)$ was given in \cite{Mart\'{i}nez}. Generalizing these two notions, we introduce the definition of the minimal binomial multiple of $f(X)$ in this paper, which is the monic binomial with the lowest degree among the binomials over $\mathbb{F}_{q}$ divided by $f(X)$. Based on the equivalent characterization of binomials via the defining sets of their radicals, we prove that a series of properties of the classical order can be naturally generalized to this case. In particular, the minimal binomial multiple of $f(X)$ is presented explicitly in terms of the defining set of the radical of $f(X)$. And a criterion for $f(X)$ being free of binomials is given. As an application, for any positive integer $N$ and nonzero element $\lambda$ in $\mathbb{F}_{q}$, the $\lambda$-constacyclic codes of length $N$ with minimal distance $2$ are determined.

math.NT↗
arXiv2025

Cyclotomy, cyclotomic cosets and arithmetic properties of some families in $\frac{\mathbb{F}_l[x]}{\langle x^{p^sq^t}-1\rangle}$

Arithmetic properties of some families in $\frac{\mathbb{F}_l[x]}{\langle x^{p^sq^t}-1\rangle}$ are obtained by using the cyclotomic classes of order 2 with respect to $n=p^sq^t$, where $p\equiv3 \mathrm{mod} 4$, $\gcd(\phi(p^s),\phi(q^t))=2$, $l$ is a primitive root modulo $q^t$ and $\mathrm{ord}_{p^s}(l)=\phi(p^s)/2$. The form of these cyclotomic classes enables us to further generalize the results obtained in \cite{ref1}. The explicit expressions of primitive idempotents of minimal ideals in $\frac{\mathbb{F}_l[x]}{\langle x^{p^sq^t}-1\rangle}$ are also obtained.

math.NT↗
arXiv2025

The Multiple Equal-Difference Structure of Cyclotomic Cosets

In this paper we introduce the definition of equal-difference cyclotomic coset, and prove that in general any cyclotomic coset can be decomposed into a disjoint union of equal-difference subsets. Among the equal-difference decompositions of a cyclotomic coset, an important class consists of those in the form of cyclotomic decompositions, called the multiple equal-difference representations of the coset. There is an equivalent correspondence between the multiple equal-difference representations of $q$-cyclotomic cosets modulo $n$ and the irreducible factorizations of $X^{n}-1$ in binomial form over finite extension fields of $\mathbb{F}_{q}$. We give an explicit characterization of the multiple equal-difference representations of any $q$-cyclotomic coset modulo $n$, through which a criterion for $X^{n}-1$ factoring into irreducible binomials is obtained. In addition, we present an algorithm to simplify the computation of the leaders of cyclotomic cosets.

math.NT↗
arXiv2024

Explicit Representatives and Sizes of Cyclotomic Cosets II: Cyclotomic Systems and Applications to Constacyclic Codes

Let $q=p^e$ be a prime power, let $\ell$ be a prime with $\gcd(\ell,qn)=1$, and let $n$ be a positive integer. While $q$-cyclotomic cosets are usually considered modulo a fixed integer, we consider their behavior along the sequence $n,\ell n,\ell^2n,\ldots$ of moduli. To describe the resulting compatibility among cyclotomic cosets, we introduce the $\ell$-adic $q$-cyclotomic system with base module $n$, defined as the projective limit of the spaces of $q$-cyclotomic cosets modulo $\ell^i n$. We associate to each $q$-cyclotomic coset modulo $n$ a cyclotomic $\ell$-adic integer and use its $\ell$-adic expansion to describe all compatible liftings through the successive levels. This leads to a classification of the corresponding sequences according to their splitting and stable behaviors, together with explicit formulas for the representatives and sizes of their components. We treat separately the cases of odd $\ell$ and $\ell=2$. These results are then used to determine representatives and sizes of $q$-cyclotomic cosets for arbitrary admissible parameters. As an application, we obtain explicit irreducible factorizations of binomials over finite fields and use them to describe the corresponding constacyclic codes.

math.NT↗
arXiv2024

Explicit Representatives and Sizes of Cyclotomic Cosets and their Application to Cyclic Codes over Finite Fields

Cyclotomic coset is a classical notion in the theory of finite field which has wide applications in various computation problems. Let $q$ be a prime power, and $n$ be a positive integer coprime to $q$. In this paper we determine explicitly the representatives and the sizes of all $q$-cyclotomic cosets modulo $n$ in the general settings. We introduce the definition of $2$-adic cyclotomic system, which is a profinite space consists of certain compatible sequences of cyclotomic cosets. A precise characterization of the structure of the $2$-adic cyclotomic system is given, which reveals the general formula for representatives of cyclotomic cosets. With the representatives and the sizes of $q$-cyclotomic cosets modulo $n$, we improve the formulas for the factorizations of $X^{n}-1$ and of $\Phi_{n}(X)$ over $\mathbb{F}_{q}$ given in \cite{Graner}. As a consequence, we classify the cyclic codes over finite fields via giving their generator polynomials. Moreover, the self-dual cyclic codes are determined and enumerated.

cs.IT↗
arXiv2011

A mean value formula for elliptic curves

It is proved in this paper that for any point on an elliptic curve, the mean value of x-coordinates of its n-division points is the same as its x-coordinate and that of y-coordinates of its n-division points is n times of its y-coordinate.

math.NT↗
arXiv2010

On the Construction of Finite Oscillator Dictionary

A finite oscillator dictionary which has important applications in sequences designs and the compressive sensing was introduced by Gurevich, Hadani and Sochen. In this paper, we first revisit closed formulae of the finite split oscillator dictionary $\mathfrak{S}^s$ by a simple proof. Then we study the non-split tori of the group $SL(2,\mathbb{F}_p)$. Finally, An explicit algorithm for computing the finite non-split oscillator dictionary $\mathfrak{S}^{ns}$ is described.

cs.IT↗
arXiv2010

Number of general Jacobi quartic curves over finite fields

In this paper the number of $\bar{\mathbb{F}}_q$-isomorphism classes of general Jacobi quartic curves, i.e., the number of general Jacobi quartic curves with distinct $j$-invariants, over the finite field $\mathbb{F}_q$ is enumerated.

math.AG↗