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 _Σ to evaluate evolving 2-threshold secret sharing schemes Σ such that a smaller K _Σ of a scheme is better. The metric K _Σ is related to the ratio of the sum of the share sizes for the first n parties in scheme Σ and the sum of the share sizes for the optimal (2, n )-threshold secret sharing scheme. Then we prove that the metric K _Σ ≥ 1.5 and construct a new prefix coding of integers, termed λ code, to achieve the metric K _Λ = 1.59375. Thus, this shows that the range of the metric K _Σ for the optimal (2,∞)-threshold secret sharing scheme is 1.5 ≤ K _Σ ≤ 1.59375. In addition, an achievable lower bound on the sum of share sizes for (2, n )-threshold secret sharing schemes is also provided.

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.

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 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.