arXiv subjects

Fang-Wei Fu

Publications and source records attributed to Fang-Wei Fu.

At least 19 recordsLinked to original sources

arXiv2026

Construction of Multi-sequences With High Nonlinear Complexity via Narrow Ray Class Fields

Nonlinear complexity is a fundamental criterion in the evaluation of pseudorandom sequences. The construction of multi-sequences with high nonlinear complexity is both theoretically and practically important in cryptography. Motivated by prior constructions of multi-sequences with high nonlinear complexity in [IEEE Trans. Inf. Theory, 60(10), 2014] and [IEEE Trans. Inf. Theory, 63(12), 2017], we provide a unified framework via narrow ray class fields, the cyclic descent due to Guruswami and Xing in [J. Combin. Theory Ser. A 129 (2015) ]. Then we can generate new multi-sequences with high nonlinear complexity over function fields with arbitrary genera.

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arXiv2026

Quasi-MSRD Codes and Their Properties

Sum-rank-metric codes have recently attracted considerable attention of many researchers, due to their applications in network coding, space-time codes and distributed storage. MSRD codes are those good codes attaining the Singleton bound in the sum-rank metric. However, MSRD codes do not exist for some dimensions. Motivated by this fact, we introduce the notion of quasi-MSRD (QMSRD) codes and provide some properties of them. What is more, we find that not every QMSRD code has a QMSRD dual code, so we give the definition of dually QMSRD codes, whose dual codes and themselves are both QMSRD. Finally, we characterize such codes, derive their support, rank-list and sum-rank distributions as well as compute their generalized sum-rank weights.

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arXiv2026

On the Minimum Field Size of Network MDS Codes for Generalized Combination Networks

This paper investigates the minimum field size required for network maximum distance separable (MDS) codes, a critical parameter affecting computational complexity at network nodes. Focusing on generalized combination networks and Zosin Khuller networks, we develop a systematic framework for both scalar and vector network MDS codes. For scalar codes on generalized combination networks, we establish an equivalence between the minimum distance of network codes and the minimum Hamming distance of classical linear codes, converting network-level MDS constraints into coding theory conditions. This yields necessary and sufficient existence conditions linked to classical MDS codes and covering Grassmannian codes. Using refined greedy constructions and MRD code designs, we obtain improved bounds on the minimum field size, outperforming the prior universal bound. For vector network codes, we develop an analogous distance equivalence and characterize existence via covering Grassmannian codes, yielding bounds on the minimum effective field size. Notably, the gap between optimal scalar and vector MDS codes vanishes for several parameter regimes; we explicitly identify a family of such networks where vector coding offers no field size advantage over scalar coding. For Zosin Khuller networks, we derive lower bounds on the minimal effective field size of vector MDS codes using hypergraph homomorphisms and subset intersection arguments, strictly improving prior scalar bounds. We further provide a necessary and sufficient condition built upon hypergraph homomorphisms for vector MDS construction, yielding an upper bound on the minimum field size. Finally, we bound the MDS gap between optimal scalar and vector solutions.

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arXiv2026

New families of asymptotically optimal codebooks from vectorial dual-bent functions

Codebooks with small maximum cross-correlation amplitudes play an important role in many applications, such as code division multiple access (CDMA) communication systems, multiple-input multiple-output (MIMO) communications, compressed sensing, and coding theory. In this paper, by using vectorial dual-bent functions, we construct several families of codebooks that asymptotically achieve the Welch bound. The maximum cross-correlation amplitudes and the distributions of the cross-correlation amplitudes of the constructed codebooks are explicitly determined. Furthermore, these codebooks have new parameters, and some of them have very small alphabet sizes.

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arXiv2026

Constructions and Characterizations of $s$-Plateaued Partitions

Bent partitions play a significant role in constructing bent functions and have rich connections with coding theory and combinatorics. In this paper, we introduce $s$-plateaued partitions, which generalize the bent partitions. Let $Γ=\{A_{i}, 1 \leq i \leq K\}$ be a partition of $V_{n}^{(p)}$, where $V_{n}^{(p)}$ is an $n$-dimensional vector space over the prime field $\mathbb{F}_{p}$ and $p \mid K$. Then $Γ$ is called an $s$-plateaued partition of $V_{n}^{(p)}$ of depth $K$ if each $p$-ary function $f: V_{n}^{(p)} \rightarrow \mathbb{F}_{p}$ for which every $j \in \mathbb{F}_{p}$ has exactly $\frac{K}{p}$ of sets $A_{i}$ in $Γ$ in its preimage set, is a $p$-ary $s$-plateaued function. By using an $s$-plateaued partition, a large number of $p$-ary $s$-plateaued functions, vectorial $s$-plateaued functions and generalized $s$-plateaued functions can be constructed. In particular, $0$-plateaued partitions are just bent partitions. In general, $s$-plateaued partitions are much more complicated than bent partitions. We analyze the possible cardinality of $A_{i}$ of an $s$-plateaued partition. We give some explicit constructions of $s$-plateaued partitions for which any generated $p$-ary $s$-plateaued function has no nonzero linear structure. We give a characterization of an $s$-plateaued partition $Γ=\{A_{i}, 1 \leq i \leq K\}$, where $p$ is odd, $K \geq 5$ and $-A_{i}=A_{i}, 1 \leq i \leq K$. Based on which, we show that if $p \geq 5$, then the preimage set partition of a $p$-ary $s$-plateaued function $f: V_{n}^{(p)} \rightarrow \mathbb{F}_{p}$ with $f(x)=f(-x)$ is an $s$-plateaued partition if and only if $f$ is of $(p-1)$-form, where $n+s$ is even.When $s=0$, we partially address an open problem on whether a bent partition $Γ$ of $V_{n}^{(p)}$ of depth $p^{\frac{n}{2}}$ must be obtained from spreads.

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arXiv2026

A New Family of Binary Sequences via Elliptic Function Fields over Finite Fields of Odd Characteristics

Motivated by the constructions of binary sequences by utilizing the cyclic elliptic function fields over the finite field $\mathbb{F}_{2^{n}}$ by Jin \textit{et al.} in [IEEE Trans. Inf. Theory 71(8), 2025], we extend the construction to the cyclic elliptic function fields with odd characteristic by using the quadratic residue map $η$ instead of the trace map used therein. For any cyclic elliptic function field with $q+1+t$ rational points and any positive integer $d$ with $\gcd(d, q+1+t)=1$, we construct a new family of binary sequences of length $q+1+t$, size $q^{d-1}-1$, balance upper bounded by $(d+1)\cdot\lfloor2\sqrt{q}\rfloor+|t|+d,$ the correlation upper bounded by $(2d+1)\cdot\lfloor2\sqrt{q}\rfloor+|t|+2d$ and the linear complexity lower bounded by $\frac{q+1+2t-d-(d+1)\cdot\lfloor2\sqrt{q}\rfloor}{d+d\cdot\lfloor2\sqrt{q}\rfloor}$ where $\lfloor x\rfloor$ stands for the integer part of $x\in\mathbb{R}$.

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arXiv2026

Reed-Solomon Codes with Optimal Repair Bandwidth: A Basis-Transformation Approach

Maximum distance separable (MDS) codes are widely used in distributed storage, but naively repairing a single failure in an $(n,k)$ MDS code requires downloading the full contents of $k$ surviving nodes. Minimum storage regenerating (MSR) codes, introduced by Dimakis et al., minimize repair bandwidth while preserving the MDS property by contacting $d>k$ helper nodes and downloading only a fraction of each helper. For scalar MDS codes, Guruswami and Wootters established a linear repair framework, and Tamo, Ye, and Barg subsequently gave the first explicit Reed-Solomon (RS) codes achieving the MSR point. Their construction yields RS-MSR codes with subpacketization $\ell=s\prod_{i=1}^n p_i$, where $s=d+1-k$ and the distinct primes $p_i$ satisfy $p_i\equiv 1\pmod{s}$. In this paper, we show that this congruence condition is not intrinsic to the RS repair problem. We develop a basis-transformation approach to the construction of repair-enabling subspaces. The approach consists of three deterministic operations -- Euclidean Square Partition, Transposition, and Column Aggregation -- which construct the required repair-enabling subspaces directly from the standard monomial basis of the repair field. Consequently, we obtain RS-MSR codes with subpacketization $\ell=s\prod_{i=1}^n p_i$ for arbitrary distinct primes $p_i>s$. For fixed $s$, this improves the subpacketization of the Tamo--Ye--Barg construction by a factor asymptotic to $φ(s)^{n+\mathrm{o}(n)}$, where $φ(\cdot)$ denotes Euler's totient function.

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arXiv2026

Constructions of locally repairable codes via concatenated codes

In recent years, locally repairable codes (LRCs) have attracted considerable attention owing to their pivotal role in distributed storage systems. Since binary linear locally repairable codes can significantly reduce the complexity of both encoding and decoding processes, the construction of binary LRCs has attracted extensive research interest. In this paper, we construct locally repairable codes via concatenated codes and present a systematic approach to select outer codes to obtain optimal binary LRCs, where the outer codes are linear codes over $\mathbb{F}_4$. The weight distributions of the resulting LRCs are determined by the weight distributions of the selected linear codes over $\mathbb{F}_4$. Furthermore, several classes of optimal binary locally repairable codes are constructed, including binary LRCs meeting the Griesmer-like bound, and binary perfect LRCs. Meanwhile, for the locality $r=2$, we improve the Johnson-like bound for binary LRCs with disjoint local repair groups established by Ma and Ge, and construct explicit LRCs that attain this new bound.

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arXiv2026

On the Number of Subsequences in the Nonbinary Deletion Channel

In the deletion channel, an important problem is to determine the number of subsequences derived from a string $U$ of length $n$ when subjected to $t$ deletions. It is well-known that the number of subsequences in the setting exhibits a strong dependence on the number of runs in the string $U$, where a run is defined as a maximal substring of identical characters. In this paper we study the number of subsequences of a non-binary string in this scenario, and propose some improved bounds on the number of subsequences of $r$-run non-binary strings. Specifically, we characterize a family of $r$-run non-binary strings with the maximum number of subsequences under any $t$ deletions, and show that this number can be computed in polynomial time.

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arXiv2026

Some New Results on Sequence Reconstruction Problem for Deletion Channels

Levenshtein first introduced the sequence reconstruction problem in $2001$. In the realm of combinatorics, the sequence reconstruction problem is equivalent to determining the value of $N(n,d,t)$, which represents the maximum size of the intersection of two metric balls of radius $t$, given that the distance between their centers is at least $d$ and the sequence length is $n$. In this paper, We present a lower bound on $N(n,3,t)$ for $n\geq \max\{13,t+8\}$ and $t \geq 4$. For $t=4$, we prove that this lower bound is tight. This settles an open question posed by Pham, Goyal, and Kiah, confirming that $N(n,3,4)=20n-166$ for all $n \geq 13$.

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arXiv2026

Secure Network Function Computation for General Target and Security Functions

Secure network function computation is a critical research direction in network coding, which aims to ensure that the target function is correctly computed at the sink node while preventing the wiretapper from obtaining any information about the security function. In this paper, we focus on the general secure network function computation model, where the target function f and the security function ζ are arbitrary, and the wiretapper can eavesdrop on any subset of edges with size at most a given security level. Using information-theoretic techniques, we establish a nontrivial upper bound on the secure computing capacity, which is applicable to arbitrary networks, arbitrary target and security functions, and arbitrary security levels. This upper bound is shown to degenerate to the existing bounds in the literature when the target and security functions are specific forms. Furthermore, we consider two specific models: one where the target function is vector-linear and the security function is the identity function, and another where both functions are vector-linear. For the former, we derive a simplified form of the upper bound on the secure computing capacity via order-theoretic methods and propose an efficient algorithm to compute this bound with linear time complexity in the number of network edges. For the latter, we characterize the equivalent conditions for the computability and security of linear secure network codes, develop two constructive schemes for such codes, and derive an upper bound on the minimal finite field size required for the constructions, thereby obtaining a nontrivial lower bound on the secure computing capacity.

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arXiv2026

Ciphertext-Policy ABE for $\mathsf{NC}^1$ Circuits with Constant-Size Ciphertexts from Succinct LWE

We construct a lattice-based ciphertext-policy attribute-based encryption (CP-ABE) scheme for $\mathsf{NC}^1$ access policies with constant-size ciphertexts. Let $λ$ be the security parameter. For an $\mathsf{NC}^1$ circuit of depth $d$ and size $s$ on $\ell$-bit inputs, our scheme has the public-key and ciphertext sizes $O(1)$ (independent of $d$), and secret-key size $O(\ell)$, where the $O(\cdot)$ hides $\operatorname{poly}(λ)$ factors. As an application, we obtain a broadcast encryption scheme for $N$ users with ciphertext size $\operatorname{poly}(λ)$ independent of $\log N$ and key sizes $\operatorname{poly}(λ,\log N)$. Our construction is selectively secure in the standard model under the $\operatorname{poly}(λ)$-succinct LWE assumption introduced by Wee (CRYPTO~2024).

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arXiv2026

A Partial-Exclusion Repair Scheme for MDS Codes

For scalar maximum distance separable (MDS) codes, the conventional repair schemes that achieve the cut-set bound with equality for the single-node repair have been proven to require a super-exponential sub-packetization level.As is well known, such an extremely high level severely limits the practical deployment of MDS codes.To address this challenge, we introduce a partial-exclusion (PE) repair scheme for scalar linear codes.In the proposed PE repair framework, each node is associated with an exclusion set.The cardinality of the exclusion set is called the flexibility of the node.The maximum value of flexibility over all nodes defines the \textit{flexibility} of the PE repair scheme. Notably, the conventional repair scheme is the special case of PE repair scheme where the flexibility is 1. Under the PE repair framework, for any valid flexibility, we establish a lower bound on the sub-packetization level of MDS codes that meet the cut-set bound with equality for single-node repair. To realize MDS codes attaining the cut-set bound under the PE repair framework, we propose two generic constructions of Reed-Solomon (RS) codes. Moreover, we demonstrate that for a sufficiently large flexibility, the sub-packetization level of our constructions is strictly lower than the known lower bound established for the conventional repair schemes.This implies that, from the perspective of sub-packetization level, our constructions outperform all existing and potential constructions designed for conventional repair schemes. Finally, we implement the repair process for these codes as executable Magma programs, thereby exhibiting the practical efficiency of our constructions.

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arXiv2026

Learning with Errors over Group Rings Constructed by Semi-direct Product

The Learning with Errors (\LWE) problem has been widely utilized as a foundation for numerous cryptographic tools over the years. In this study, we focus on an algebraic variant of the \LWE problem called \emph{Group ring} \LWE ($\GRLWE$). We select group rings (or their direct summands) that underlie specific families of finite groups constructed by taking the semi-direct product of two cyclic groups. Unlike the Ring-\LWE problem described in \cite{lyubashevsky2010ideal}, the multiplication operation in the group rings considered here is non-commutative. As an extension of Ring-$\LWE$, it maintains computational hardness and can be potentially applied in many cryptographic scenarios. In this paper, we present two polynomial-time quantum reductions. Firstly, we provide a quantum reduction from the worst-case shortest independent vectors problem (\SIVP) in ideal lattices with polynomial approximate factor to the search version of $\GRLWE$. This reduction requires that the underlying group ring possesses certain mild properties; Secondly, we present another quantum reduction for two types of group rings, where the worst-case \SIVP problem is directly reduced to the (average-case) decision $\GRLWE$ problem. The pseudorandomness of $\GRLWE$ samples guaranteed by this reduction can be consequently leveraged to construct semantically secure public-key cryptosystems.

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arXiv2026

A class of pseudorandom sequences From Function Fields

Motivated by the constructions of pseudorandom sequences over the cyclic elliptic function fields by Hu \textit{et al.} in \text{[IEEE Trans. Inf. Theory, 53(7), 2007]} and the constructions of low-correlation, large linear span binary sequences from function fields by Xing \textit{et al.} in \text{[IEEE Trans. Inf. Theory, 49(6), 2003]}, we utilize the bound derived by Weil \text{[Basic Number Theory, Grund. der Math. Wiss., Bd 144]} and Deligne \text{[ Lecture Notes in Mathematics, vol. 569 (Springer, Berlin, 1977)]} for the exponential sums over the general algebraic function fields and study the periods, linear complexities, linear complexity profiles, distributions of $r-$patterns, period correlation and nonlinear complexities for a class of $p-$ary sequences that generalize the constructions in \text{[IEEE Trans. Inf. Theory, 49(6), 2003]} and [IEEE Trans. Inf. Theory, 53(7), 2007].

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arXiv2026

Extended Gabidulin-Kronecker Product Codes and Their Application to Cryptosystems

In this paper, we initiate the study of Extended Gabidulin codes with a Kronecker product structure and propose three enhanced variants of the Rank Quasi-Cyclic (RQC) (Melchor et.al., IEEE IT, 2018) cryptosystem. First, we establish precise bounds on the minimum rank distance of Gabidulin-Kronecker product codes under two distinct parameter regimes. Specifically, when $n_{1}=k_{1}$ and $n_{2}=m<n_{1}n_{2}$, the minimum rank distance is exactly $n_{2}-k_{2}+1$. This yields a new family of Maximum Rank Distance (MRD) codes, which are distinct from classical Gabidulin codes. For the case of $k_{1}\leq n_{1},k_{2}\leq n_{2},n_{1}n_{2}\leq m$, the minimum rank distance $d$ of Gabidulin-Kronecker product codes satisfies a tight upper and lower bound, i.e., $n_{2}-k_{2}+1 \leq d \leq (n_{1}-k_{1}+1)(n_{2}-k_{2}+1)$. Second, we introduce a new class of decodable rank-metric codes, namely Extended Gabidulin-Kronecker product (EGK) codes, which generalize the structure of Gabidulin-Kronecker product (GK) codes. We also propose a decoding algorithm that directly retrieves the codeword without recovering the error vector, thus improving efficiency. This algorithm achieves zero decoding failure probability when the error weight is within its correction capability. Third, we propose three enhanced variants of the RQC cryptosystem based on EGK codes, each offering a distinct trade-off between security and efficiency. For 128-bit security, all variants achieve significant reductions in public key size compared to the Multi-UR-AG (Bidoux et.al., IEEE IT, 2024) while ensuring zero decryption failure probability--a key security advantage over many existing rank-based schemes.

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arXiv2025

Sequence Reconstruction Problem for Ternary Deletion Channels

The sequence reconstruction problem was proposed by Levenshtein in 2001. In this model, a sequence from a code is transmitted over several channels, and the decoder receives the distinct outputs from each channel. The main problem is to determine the minimum number of channels required to reconstruct the transmitted sequence. In the combinatorial context, the sequence reconstruction problem is equivalent to finding the value of $N_q(n,d,t)$, defined as the size of the largest intersection of two metric balls of radius $t$, where the distance between their centers is at least $d$ and the sequences are $q$-ary sequences of the length $n$. Levenshtein first discussed this problem in the uncoded sequence setting and determined the value of $N_q(n,1,t)$ for any $n\geqslant t$. Moreover, Gabrys and Yaakobi studied this problem in the context of binary one-deletion-correcting codes and determined the value of $N_2(n,2,t)$ for $t\geqslant 2$. In this paper we study this problem for $3$-ary sequences of length $n$ over the deletion channel, where the transmitted sequence belongs to a one-deletion-correcting code and there are $t$ deletions in every channel. Specifically, we determine $N_3(n,2,t)$ for $t\geqslant 2$.

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arXiv2025

Further results on bent partitions

Bent partitions of $V_{n}^{(p)}$ play an important role in constructing (vectorial) bent functions, partial difference sets, and association schemes, where $V_{n}^{(p)}$ denotes an $n$-dimensional vector space over the finite field $\mathbb{F}_{p}$, $n$ is an even positive integer, and $p$ is a prime. For bent partitions, there remains a challenging open problem: Whether the depth of any bent partition of $V_{n}^{(p)}$ is always a power of $p$. Notably, the depths of all current known bent partitions of $V_{n}^{(p)}$ are powers of $p$. In this paper, we prove that for a bent partition $Γ$ of $V_{n}^{(p)}$ for which all the $p$-ary bent functions generated by $Γ$ are regular or all are weakly regular but not regular, the depth of $Γ$ must be a power of $p$. We present new constructions of bent partitions that (do not) correspond to vectorial dual-bent functions. In particular, a new construction of vectorial dual-bent functions is provided. Additionally, for general bent partitions of $V_{n}^{(2)}$, we establish a characterization in terms of Hadamard matrices.

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