Strong security of linear ramp secret sharing schemes with general access structures

Linear Algebraic Approach to Strongly Secure Ramp Secret Sharing for General Access Structures with Application to Symmetric PIR

“…Initial scheme Field size 𝑞 ′ Design [7] Partially decryptable 2𝐿 Explicit [12] Multiple assignment 𝐿 + 𝑚 * Optimization [8] Linear 𝑞 𝑂 (𝐿 In a real-world situation, there are many users having different response patterns that reflect their individual requirements, e.g., some user has access to a smaller number of servers than others. We consider a system model, in which there are non-colluding users each of whom has one of 𝑀 response patterns Φ 1 ⊇ • • • ⊇ Φ 𝑀 .…”

“…Iwamoto and Yamamoto [7] proposed an explicit method to make a given ramp scheme for a general access structure strongly secure, assuming that the initial scheme satisfies a special property of partial decryptability. Then Eriguchi and Kunihiro [8] gave probabilistic methods assuming that the initial scheme satisfies linearity, which is a more common property of secret sharing. However, their methods are non-constructive and there is no method to verify the resulting scheme is indeed strongly secure except for the brute-force approach, which involves checking exponentially many matrices for non-singularity.…”

“…In this section, we give a more detailed explanation of Table 1. We compare our transformations in Theorems 1 and 2 with [7], [8], [12] in terms of the assumption on an initial scheme, the design of transformations, and the field size.…”

“…(1) Assumption on an Initial Scheme Theorems 1, 2, and [8] only require that an initial scheme be linear, which is satisfied by most of the previously proposed secret sharing schemes. To be precise, let Σ be an F 𝑞 -linear ramp secret sharing scheme realizing an 𝐿-level access structure A 𝐿 = (Γ 𝑗 ) 0≤ 𝑗 ≤𝐿 .…”

“…However, it still works since 𝑛 is fairly small in application to distributed data storage (e.g., 𝑛 = 6) and also since once 𝑻 is found, we can efficiently share and reconstruct secrets using the resulting strongly secure scheme. Our constructions being explicit and providing a deterministic algorithm are preferable to the non-constructive methods of [8], which fail to find 𝑻 with non-zero probability. Moreover, they provide no method to verify the resulting scheme is indeed strongly secure except for the brute-force approach of checking whether all the linear codes C 𝐴 (Σ 𝑻 ) are MDS.…”