Asymmetric numeral systems (ANS) is a widely used entropy coding method in commercial compressors due to its high performance. Joint compression and encryption techniques can offer reliability and cost-effectiveness for secure Big Data storage. However, existing joint compression-encryption schemes for ANS coding often suffer from either increased storage space requirements or limited security. To address these issues, this paper proposes two ANS-based joint compression-encryption algorithms that provide considerable security with almost no compression loss. The first scheme, based on interval swapping, employs a cryptographically secure ChaCha20 generator to perturb the order of contiguous intervals, thereby introducing controlled randomness into the encoding process. The second scheme, based on interval splitting, discards the conventional assumption of representing each symbol with a single contiguous interval, instead assigning multiple sub-intervals to enhance both security and flexibility. In addition, a sequence of output permutations is applied to further strengthen resistance against attacks. Experimental results show that the proposed methods reduce compression loss by approximately 3.83% compared with existing schemes, while the interval swapping scheme achieves a 46.9% reduction in time cost. Security analysis confirms that the enlarged key space significantly increases robustness against brute-force attacks. These results demonstrate that the proposed approaches effectively balance compression efficiency and encryption strength, offering a lightweight and secure solution for Big Data storage.

The Universal Coding of Integers (UCI) is suitable for discrete memoryless sources with unknown probability distributions and infinitely countable alphabet sizes. A UCI is a class of prefix codes for which the ratio of the average codeword length to max{1,H(P)} is within a constant expansion factorCCfor any decreasing probability distributionP, whereH(P)is the entropy ofP. For any UCI codeC, the minimum expansion factor C∗Cis defined to represent the infimum of the set of extension factors ofC. EachChas a unique correspondingC∗C, and the smallerC∗Cis, the better the compression performance ofCis. The class of UCIsC(or a family {Ci}∞i=1) that achieves the smallestC∗Cis defined as theoptimal UCI. The best current result is that the range ofC∗Cfor the optimal UCI is 2 ≤C∗C≤ 2.5. In this paper, we prove a tighter probability inequality for decreasing distributions, which serves as a new tool for studying the properties of UCIs. On the basis of this inequality, we prove that there exists a class of near-optimal UCIs, called the ν code, achievingCν= 2.0386. This narrows the range of the minimum expansion factor for the optimal UCI to 2 ≤C∗C≤ 2.0386. We show that the ν code is currently optimal in terms of the minimum expansion factor. In addition, we propose a new proof showing that the minimum expansion factor of the optimal UCI is lower bounded by 2.

In the field of lossless source coding, universal coding of integers (UCI) and generalized universal coding of integers (GUCI) are binary codes that are suitable for probability distributions without prior knowledge. UCICis defined as a prefix coding in which the constant expansion factor KC times max{1,H(P)} is greater than or equal to the expected codeword length, wherePis the decreasing probability distribution of the source andH(P)is the entropy ofP. SincePis decreasing, when the set of codewords of the prefix codeCis determined, the length of then+1-th codeword of the prefix codeCis greater than or equal to the length of then-th codeword for any positive integer n, at which time the expected codeword length ofCis minimized, andCis said to be minimal. GUCIGis defined as a prefix variable-to-variable length (VV) coding for which the constant expansion factor KG timesH(P)is greater than or equal to the coding rate. In this paper, we prove two important theorems for UCI. First, we provide and prove the necessary and sufficient conditions for a minimal prefix code to be UCI. Second, we provide the first proof of an essential theorem for VV codes. This theorem can reveal the connection between UCI and GUCI and prove the converse part of Shannon’s first theorem concerning VV codes.

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 bounded by a constant factor, whereH(P) is the Shannon entropy of the decreasing probability distributionP. However, if we consider the ratio of the expected codeword length toH(P) for UCI, the ratio tends to infinity whenH(P) tends to zero. To resolve this issue, we introduce a class of codes, called generalized universal coding of integers (GUCI), where the ratio of the expected codeword length toH(P) is bounded by a constant factorK. First, the definition of GUCI is proposed. The coding structure of GUCI is introduced. Next, we propose a class of GUCIsCto reach the expansion factorKC= 2, and we show that the smallest minimum expansion factor is in the range 1 ≤K* ≤ 2. Then, by comparing UCI and GUCI, we show that when the entropy is very large orP(0) is not large, there are also cases where the average codeword length of GUCI is shorter. Finally, the asymptotically optimal GUCI is presented.

The Asmuth and Bloom threshold secret sharing (AB-SS) is a classical introduction of the Chinese remainder theorem (CRT) to secret sharing, offering low computational complexity compared to other branches of secret sharing. For decades, numerous schemes have been proposed for practical applications of AB-SS, such as secret image sharing (SIS). However, in terms of security, AB-SS has proved to be neither ideal nor perfect, and its derivatives in image sharing exhibit vulnerabilities associated with secret leakage. This paper studies the security issues in the SIS schemes derived from AB-SS and improves the core sharing principle of AB-SS to enhance security in image protection. First, for$(2,n)$-CRTSIS schemes, we exploit the vulnerability in a single share image to crack the confidential information of the original image, including secret pixel values and the ratio of different pixels. Then, by employing the XOR operation, we introduce a chain obfuscation technology and propose a secure image sharing scheme based on the Chinese remainder theorem (COxor-CRTSIS). The COxor-CRTSIS scheme utilizes integer linear programming for achieving lossless recovery without segmentation and eliminates potential secret disclosure risks without additional encryption. Furthermore, to comprehensively evaluate the security of existing schemes, this paper presents three metrics, information loss rate, fluctuation degree, and coverage rate, enabling a quantitative comparison of security for the first time. Theoretical analyses and experiments are conducted to validate the effectiveness of our scheme.

$(k,n)$Threshold secret image sharing (SIS) hides a secret image within$n$shadows, and at least$k$shadows are needed for recovery. Due to the popularity and frequency of JPEG recompression, there is a need for robust secret image sharing (ROSIS) designed for JPEG images that is resilient to recompression for practical SIS applications. The current state-of-the-art ROSIS, which relies on error-correcting codes (ECC), is effective only for JPEG compression with quality factors (QFs) of 99 and 100. However, it generates noise-like shadow images that are confined to the spatial domain. In this paper, we present SBC-ROSIS (Robust Secret Image Sharing Scheme Resistant to JPEG Recompression Based on Stable Block Condition), a novel ROSIS scheme that utilizes a stable block condition to guarantee the invariance of discrete cosine transform (DCT) coefficients during JPEG recompression, significantly enhancing the robustness of the scheme. By employing a polynomial-based secret sharing (SS) algorithm, we construct DCT blocks that adhere to stable block condition either directly or through strategic global regulation. Additionally, we carefully consider the similarity between the generated DCT blocks and the original cover DCT blocks. Furthermore, we devised a tailored evaluation methodology specifically for ROSIS. Extensive experimental results indicate that SBC-ROSIS can effectively process JPEG images, achieving a balance among security, robustness, concealment, and adherence to the$(k, n)$threshold, without relying on steganography, ECC, or pixel expansion, and demonstrating robust performance in realistic recompression scenarios.

Ternary AIFV codes are almost instantaneous fixed-to-variable length codes, and are constructed based on two code trees. It is known that the redundancy of ternary AIFV codes is no more than one. In this paper, we provide a tighter upper bound on the redundancy of the ternary Huffman codes when the greatest probability of the source$p_{max}$is known. As a result, the redundancy of the optimal ternary AIFV codes is bounded by Huffman codes, since the ternary Huffman codes can be seen as the special AIFV codes. To achieve lower redundancy than Huffman codes, we also propose a method to construct a class of ternary AIFV codes with time complexity$O(n)$for$n$source symbols. In addition, the redundancy of the proposed AIFV codes is analyzed and compared with Huffman codes. Analyzing the ternary AIFV codes constructed by the algorithm, we derive a tighter redundancy upper bounds under some conditions, which are superior to Huffman codes.

Evolving secret sharing schemes do not require prior knowledge of the number of parties$n$, which 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 unknown what prefix coding of integers should be used to construct a better secret sharing scheme. In this paper, we introduce a metric$K_{\Sigma }$to evaluate evolving 2-threshold secret sharing schemes$\Sigma $such that a smaller$K_{\Sigma }$of a scheme is better. The metric$K_{\Sigma }$is related to the ratio of the sum of the share sizes for the first$n$parties in scheme$\Sigma $and the sum of the share sizes for the optimal$(2,n)$-threshold secret sharing scheme. Then we prove that the metric$K_{\Sigma }\geq 1.5$and construct a new prefix coding of integers, termed$\lambda $code, to achieve the metric$K_{\Lambda }=1.59375$. Thus, this shows that the range of the metric$K_{\Sigma }$for the optimal$(2,\infty)$-threshold secret sharing scheme is$1.5\leq K_{\Sigma }\leq 1.59375$. In addition, an achievable lower bound on the sum of share sizes for$(2,n)$-threshold secret sharing schemes is also provided.

Entropy coding is a widely used technique for lossless data compression. The entropy coding schemes supporting the direct access capability on the encoded stream have been investigated in recent years. However, all prior schemes require auxiliary space to support the direct access ability. This paper proposes a rearranging method for prefix codes to support a certain level of direct access to the encoded stream without requiring additional data space. Then, an efficient decoding algorithm is proposed based on lookup tables. The simulation results show that when the encoded stream does not allow additional space, the number of bits per access read of the proposed method is above two orders of magnitude less than the conventional method. In contrast, the alternative solution consumes at least one more bit per symbol on average than the proposed method to support direct access. This indicates that the proposed scheme can achieve a good trade-off between space usage and access performance. In addition, if a small amount of additional storage space is allowed (it is approximately 0.057% in the simulation), the number of bits per access read in our proposal can be significantly reduced by 90%.

The codebook is important for canonical Huffman coding, which needs to contain the number of leaves in each layer of the canonical Huffman tree and the corresponding symbols. Specifically, as two conventional methods in [1], [2], only the number of leaves in each level of the canonical Huffman tree is needed to store. However, we provide a new method to store the number of internal nodes in each layer and compactly encode the string of numbers according to the specific property between the internal nodes.

In the conventional range asymmetric numeral systems (rANS), state$x$becomes larger after encoding a symbol$s$. In contrast, the proposed scheme directly outputs an$n$-bit digit$cdf_{s}+x\ (\text{mod}\ f_{s})$for symbol$s$, and decrease$x$via$x\leftarrow\lfloor x/f_{s}\rfloor$, where$2^{n}$denotes the denominator of the quantized frequency distribution,$f_{s}$and$cdf_{s}= \sum\nolimits_{i=0}^{s-1}f_{i}$represent the frequency of symbol$s$and the cumulative frequency counts, respectively. Therefore,$x$will become too small after encoding several symbols. To solve this issue, our proposal forces the state$x$always at a specific interval$I= [2^{T-vn}, 2^{T})$, and$I_{s}:=\left[f_{s}\times 2^{T-vn}, 2^{T}\right)$indicates the interval corresponding to symbol$s$, where$T, v\in \mathbb{N}$. The specific algorithm can be implemented based on the deque. Precisely, for a symbol$s$to be encoded, if the current$x$is within$I_{s}$, we encode it to an$n$-bit digit$cdf_{s}+x\ (\text{mod}\ f_{s})$and push the digit to deque. Otherwise, we first pop data from the deque to enlarge$x$before encoding. Finally, the remaining data in the deque is the desired encoded bit sequence.

In this paper, we study the straggler mitigation via coded computing in distributed computations. In particular, we consider the coded matrix-vector multiplication where the matrix is sparse and coding may break the sparsity of the matrix. We construct a class of sparse universal decodable matrices (UDMs) for coded computing. In simulations, it shows that the proposed code possesses better sparsity than other schemes with randomly generated sparse matrices. Besides, the proposed code performs well in numerical stability.

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 construct a class of UCI, termed$\iota $code, to achieve$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. Then, tighter upper bounds of the expansion factors are derived for several classic UCIs. Finally, we prove that the length of the first codeword of optimal UCI is one.

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 UCI, such that the ratio of the expected codeword length to H(P) is within a constant factor K. The definition of generalized UCI is proposed, and then the coding structure of generalized UCI is introduced. Finally, the asymptotically optimal generalized UCI is presented.

Universal coding of integers (UCI) is a prefix coding suitable for probability distributions without prior knowledge. UCI has the property that the expected codeword length is no more than constant$K_{C}$times$\max \{1,H(P)\}$, where$K_{C}$is called the expansion factor and$H(P)$is the entropy of source$P$. A class of UCI$C$with a smaller$K_{C}$is preferred, but the minimum value of$K_{C}$(as well as its code construction) is still unknown. Thus, this paper provides a range of the minimum values of$K_{C}$. First, we show that$K_{C}\geq 2$for each UCI$C$. Then, for the upper bound, we provide a class of UCIs, termed$\eta $code, to achieve$K_{C}=2.75$. This approach improves the prior result$K_{C}=3$achieved by Elias$\gamma $coding. Next, we propose an asymptotically optimal UCI, termed$\theta $code, to achieve$K_{C}=3.5$. Finally, we compare the range of the minimum value of$K_{C}$of$\eta $code,$\theta $code and some other UCIs.

In this paper, a variant of the Reed-Muller code, termed symmetric Reed-Muller (SRM) code, is proposed. We show that the bivariate version of SRM code forms the best-known locally-correctable codes in certain parameter regimes. Then, the minimum distance of SRM codes is provided in certain cases. Finally, the multivariate version of SRM codes is introduced.