On secret-sharing matroids

AbstractA matroid M is secret-sharing if there is a finite set S and a matrix A = (aij: i ∈ I, j ∈ E(M)) with entries in S, such that for all X ⊇ E(M), the submatrix (aij : i ∈ I, j ∈ X) has precisely |S|rk(χ) distinct rows. Such matroids occur naturally in the study of secret-sharing schemes in cryptography. Brickell and Davenport (J. Cryptography, to appear) asked if every matroid is a secret-sharing matroid. We answer this negatively, by showing that the Vamos matroid is not