arXiv subjects

Sian-Jheng Lin

Publications and source records attributed to Sian-Jheng Lin.

17 recordsLinked to original sources

arXiv2024

A Construction of Evolving $3$-threshold Secret Sharing Scheme with Perfect Security and Smaller Share Size

The evolving $k$-threshold secret sharing scheme allows the dealer to distribute the secret to many participants such that only no less than $k$ shares together can restore the secret. In contrast to the conventional secret sharing scheme, the evolving scheme allows the number of participants to be uncertain and even ever-growing. In this paper, we consider the evolving secret sharing scheme with $k=3$. First, we point out that the prior approach has risks in the security. To solve this issue, we then propose a new evolving $3$-threshold scheme with perfect security. Given a $\ell$-bit secret, the $t$-th share of the proposed scheme has $\lceil\log_2 t\rceil +O({\lceil \log_4 \log_2 t\rceil}^2)+\log_2 p(2\lceil \log_4 \log_2 t\rceil-1)$ bits, where $p$ is a prime. Compared with the prior result $2 \lfloor\log_2 t\rfloor+O(\lfloor\log_2 t\rfloor)+\ell$, the proposed scheme reduces the leading constant from $2$ to $1$. Finally, we propose a conventional $3$-threshold secret sharing scheme over a finite field. Based on this model of the revised scheme and the proposed conventional $3$-threshold scheme, we present a brand-new and more concise evolving $3$-threshold secret sharing scheme.

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arXiv2024

A Construction of Evolving $k$-threshold Secret Sharing Scheme over A Polynomial Ring

The threshold secret sharing scheme allows the dealer to distribute the share to every participant such that the secret is correctly recovered from a certain amount of shares. The traditional $(k, n)$-threshold secret sharing scheme requests that the number of participants $n$ is known in advance. In contrast, the evolving secret sharing scheme allows that $n$ can be uncertain and even ever-growing. In this paper, we consider the evolving secret sharing scenario. Using the prefix codes and the properties of the polynomial ring, we propose a brand-new construction of evolving $k$-threshold secret sharing scheme for an $\ell$-bit secret over a polynomial ring, with correctness and perfect security. The proposed schemes establish the connection between prefix codes and the evolving schemes for $k\geq2$, and are also first evolving $k$-threshold secret sharing schemes by generalizing Shamir's scheme onto a polynomial ring. Specifically, the proposal also provides an unified mathematical decryption for prior evolving $2$-threshold secret sharing schemes. Besides, the analysis of the proposed schemes show that the size of the $t$-th share is $(k-1)(\ell_t-1)+\ell$ bits, where $\ell_t$ denotes the length of a binary prefix code of encoding integer $t$. In particular, when $δ$ code is chosen as the prefix code, the share size achieves $(k-1)\lfloor\lg t\rfloor+2(k-1)\lfloor\lg ({\lfloor\lg t\rfloor+1}) \rfloor+\ell$, which improves the prior best result $(k-1)\lg t+6k^4\ell\lg{\lg t}\cdot\lg{\lg {\lg t}}+ 7k^4\ell\lg k$, where $\lg$ denotes the binary logarithm. When $k=2$, the proposed scheme also achieves the minimal share size for single-bit secret, which is the same as the best known scheme.

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arXiv2023

An Entropy Coding Based on Binary Encoding for Mixed-Radix Digits

The necessity of radix conversion of numeric data is an indispensable component in any complete analysis of digital computation. In this paper, we propose a binary encoding for mixed-radix digits. Second, a variant of rANS coding based on this conversion is given, which supports parallel decoding. The simulations show that the proposed coding in serial mode has a higher throughput than the baseline (with the speed-up factor about 2X) without loss of compression ratio, and it outperforms the existing 2-way interleaving implementation.

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arXiv2023

A Novel Reversible Data Hiding Scheme Based on Asymmetric Numeral Systems

Reversible data hiding (RDH) has been extensively studied in the field of information security. In our previous work [1], an explicit implementation approaching the rate-distortion bound of RDH has been proposed. However, there are two challenges left in our previous method. Firstly, this method suffers from computing precision problem due to the use of arithmetic coding, which may cause the further embedding impossible. Secondly, it had to transmit the probability distribution of the host signals during the embedding/extraction process, yielding quite additional overhead and application limitations. In this paper, we first propose an RDH scheme that employs our recent asymmetric numeral systems (ANS) variant as the underlying coding framework to avoid the computing precision problem. Then, we give a dynamic implementation that does not require transmitting the host distribution in advance. The simulation results show that the proposed static method provides slightly higher peak signal-to-noise ratio (PSNR) values than our previous work, and larger embedding capacity than some state-of-the-art methods on gray-scale images. In addition, the proposed dynamic method totally saves the explicit transmission of the host distribution and achieve data embedding at the cost of a small image quality loss.

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arXiv2023

Computation of Rate-Distortion-Perception Functions With Wasserstein Barycenter

The nascent field of Rate-Distortion-Perception (RDP) theory is seeing a surge of research interest due to the application of machine learning techniques in the area of lossy compression. The information RDP function characterizes the three-way trade-off between description rate, average distortion, and perceptual quality measured by discrepancy between probability distributions. However, computing RDP functions has been a challenge due to the introduction of the perceptual constraint, and existing research often resorts to data-driven methods. In this paper, we show that the information RDP function can be transformed into a Wasserstein Barycenter problem. The nonstrictly convexity brought by the perceptual constraint can be regularized by an entropy regularization term. We prove that the entropy regularized model converges to the original problem. Furthermore, we propose an alternating iteration method based on the Sinkhorn algorithm to numerically solve the regularized optimization problem. Experimental results demonstrate the efficiency and accuracy of the proposed algorithm.

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arXiv2022

A New Metric and Its Scheme Construction for Evolving $2$-Threshold Secret Sharing Schemes

Evolving secret sharing schemes do not require prior knowledge of the number of parties $n$ and $n$ may be infinitely countable. It is known that the evolving $2$-threshold secret sharing scheme and prefix coding of integers have a one-to-one correspondence. However, it is not known what prefix coding of integers to use to construct the scheme better. In this paper, we propose a new metric $K_Σ$ for evolving $2$-threshold secret sharing schemes $Σ$. We prove that the metric $K_Σ\geq 1.5$ and construct a new prefix coding of integers, termed $λ$ code, to achieve the metric $K_Λ=1.59375$. Thus, it is proved that the range of the metric $K_Σ$ for the optimal $(2,\infty)$-threshold secret sharing scheme is $1.5\leq K_Σ\leq1.59375$. In addition, the reachable lower bound of the sum of share sizes for $(2,n)$-threshold secret sharing schemes is proved.

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arXiv2022

Generalized Universal Coding of Integers

Universal coding of integers~(UCI) is a class of variable-length code, such that the ratio of the expected codeword length to $\max\{1,H(P)\}$ is within a constant factor, where $H(P)$ is the Shannon entropy of the decreasing probability distribution $P$. However, if we consider the ratio of the expected codeword length to $H(P)$, the ratio tends to infinity by using UCI, when $H(P)$ tends to zero. To solve this issue, this paper introduces a class of codes, termed generalized universal coding of integers~(GUCI), such that the ratio of the expected codeword length to $H(P)$ is within a constant factor $K$. First, the definition of GUCI is proposed and the coding structure of GUCI is introduced. Next, we propose a class of GUCI $\mathcal{C}$ to achieve the expansion factor $K_{\mathcal{C}}=2$ and show that the optimal GUCI is in the range $1\leq K_{\mathcal{C}}^{*}\leq 2$. Then, by comparing UCI and GUCI, we show that when the entropy is very large or $P(0)$ is not large, there are also cases where the average codeword length of GUCI is shorter. Finally, the asymptotically optimal GUCI is presented.

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arXiv2021

A Tighter Upper Bound of the Expansion Factor for Universal Coding of Integers and Its Code Constructions

In entropy coding, universal coding of integers~(UCI) is a binary universal prefix code, such that the ratio of the expected codeword length to $\max\{1, H(P)\}$ is less than or equal to a constant expansion factor $K_{\mathcal{C}}$ for any probability distribution $P$, where $H(P)$ is the Shannon entropy of $P$. $K_{\mathcal{C}}^{*}$ is the infimum of the set of expansion factors. The optimal UCI is defined as a class of UCI possessing the smallest $K_{\mathcal{C}}^{*}$. Based on prior research, the range of $K_{\mathcal{C}}^{*}$ for the optimal UCI is $2\leq K_{\mathcal{C}}^{*}\leq 2.75$. Currently, the code constructions achieve $K_{\mathcal{C}}=2.75$ for UCI and $K_{\mathcal{C}}=3.5$ for asymptotically optimal UCI. In this paper, we propose a class of UCI, termed $ι$ code, to achieve $K_{\mathcal{C}}=2.5$. This further narrows the range of $K_{\mathcal{C}}^{*}$ to $2\leq K_{\mathcal{C}}^{*}\leq 2.5$. Next, a family of asymptotically optimal UCIs is presented, where their expansion factor infinitely approaches $2.5$. Finally, a more precise range of $K_{\mathcal{C}}^{*}$ for the classic UCIs is discussed.

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arXiv2020

Update Bandwidth for Distributed Storage

In this paper, we consider the update bandwidth in distributed storage systems~(DSSs). The update bandwidth, which measures the transmission efficiency of the update process in DSSs, is defined as the total amount of data symbols transferred in the network when the data symbols stored in a node are updated. This paper contains the following contributions. First, we establish the closed-form expression of the minimum update bandwidth attainable by irregular array codes. Second, after defining a class of irregular array codes, called Minimum Update Bandwidth~(MUB) codes, which achieve the minimum update bandwidth of irregular array codes, we determine the smallest code redundancy attainable by MUB codes. Third, the code parameters, with which the minimum code redundancy of irregular array codes and the smallest code redundancy of MUB codes can be equal, are identified, which allows us to define MR-MUB codes as a class of irregular array codes that simultaneously achieve the minimum code redundancy and the minimum update bandwidth. Fourth, we introduce explicit code constructions of MR-MUB codes and MUB codes with the smallest code redundancy. Fifth, we establish a lower bound of the update complexity of MR-MUB codes, which can be used to prove that the minimum update complexity of irregular array codes may not be achieved by MR-MUB codes. Last, we construct a class of $(n = k + 2, k)$ vertical maximum-distance separable (MDS) array codes that can achieve all of the minimum code redundancy, the minimum update bandwidth and the optimal repair bandwidth of irregular array codes.

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arXiv2020

On the Exact Lower Bounds of Encoding Circuit Sizes of Hamming Codes and Hadamard Codes

In this paper, we investigate the encoding circuit size of Hamming codes and Hadamard codes. To begin with, we prove the exact lower bound of circuit size required in the encoding of (punctured)~Hadamard codes and (extended)~Hamming codes. Then the encoding algorithms for (punctured)~Hadamard codes are presented to achieve the derived lower bounds. For (extended)~Hamming codes, we also propose encoding algorithms that achieve the lower bounds.

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arXiv2017

On the Local Correctabilities of Projective Reed-Muller Codes

In this paper, we show that the projective Reed-Muller~(PRM) codes form a family of locally correctable codes~(LCC) in the regime of low query complexities. A PRM code is specified by the alphabet size $q$, the number of variables $m$, and the degree $d$. When $d\leq q-1$, we present a perfectly smooth local decoder to recover a symbol by accessing $γ\leq q$ symbols to the coordinates fall on a line. There are three major parameters considered in LCCs, namely the query complexity, the message length and the code length. This paper shows that PRM codes are shorter than generalized Reed-Muller~(GRM) codes in LCCs. Precisely, given a GRM code over a field of size $q$, there exists a class of shorter codes over a field of size $q-1$, while maintaining the same values on the query complexities and the message lengths.

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arXiv2016

On locality of Generalized Reed-Muller codes over the broadcast erasure channel

One to Many communications are expected to be among the killer applications for the currently discussed 5G standard. The usage of coding mechanisms is impacting broadcasting standard quality, as coding is involved at several levels of the stack, and more specifically at the application layer where Rateless, LDPC, Reed Solomon codes and network coding schemes have been extensively studied, optimized and standardized in the past. Beyond reusing, extending or adapting existing application layer packet coding mechanisms based on previous schemes and designed for the foregoing LTE or other broadcasting standards; our purpose is to investigate the use of Generalized Reed Muller codes and the value of their locality property in their progressive decoding for Broadcast/Multicast communication schemes with real time video delivery. Our results are meant to bring insight into the use of locally decodable codes in Broadcasting.

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arXiv2016

FFT Algorithm for Binary Extension Finite Fields and its Application to Reed-Solomon Codes

Recently, a new polynomial basis over binary extension fields was proposed such that the fast Fourier transform (FFT) over such fields can be computed in the complexity of order $\mathcal{O}(n\lg(n))$, where $n$ is the number of points evaluated in FFT. In this work, we reformulate this FFT algorithm such that it can be easier understood and be extended to develop frequency-domain decoding algorithms for $(n=2^m,k)$ systematic Reed-Solomon~(RS) codes over $\mathbb{F}_{2^m},m\in \mathbb{Z}^+$, with $n-k$ a power of two. First, the basis of syndrome polynomials is reformulated in the decoding procedure so that the new transforms can be applied to the decoding procedure. A fast extended Euclidean algorithm is developed to determine the error locator polynomial. The computational complexity of the proposed decoding algorithm is $\mathcal{O}(n\lg(n-k)+(n-k)\lg^2(n-k))$, improving upon the best currently available decoding complexity $\mathcal{O}(n\lg^2(n)\lg\lg(n))$, and reaching the best known complexity bound that was established by Justesen in 1976. However, Justesen's approach is only for the codes over some specific fields, which can apply Cooley-Tucky FFTs. As revealed by the computer simulations, the proposed decoding algorithm is $50$ times faster than the conventional one for the $(2^{16},2^{15})$ RS code over $\mathbb{F}_{2^{16}}$.

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arXiv2014

Novel Polynomial Basis and Its Application to Reed-Solomon Erasure Codes

In this paper, we present a new basis of polynomial over finite fields of characteristic two and then apply it to the encoding/decoding of Reed-Solomon erasure codes. The proposed polynomial basis allows that $h$-point polynomial evaluation can be computed in $O(h\log_2(h))$ finite field operations with small leading constant. As compared with the canonical polynomial basis, the proposed basis improves the arithmetic complexity of addition, multiplication, and the determination of polynomial degree from $O(h\log_2(h)\log_2\log_2(h))$ to $O(h\log_2(h))$. Based on this basis, we then develop the encoding and erasure decoding algorithms for the $(n=2^r,k)$ Reed-Solomon codes. Thanks to the efficiency of transform based on the polynomial basis, the encoding can be completed in $O(n\log_2(k))$ finite field operations, and the erasure decoding in $O(n\log_2(n))$ finite field operations. To the best of our knowledge, this is the first approach supporting Reed-Solomon erasure codes over characteristic-2 finite fields while achieving a complexity of $O(n\log_2(n))$, in both additive and multiplicative complexities. As the complexity leading factor is small, the algorithms are advantageous in practical applications.

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arXiv2013

Novel Repair-by-Transfer Codes and Systematic Exact-MBR Codes with Lower Complexities and Smaller Field Sizes

The $(n,k,d)$ regenerating code is a class of $(n,k)$ erasure codes with the capability to recover a lost code fragment from other $d$ existing code fragments. This paper concentrates on the design of exact regenerating codes at Minimum Bandwidth Regenerating (MBR) points. For $d=n-1$, a class of $(n,k,d=n-1)$ Exact-MBR codes, termed as repair-by-transfer codes, have been developed in prior work to avoid arithmetic operations in node repairing process. The first result of this paper presents a new class of repair-by-transfer codes via congruent transformations. As compared with the prior works, the advantages of the proposed codes include: i) The minimum of the finite field size is significantly reduced from $n \choose 2$ to $n$. ii) The encoding complexity is decreased from $n^4$ to $n^3$. As shown in simulations, the proposed repair-by-transfer codes have lower computational overhead when $n$ is greater than a specific constant. The second result of this paper presents a new form of coding matrix for product-matrix Exact-MBR codes. The proposed coding matrix includes a number of advantages: i). The minimum of the finite field size is reduced from $n-k+d$ to $n$. ii). The fast Reed-Solomon erasure coding algorithms can be applied on the Exact-MBR codes to reduce the time complexities.

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arXiv2013

Symbol and Bit Mapping Optimization for Physical-Layer Network Coding with Pulse Amplitude Modulation

In this paper, we consider a two-way relay network in which two users exchange messages through a single relay using a physical-layer network coding (PNC) based protocol. The protocol comprises two phases of communication. In the multiple access (MA) phase, two users transmit their modulated signals concurrently to the relay, and in the broadcast (BC) phase, the relay broadcasts a network-coded (denoised) signal to both users. Nonbinary and binary network codes are considered for uniform and nonuniform pulse amplitude modulation (PAM) adopted in the MA phase, respectively. We examine the effect of different choices of symbol mapping (i.e., mapping from the denoised signal to the modulation symbols at the relay) and bit mapping (i.e., mapping from the modulation symbols to the source bits at the user) on the system error-rate performance. A general optimization framework is proposed to determine the optimal symbol/bit mappings with joint consideration of noisy transmissions in both communication phases. Complexity-reduction techniques are developed for solving the optimization problems. It is shown that the optimal symbol/bit mappings depend on the signal-to-noise ratio (SNR) of the channel and the modulation scheme. A general strategy for choosing good symbol/bit mappings is also presented based on a high-SNR analysis, which suggests using a symbol mapping that aligns the error patterns in both communication phases and Gray and binary bit mappings for uniform and nonuniform PAM, respectively.

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arXiv2013

Energy Efficient Transmission over Space Shift Keying Modulated MIMO Channels

Energy-efficient communication using a class of spatial modulation (SM) that encodes the source information entirely in the antenna indices is considered in this paper. The energy-efficient modulation design is formulated as a convex optimization problem, where minimum achievable average symbol power consumption is derived with rate, performance, and hardware constraints. The theoretical result bounds any modulation scheme of this class, and encompasses the existing space shift keying (SSK), generalized SSK (GSSK), and Hamming code-aided SSK (HSSK) schemes as special cases. The theoretical optimum is achieved by the proposed practical energy-efficient HSSK (EE-HSSK) scheme that incorporates a novel use of the Hamming code and Huffman code techniques in the alphabet and bit-mapping designs. Experimental studies demonstrate that EE-HSSK significantly outperforms existing schemes in achieving near-optimal energy efficiency. An analytical exposition of key properties of the existing GSSK (including SSK) modulation that motivates a fundamental consideration for the proposed energy-efficient modulation design is also provided.

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