Connection between Coding Theory and Secret Sharing Sample Clauses

Connection between Coding Theory and Secret Sharing. ‌ Let F be a finite field of size q, n a positive integer, and let V be a finite- dimensional F-vector space. In the ambient space Fn+1, coordinates are in- { } dexed by 0, 1, . . . , n . First observe that we can naturally associate a code to a secret sharing scheme. Definition 2.6.12. Let Σ = (π0, . . . , πn) ⊆ V ∗ be a linear secret sharing scheme. The code associated to Σ is the code C(Σ) := {(π0(x), . . . , πn(x)) : x ∈ V } ⊆ Fn+1. This is indeed a linear space by linearity of Σ. It is a code of length n + 1 and dimension dim C(Σ) = dim⟨π1, . . . , πn⟩. /∈ /∈ Moreover the properties (i), (ii) required by Definition 2.6.4 are equivalent to e0 C(Σ)⊥ and e0 C(Σ) respectively, where e0 denotes the 0-th unit vector of Fn+1. One can view C(Σ) as the set of all (n + 1)-tuples (s, x1, . . . , xn) where s is a secret and x1, . . . , xn is a valid set of shares for s in the scheme Σ. The parameters of a secret sharing scheme give an estimate of the dimension of the associated code as follows. Theorem 2.6.13. If the secret sharing scheme Σ has t-privacy and r-recon- struction then the code C(Σ) has dimension t < dim C(Σ) ≤ r.