On secret-sharing matroids

Abstract

A matroid M is secret-sharing if there is a finite set S and a matrix A = ( a ij : i ∈ I, j ∈ E( M)) with entries in S, such that for all X ⊇ E( M), the submatrix ( a ij : 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.