Symmetric multilevel diversity coding (SMDC) is a multi-source coding problem where the independent sources are ordered according to their importance. Prior work demonstrated thatsuperposition coding, where sources are encoded independently, is optimal. This paper investigates the(L,s)sliding secure SMDC problem, whereLrepresents the number of encoders andsis the security threshold. The security requirement dictates that each sourceXαmust be kept perfectly secure if no more than α –sencoders are accessible. The problem is specialized to the(L,s)multilevel secret sharingproblem when the firsts – 1sources are constants. Fors= 1, the two problems coincide, and we show that superposition coding is optimal. The rate regions for the(L,s)=(3,2)problems are characterized, which implies that superposition coding is suboptimal for the general case. The core insight for achieving lower rates through joint encoding is leveraging less important sources likeXα–1as secret keys for more important sources likeXα. Based on this idea, we propose a joint coding scheme that achieves the minimum sum rate of the general(L,s)multilevel secret sharing problem. Moreover, a pseudo-superposition coding scheme is proposed to achieve the minimum sum rate of the general sliding secure SMDC problem, which uses superposition coding for thessets of sourcesX1, X2,..., Xs–1, (Xs,Xs+1,XL)and joint coding amongXs,Xs+1,XL.

This paper investigates the lazy secret sharing problem from an information-theoretic perspective. The participants are classified into two categories: Lazy-Participants and Share-Participants. The objective is to guarantee the perfect secret recovery from any t participants, while ensuring security exclusively for Share-Participants. The optimal coding rate region for lazy secret sharing is characterized. We further consider the imperfect security formulation, formulating security through information leakage constraints rather than strict security. The optimal rate region in the imperfect formulation is also established for a specific symmetric scenario.

We present an algebraic-geometric framework for analyzing information-theoretic inequalities under Markov chain constraints. By the nonnegativity of the I -Measure in Markov chains, we establish a polynomial representation of the joint entropies. This polynomial representation unifies the algebraic manipulation of entropy terms, enabling the automated verification of inequalities via Fourier-Motzkin Elimination. The experimental validation against the PSITIP toolkit demonstrates that the framework effectively resolves the complex dependencies, particularly for large systems, with superior efficiency and scalability.

The proof of information inequalities and identities under linear constraints on the information measures is an important problem in information theory. For this purpose, ITIP and other variant algorithms have been developed and implemented, which are all based on solving a linear program (LP). Building on our recent work (Guo et al., 2023), we developed in this paper an enhanced approach for solving this problem. Experimental results show that our new approach improves the time complexity by over 500 times compared with Guo et al. (2023) for the problem studied by Tian (2014).

In this paper, we investigate the completely monotone conjecture along the heat flow for the Rényi entropy. We confirm this conjecture for the order of derivative up to 4, when the order of Rényi entropy is in certain regimes. We also investigate concavity of Rényi entropy power and the complete monotonicity of Tsallis entropy. We recover and slightly extend Hung’s result on the fourth-order derivative of the Tsallis entropy, and observe that the complete monotonicity holds for Tsallis entropy of order 2, which is equivalent to that the noise stability with respect to the heat semigroup is completely monotone. Based on this observation, we conjecture that the complete monotonicity holds for Tsallis entropy of all orders α ∈ (1, 2). Our proofs in this paper are based on the techniques of integration-by-parts, sum-of-squares, and curve-fitting.

Symmetric multilevel diversity coding (SMDC) is a multi-source coding problem where the independent sources are ordered according to their importance. It was shown that sepa-rately encoding independent sources, referred to as superposition coding, is optimal. In this paper, an (L, s) sliding secure SMDC problem is considered, where$L$is the number of encoders and$s$is the security threshold, which means that each source$X$ais kept perfectly secure if no more than a -$s$encoders are accessible. It is shown that superposition coding is optimal for$s$= 1. The rate region for (L, s) = (3, 2) is characterized, which implies the suboptimality of superposition coding for the general problem. The main idea that joint coding can reduce rates is that we can use the previous source X a -1 as the secret key of X a. Based on this idea, a pseudo-superposition coding scheme is proposed to achieve the minimum sum rate, which uses superposition for the$s$sets of sources Xl, X2,‥ Xs-1, (Xs, Xs+1,”, XL). and joint encoding among Xs, Xs+1,”, XL.

The proof of information inequalities under linear constraints on the information measures is an important problem in information theory. For this purpose, ITIP and other variant algorithms have been developed and implemented, which are all based on solving a linear program (LP). Building on our recent work [13], we develop in this paper an enhanced approach for solving this problem.

Proving linear inequalities and identities of Shannon’s information measures, possibly with linear constraints on the information measures, is an important problem in information theory. For this purpose, ITIP and other variant algorithms have been developed and implemented, which are all based on solving a linear program (LP). In particular, an identity$f = 0$is verified by solving two LPs, one for$f \ge 0$and one for$f \le 0$. In this paper, we develop a set of algorithms that can be implemented by symbolic computation. Based on these algorithms, procedures for verifying linear information inequalities and identities are devised. Compared with LP-based algorithms, our procedures can produce analytical proofs that are both human-verifiable and free of numerical errors. Our procedures are also more efficient computationally. For constrained inequalities, by taking advantage of the algebraic structure of the problem, the size of the LP that needs to be solved can be significantly reduced. For identities, instead of solving two LPs, the identity can be verified directly with very little computation.

Proving linear inequalities and identities of Shannon’s information measures, possibly with linear constraints on the information measures, is an important problem in information theory. For this purpose, ITIP and other variant algorithms have been developed and implemented, which are all based on solving a linear program (LP). In particular, an identity f = 0 is verified by solving two LPs, one for f ≥ 0 and one for f ≤ 0. In this paper, we develop a set of algorithms that can be implemented by symbolic computation. Based on these algorithms, procedures for verifying linear information inequalities and identities are devised. Compared with LP-based algorithms, our procedures can produce analytical proofs that are both human-verifiable and free of numerical errors. Our procedures are also more efficient computationally. For constrained inequalities, by taking advantage of the algebraic structure of the problem, the size of the LP that needs to be solved can be significantly reduced. For identities, instead of solving two LPs, the identity can be verified directly with very little computation.

Let$H(X_{t})$be the differential entropy of an$n$-dimensional random vector$X_{t}$introduced by Costa. Cheng and Geng conjectured that$C_{1}(m, n): (-1)^{m+1}(\mathrm{d}^{m}/\mathrm{d}^{m}t)H(X_{t})\geq 0$. McKean conjectured that$C_{1}(m, n): (-1)^{m+1}(\mathrm{d}^{m}/\mathrm{d}^{m}t)H(X_{t})\geq 0 (-1)^{m+1}(\mathrm{d}^{m}/\mathrm{d}^{m}t)H(X_{Gt})$. McKean's conjecture was only considered in the univariate case before:$C_{2}(1,1)$and$C_{2}(2,1)$were proved by McKean and$C_{2}(i, 1), i=3,4,5$were proved by Zhang-Anantharam-Geng under the log-concave condition. In this paper, we prove$C_{2}(1, n),\ C_{2}(2, n)$and observe that McKean's conjecture might not be true for$n\ > \ 1$and$m > 2$. We further propose a weaker conjecture$C_{3}(m, n): (-1)^{m+1}(\mathrm{d}^{m}/\mathrm{d}^{m}t)H(X_{t}) \ \geq\ (-1)^{m+1}\frac{1}{n}(\mathrm{d}^{m}/\mathrm{d}^{m}t)H(X_{Gt})$and prove$C_{3}(3,2), C_{3}(3,3), C_{3}(3,4)$under the log-concave condition. A systematic procedure to prove$C_{l}(m, n)$is proposed and the results mentioned above are proved using this procedure.