Vamos Matroid Research Articles

Improving the Linear Programming Technique in the Search for Lower Bounds in Secret Sharing

We present a new improvement in the linear programming technique to derive lower bounds on the information ratio of secret sharing schemes. We obtain non-Shannon-type bounds without using information inequalities explicitly. Our new technique makes it possible to determine the optimal information ratio of linear secret sharing schemes for all access structures on 5 participants and all graph-based access structures on 6 participants. In addition, new lower bounds are presented also for some small matroid ports and, in particular, the optimal information ratios of the linear secret sharing schemes for the ports of the Vamos matroid are determined.

A Matroid Extension Result

Adding elements to matroids can be fraught with difficulty. In the Vámos matroid $V_8$, there are four pairs $X_1,X_2, X_3,$ and $X_4$ that partition $E(V_8)$ such that $(X_1 \cup X_2,X_3 \cup X_4)$ is a $3$-separation while exactly three of the local connectivities $\sqcap(X_1,X_{3})$, $\sqcap(X_1,X_{4})$, $\sqcap(X_2,X_{3})$, and $\sqcap(X_2,X_{4})$ are one, with the fourth being zero. As is well known, there is no extension of $V_8$ by a nonloop element $p$ such that $X_j \cup p$ is a circuit for all $j$. This paper proves that a matroid can be extended by a fixed element in the guts of a $3$-separation provided no Vámos-like structure is present.

The Facets of the Bases Polytope of a Matroid and Two Consequences

Let M be a matroid defined on a finite set E and L ⊂ E . L is locked in M if and are 2-connected, and . In this paper, we prove that the nontrivial facets of the bases polytope of M are described by the locked subsets. We deduce that finding the maximum-weight basis of M is a polynomial time problem for matroids with a polynomial number of locked subsets. This class of matroids is closed under 2-sums and contains the class of uniform matroids, the Vamos matroid and all the excluded minors of 2-sums of uniform matroids. We deduce also a matroid oracle for testing uniformity of matroids after one call of this oracle.

Hyperbolic polynomials, interlacers, and sums of squares

Hyperbolic polynomials are real polynomials whose real hypersurfaces are nested ovaloids, the inner most of which is convex. These polynomials appear in many areas of mathematics, including optimization, combinatorics and differential equations. Here we investigate the special connection between a hyperbolic polynomial and the set of polynomials that interlace it. This set of interlacers is a convex cone, which we write as a linear slice of the cone of nonnegative polynomials. In particular, this allows us to realize any hyperbolicity cone as a slice of the cone of nonnegative polynomials. Using a sums of squares relaxation, we then approximate a hyperbolicity cone by the projection of a spectrahedron. A multiaffine example coming from the Vamos matroid shows that this relaxation is not always exact. Using this theory, we characterize the real stable multiaffine polynomials that have a definite determinantal representation and construct one when it exists.

Multipartite matroids and secret sharing

In a secret-sharing scheme, a secret value is distributed among a set of participants by giving each participant a share. The requirement is that only predefined subsets of participants can recover the secret from their shares. The family of the predefined authorized subsets is called the access structure. An access structure is ideal if there exists a secret-sharing scheme realizing it in which the shares have optimal length, that is, in which the shares are taken from the same domain as the secrets. Brickell and Davenport proved that ideal access structures are induced by matroids. Subsequently, ideal access structures and access structures induced by matroids have received a lot of attention. Seymour gave the first example of an access structure induced by a matroid namely the Vamos matroid, that is non-ideal. Since every matroid is multipartite and has the associated discrete polymatroid, in this paper, by dealing with the rank functions of discrete polymatroids, we obtain a sufficient condition for a multipartite access structure to be ideal. Furthermore, we give a new proof that all access structures related to bipartite and tripartite matroids coincide with the ideal ones. Our results give new contributions to the open problem, that is, which matroids induce ideal access structures.

On Matroids and Nonideal Secret Sharing

Secret-sharing schemes are a tool used in many cryptographic protocols. In these schemes, a dealer holding a secret string distributes shares to the parties such that only authorized subsets of participants can reconstruct the secret from their shares. The collection of authorized sets is called an access structure. An access structure is ideal if there is a secret-sharing scheme realizing it such that the shares are taken from the same domain as the secrets. Brickell and Davenport (Journal of Cryptology, 1991) have shown that ideal access structures are closely related to matroids. They give a necessary condition for an access structure to be ideal-the access structure must be induced by a matroid. Seymour (Journal of Combinatorial Theory B, 1992) has proved that the necessary condition is not sufficient: There exists an access structure induced by a matroid that does not have an ideal scheme. The research on access structures induced by matroids is continued in this work. The main result in this paper is strengthening the result of Seymour. It is shown that in any secret-sharing scheme realizing the access structure induced by the Vamos matroid with domain of the secrets of size k, the size of the domain of the shares is at least k + Omega(radic(k)). The second result considers nonideal secret-sharing schemes realizing access structures induced by matroids. It is proved that the fact that an access structure is induced by a matroid implies lower and upper bounds on the size of the domain of shares of subsets of participants even in nonideal schemes (as long as the shares are still relatively short). This generalized results of Brickell and Davenport for ideal schemes. Finally, an example of a nonideal access structure that is nearly ideal is presented.

Networks, Matroids, and Non-Shannon Information Inequalities

We define a class of networks, called matroidal networks, which includes as special cases all scalar-linearly solvable networks, and in particular solvable multicast networks. We then present a method for constructing matroidal networks from known matroids. We specifically construct networks that play an important role in proving results in the literature, such as the insufficiency of linear network coding and the unachievability of network coding capacity. We also construct a new network, from the Vamos matroid, which we call the Vamos network, and use it to prove that Shannon-type information inequalities are in general not sufficient for computing network coding capacities. To accomplish this, we obtain a capacity upper bound for the Vamos network using a non-Shannon-type information inequality discovered in 1998 by Zhang and Yeung, and then show that it is smaller than any such bound derived from Shannon-type information inequalities. This is the first application of a non-Shannon-type inequality to network coding. We also compute the exact routing capacity and linear coding capacity of the Vamos network. Finally, using a variation of the Vamos network, we prove that Shannon-type information inequalities are insufficient even for computing network coding capacities of multiple-unicast networks.

On secret-sharing matroids

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.