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Haibin Kan

Publications and source records attributed to Haibin Kan.

At least 19 recordsLinked to original sources
arXiv2026

$r$-Minimal Poset Codes

In this paper, we propose and study $r$-minimal codes with respect to $\mathbf{P}$-support, where $\mathbf{P}=(\Omega,\preccurlyeq_{\mathbf{P}})$ is a poset defined on the coordinate set of the ambient space $\mathbf{H}$. $r$-Minimal $\mathbf{P}$-codes are natural extensions of Hamming metric minimal codes that have been extensively studied in the literature. We characterize $r$-minimal $\mathbf{P}$-codes in terms of the notion so called cutting $r$-blocking maps, which generalizes the well-known equivalence between minimal Hamming metric codes and cutting blocking sets. We also give a necessary and sufficient condition for $r$-minimality in terms of $(\mathbf{P},\omega)$-weight defined on $\mathbf{H}$, where $\omega:\Omega\longrightarrow\mathbb{R}^{+}$ is an arbitrary weight function. This leads to a generalization of the well-known Ashikhmin-Barg criterion for Hamming metric minimal codes. We then prove two existence results for $r$-minimal $\mathbf{P}$-codes, both for general $\mathbf{P}$ and for the special case that $\mathbf{P}$ is a disjoint union of chains. When $\mathbf{P}$ is hierarchical, we characterize $r$-minimal $\mathbf{P}$-codes in terms of $r$-minimal Hamming metric codes. Finally, we characterize cutting $r$-blocking sets induced by hierarchical posets with two levels, which further enables us to answer a question raised in Hyun, Kim, Wu and Yue \cite{28}.

cs.IT↗
arXiv2026

List-Decoding Counterexamples Yield Lower Bounds on Mutual Correlated Agreement Error

Mutual correlated agreement captures whether a random linear combination of received words can create a new large agreement with a code, a property relevant to the soundness of batched proximity testing. We show constructively that list-decoding counterexamples yield lower bounds on the mutual correlated agreement error. Given an explicit counterexample to the $(p,L)$-list-decodability of a linear code over $\mathbb{F}_q$, we construct a related code $C'$ of the same length and dimension such that $\operatorname{err}_{\mathrm{MCA}}(C',p)\ge\frac{1}{q}\left\lceil\frac{(L+1)q}{q+L}\right\rceil$, while decreasing its minimum distance by at most one. The construction also produces an explicit pair of words witnessing this error. We further give a structure-preserving version for code families whose coordinates are indexed by a finite set $\Omega$, with each index determining a generator-matrix column through a map $v:\Omega\to\mathbb{F}_q^k$. The construction changes at most one coordinate index and ensures that the output code remains in the same indexed family. As applications, we instantiate this principle for algebraic-geometry (AG) evaluation codes and Reed--Solomon codes. For AG codes, if $G$ is the divisor defining the underlying Riemann--Roch space and $N$ is the number of rational places outside $\operatorname{supp}(G)$ available for evaluation, the resulting code remains over the same function field and Riemann--Roch space, with a modified set of evaluation places. Its mutual correlated agreement error is at least $\frac{1}{q}\left\lceil\frac{(L+1)N}{N+L\mathrm{deg} G}\right\rceil$. The Reed--Solomon conclusion follows as the Vandermonde-column specialization.

cs.IT↗
arXiv2026

Generalized Rank Weight and Extended Generalized Poset Weight Defined For Codes Over Rings: A Galois Connection Approach

In this paper, we study generalized rank weights (GRWs) and extended generalized poset weight (EGPWs) of codes over rings via a Galois connection approach. First, we show that various coding-theoretic properties related to generalized weights, including security drops of a code employed in wire-tap channel of type II, connections between generalized weights of a Gabidulin code and its associated Delsarte code, (generalized) Singleton bound, MDS discrepancy of a code, characterizations of MDS, near MDS, $i$-MDS, MRD, near MRD, $i$-MRD, (dually) quasi-MRD codes as well as evasive property of subspaces, can be reformulated in terms of Galois connections. Next, we study GRWs and rank profiles defined for modules over principal ideal rings, especially those over chain rings. Generalizing GRWs defined for vector spaces over fields, we establish a singleton bound and a Wei-type duality theorem, characterize MRD, near MRD and dually quasi-MRD codes and determine their GRWs; moreover, we characterize $i$-MRD codes and establish a scattered bound for $(h,h)$-evasive codes over chain rings, generalizing counterpart result established for vector space over finite fields. Finally, we propose and study EGPWs and extended poset profiles defined for modules with a composition series, which in fact form a Galois connection. Generalizing EGPWs defined for modules over finite Galois rings, we establish a Wei-type duality theorem for modules over arbitrary quasi-Frobenius rings, which unifies the two Wei-type duality theorems derived in both \cite{32} and \cite{33}.

cs.IT↗
arXiv2025

An Elementary Approach to MacWilliams Extension Property and Constant Weight Code with Respect to Weighted Hamming Metric

In this paper, we characterize the MacWilliams extension property (MEP) and constant weight codes with respect to $\omega$-weight defined on $\mathbb{F}^{\Omega}$ via an elementary approach, where $\mathbb{F}$ is a finite field, $\Omega$ is a finite set, and $\omega:\Omega\longrightarrow\mathbb{R}^{+}$ is a weight function. Our approach relies solely on elementary linear algebra and two key identities for $\omega$-weight of subspaces derived from a double-counting argument. When $\omega$ is the constant $1$ map, our results recover two well-known results for Hamming metric code: (1) any Hamming weight preserving map between linear codes extends to a Hamming weight isometry of the entire ambient space; and (2) any constant weight Hamming metric code is a repetition of the dual of Hamming code.

cs.IT↗
arXiv2025

Reflexive Partitions Induced by Rank Support and Non-Reflexive Partitions Induced by Rank Weight

In this paper, we study partitions of finite modules induced by rank support and rank weight. First, we show that partitions induced by rank support are mutually dual with respect to suitable non-degenerate pairings, and hence are reflexive; moreover, we compute the associated generalized Krawtchouk matrices. Similar results are established for partitions induced by isomorphic relation of rank support. These results generalize counterpart results established for row space partitions and rank partitions of matrix spaces over finite fields. Next, we show that partitions of free modules over a finite chain ring $R$ induced by rank weight are non-reflexive provided that $R$ is not a field; moreover, we characterize the dual partitions explicitly. As a corollary, we show that rank partitions of matrix spaces over $R$ are reflexive if and only if $R$ is a field; moreover, two matrices belong to the same member of the dual partition if and only if their transposes are equivalent. In particular, we show that opposite to matrices over finite fields, rank metric does not induce an association scheme provided that $R$ is not a field, which further settles an open question proposed by Blanco-Chac\'{o}n, Boix, Greferath and Hieta-Aho in \cite{2}.

cs.IT↗
arXiv2024

Data sharing in the metaverse with key abuse resistance based on decentralized CP-ABE

Data sharing is ubiquitous in the metaverse, which adopts blockchain as its foundation. Blockchain is employed because it enables data transparency, achieves tamper resistance, and supports smart contracts. However, securely sharing data based on blockchain necessitates further consideration. Ciphertext-policy attribute-based encryption (CP-ABE) is a promising primitive to provide confidentiality and fine-grained access control. Nonetheless, authority accountability and key abuse are critical issues that practical applications must address. Few studies have considered CP-ABE key confidentiality and authority accountability simultaneously. To our knowledge, we are the first to fill this gap by integrating non-interactive zero-knowledge (NIZK) proofs into CP-ABE keys and outsourcing the verification process to a smart contract. To meet the decentralization requirement, we incorporate a decentralized CP-ABE scheme into the proposed data sharing system. Additionally, we provide an implementation based on smart contract to determine whether an access control policy is satisfied by a set of CP-ABE keys. We also introduce an open incentive mechanism to encourage honest participation in data sharing. Hence, the key abuse issue is resolved through the NIZK proof and the incentive mechanism. We provide a theoretical analysis and conduct comprehensive experiments to demonstrate the feasibility and efficiency of the data sharing system. Based on the proposed accountable approach, we further illustrate an application in GameFi, where players can play to earn or contribute to an accountable DAO, fostering a thriving metaverse ecosystem.

cs.CR↗
arXiv2024

Registered Attribute-Based Encryption with Reliable Outsourced Decryption Based on Blockchain

Decentralized data sovereignty and secure data exchange are regarded as foundational pillars of the new era. Attribute-based encryption (ABE) is a promising solution that enables fine-grained access control in data sharing. Recently, Hohenberger et al. (Eurocrypt 2023) introduced registered ABE (RABE) to eliminate trusted authority and gain decentralization. Users generate their own public and secret keys and then register their keys and attributes with a transparent key curator. However, RABE still suffers from heavy decryption overhead. A natural approach to address this issue is to outsource decryption to a decryption cloud server (DCS). In this work, we propose the first auditable RABE scheme with reliable outsourced decryption (ORABE) based on blockchain. First, we achieve verifiability of transform ciphertext via a verifiable tag mechanism. Then, the exemptibility, which ensures that the DCS escapes false accusations, is guaranteed by zero knowledge fraud proof under the optimistic assumption. Additionally, our system achieves fairness and auditability to protect the interests of all parties through blockchain. Finally, we give concrete security and theoretical analysis and evaluate our scheme on Ethereum to demonstrate feasibility and efficiency.

cs.CR↗
arXiv2024

Attribute-Based Encryption With Payable Outsourced Decryption Using Blockchain and Responsive Zero Knowledge Proof

Attribute-Based Encryption (ABE) is a promising solution for access control in cloud services. However, the heavy decryption overhead hinders its widespread adoption. A general approach to address this issue is to outsource decryption to decryption cloud service(DCS). Existing schemes have utilized various methods to enable users to verify outsourced results; however, they lack an effective mechanism to achieve exemptibility which enables the honest DCS to escape from wrong claims. And it is impractical to assume that the DCS will provide free services. In this paper, we propose a blockchain-based payable outsourced decryption ABE scheme that achieves both verifiability and exemptibility without adding redundant information to ABE ciphertext. We use zero-knowledge proof to verify outsourced results on blockchain and introduce an optional single-round challenge game under optimistic assumption to address the high cost of proof generation. Moreover, our system achieves fairness and decentralized outsourcing to protect the interests of all parties. Finally, we implement and evaluate our scheme on Ethereum to demonstrate its feasibility and efficiency, the gas usage in attribute numbers from 5 to 60 is 11$\times$ to 140$\times$ in the happy case and 4$\times$ to 55$\times$ in the challenge case lower than the scheme of Ge et al. (TDSC'23).

cs.CR↗
arXiv2024

$r$-Minimal Codes with Respect to Rank Metric

In this paper, we propose and study $r$-minimal codes, a natural extension of minimal codes which have been extensively studied with respect to Hamming metric, rank metric and sum-rank metric. We first propose $r$-minimal codes in a general setting where the ambient space is a finite dimensional left module over a division ring and is supported on a lattice. We characterize minimal subcodes and $r$-minimal codes, derive a general singleton bound, and give existence results for $r$-minimal codes by using combinatorial arguments. We then consider $r$-minimal rank metric codes over a field extension $\mathbb{E}/\mathbb{F}$ of degree $m$, where $\mathbb{E}$ can be infinite. We characterize these codes in terms of cutting $r$-blocking sets, generalized rank weights of the codes and those of the dual codes, and classify codes whose $r$-dimensional subcodes have constant rank support weight. Next, with the help of the evasiveness property of cutting $r$-blocking sets and some upper bounds for the dimensions of evasive subspaces, we derive several lower and upper bounds for the minimal length of $r$-minimal codes. Furthermore, when $\mathbb{E}$ is finite, we establish a general upper bound which generalizes and improves the counterpart for minimal codes in the literature. As a corollary, we show that if $m=3$, then for any $k\geqslant2$, the minimal length of $k$-dimensional minimal codes is equal to $2k$. To the best of our knowledge, when $m\geqslant3$, there was no known explicit formula for the minimal length of $k$-dimensional minimal codes for arbitrary $k$ in the literature.

cs.IT↗
arXiv2024

Minimal Ternary Linear Codes from Vectorial Functions

The study on minimal linear codes has received great attention due to their significant applications in secret sharing schemes and secure two-party computation. Until now, numerous minimal linear codes have been discovered. However, to the best of our knowledge, no infinite family of minimal ternary linear codes was found from vectorial functions. In this paper, we present a necessary and sufficient condition for a large class of ternary linear codes from vectorial functions such that those codes are minimal. Based on that, we construct several minimal ternary linear codes with three-weight from vectorial regular plateaued functions, and determine their weight distributions. Moreover, we also give a necessary and sufficient condition for a large family of ternary linear codes from vectorial functions such that the codes are minimal and violate the AB condition simultaneously. According to this characterization, we find several minimal ternary linear codes violating the AB condition. Notably, our results show that our method can be applied to solve a problem on minimal linear codes proposed by Li et al.

cs.IT↗
arXiv2023

Trace Monomial Boolean Functions with Large High-Order Nonlinearities

Exhibiting an explicit Boolean function with a large high-order nonlinearity is an important problem in cryptography, coding theory, and computational complexity. We prove lower bounds on the second-order, third-order, and higher-order nonlinearities of some trace monomial Boolean functions. We prove lower bounds on the second-order nonlinearities of functions $\mathrm{tr}_n(x^7)$ and $\mathrm{tr}_n(x^{2^r+3})$ where $n=2r$. Among all trace monomials, our bounds match the best second-order nonlinearity lower bounds by \cite{Car08} and \cite{YT20} for odd and even $n$ respectively. We prove a lower bound on the third-order nonlinearity for functions $\mathrm{tr}_n(x^{15})$, which is the best third-order nonlinearity lower bound. For any $r$, we prove that the $r$-th order nonlinearity of $\mathrm{tr}_n(x^{2^{r+1}-1})$ is at least $2^{n-1}-2^{(1-2^{-r})n+\frac{r}{2^{r-1}}-1}- O(2^{\frac{n}{2}})$. For $r \ll \log_2 n$, this is the best lower bound among all explicit functions.

cs.CR↗
arXiv2023

A Gr\"{o}bner Basis Approach to Combinatorial Nullstellensatz

In this paper, using some conditions that arise naturally in Alon's combinatorial Nullstellensatz as well as its various extensions and generalizations, we characterize Gr\"{o}bner bases consisting of monic polynomials, which helps us to establish a Nullstellensatz from a Gr\"{o}bner basis perspective. As corollaries of this general Nullstellensatz, we establish four special Nullstellensatz, which, among others, include a common generalization of the Nullstellensatz for multisets established in K\'{o}s, R\'{o}nyai and M\'{e}sz\'{a}ros \cite{23,24} and the Nullstellensatz with multiplicity established in Ball and Serra \cite{9}, and include a punctured Nullstellensatz, generalizing several existing results in the literature. As applications of our punctured Nullstellensatz, we extend some results on hyperplane covering in \cite{9,23,24} to wider settings, and give an alternative proof of the generalized Alon-F\"{u}redi theorem established in Bishnoi, Clark, Potukuchi and Schmitt \cite{12}. Unless specified otherwise, all our results are established over an arbitrary commutative ring $R$.

math.CO↗
arXiv2022

Hitting Times of Random Walks on Edge Corona Product Graphs

Graph products have been extensively applied to model complex networks with striking properties observed in real-world complex systems. In this paper, we study the hitting times for random walks on a class of graphs generated iteratively by edge corona product. We first derive recursive solutions to the eigenvalues and eigenvectors of the normalized adjacency matrix associated with the graphs. Based on these results, we further obtain interesting quantities about hitting times of random walks, providing iterative formulas for two-node hitting time, as well as closed-form expressions for the Kemeny's constant defined as a weighted average of hitting times over all node pairs, as well as the arithmetic mean of hitting times of all pairs of nodes.

cs.SI↗
arXiv2022

Resistance Distances in Simplicial Networks

It is well known that in many real networks, such as brain networks and scientific collaboration networks, there exist higher-order nonpairwise relations among nodes, i.e., interactions between among than two nodes at a time. This simplicial structure can be described by simplicial complexes and has an important effect on topological and dynamical properties of networks involving such group interactions. In this paper, we study analytically resistance distances in iteratively growing networks with higher-order interactions characterized by the simplicial structure that is controlled by a parameter q. We derive exact formulas for interesting quantities about resistance distances, including Kirchhoff index, additive degree-Kirchhoff index, multiplicative degree-Kirchhoff index, as well as average resistance distance, which have found applications in various areas elsewhere. We show that the average resistance distance tends to a q-dependent constant, indicating the impact of simplicial organization on the structural robustness measured by average resistance distance.

cs.SI↗
arXiv2022

On the Uniqueness of Balanced Complex Orthogonal Design

Complex orthogonal designs (CODs) play a crucial role in the construction of space-time block codes. Their real analog, real orthogonal designs (or equivalently, sum of squares composition formula) have a long history. Adams et al. (2011) introduced the concept of balanced complex orthogonal designs (BCODs) to address practical considerations. BCODs have a constant code rate of $1/2$ and a minimum decoding delay of $2^m$, where $2m$ is the number of columns. Understanding the structure of BCODs helps design space-time block codes, and it is also fascinating in its own right. We prove, when the number of columns is fixed, all (indecomposable) balanced complex orthogonal designs (BCODs) have the same parameters $[2^m, 2m, 2^{m-1}]$, and moreover, they are all equivalent.

cs.IT↗
arXiv2022

Minimal Binary Linear Codes from Vectorial Boolean Functions

Recently, much progress has been made to construct minimal linear codes due to their preference in secret sharing schemes and secure two-party computation. In this paper, we put forward a new method to construct minimal linear codes by using vectorial Boolean functions. Firstly, we give a necessary and sufficient condition for a generic class of linear codes from vectorial Boolean functions to be minimal. Based on that, we derive some new three-weight minimal linear codes and determine their weight distributions. Secondly, we obtain a necessary and sufficient condition for another generic class of linear codes from vectorial Boolean functions to be minimal and to be violated the AB condition. As a result, we get three infinite families of minimal linear codes violating the AB condition. To the best of our knowledge, this is the first time that minimal liner codes are constructed from vectorial Boolean functions. Compared with other known ones, in general the minimal liner codes obtained in this paper have higher dimensions.

cs.IT↗
arXiv2022

The Covering Radius of the Third-Order Reed-Muller Code RM(3,7) is 20

We prove the covering radius of the third-order Reed-Muller code RM(3,7) is 20, which was previously known to be between 20 and 23 (inclusive). The covering radius of RM(3, 7) is the maximum third-order nonlinearity among all 7-variable Boolean functions. It was known that there exist 7-variable Boolean functions with third-order nonlinearity 20. We prove the third-order nonlinearity cannot achieve 21. According to the classification of the quotient space of RM(6,6)/RM(3,6), we classify all 7-variable Boolean functions into 66 types. Firstly, we prove 62 types (among 66) cannot have third-order nonlinearity 21; Secondly, we prove function of the remaining 4 types can be transformed into a type (6, 10) function, if its third-order nonlinearity is 21; Finally, we transform type (6, 10) functions into a specific form, and prove the functions in that form cannot achieve third-order nonlinearity 21 (with the assistance of computers). By the way, we prove that the affine transformation group over any finite field can be generated by two elements.

cs.IT↗