Robust Secret Sample Clauses

Robust Secret. Sharing We recall the definition of a robust secret sharing scheme, slightly simplified for our purposes from Xxxxxx et al. [22]. For a vector c ∈ Fn and a set A ⊆ [n], we denote with c the projection Fn → F|A|, q

Robust Secret. Sharing We recall the definition of a robust secret sharing scheme, slightly simplified for our purposes from Xxxxxx et al. [CDD+15]. For a vector c ∈ Fn and a set A ⊆ [n], we denote with c the projection Fn → F|A|, i.e., the sub-vector q (ci)i∈A. A q q Definition 2. Let λ ∈ N, q a λ-bit prime, Fq a finite field and n, t, m, r ∈ N with t < r ≤ n and m < r. An (n, t, r)q robust secret sharing scheme (RSS) consists of two probabilistic algorithms Share : Fq → Fn and Reconstruct : Fn → Fq with the following q q properties: – t-privacy: for any s, sj ∈ Fq, A ⊂ [n] with |A| ≤ t, the projections cA of c ←$ Share(s) and cjA of cj ← Share(sj) are identically distributed. $ ∈ ⊂ | | ≥ – r-robustness: for any s Fq, A [n] with A r, any c output by Share(s), and any c˜ such that cA = c˜A, it holds that Reconstruct(c˜) = s. − In other words, an (n, t, r)q-RSS is able to reconstruct the shared secret even if the adversary tampered with up to n r shares, while each set of t shares is distributed independently of the shared secret s and thus reveals nothing about it. We note that we allow for a gap, i.e., r ≥ t + 1. Schemes with r > t + 1 are called ramp RSS.