Sliding Secure Symmetric Multilevel Diversity Coding

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.