Skip to content
BY 4.0 license Open Access Published by De Gruyter September 25, 2023

Access structures determined by uniform polymatroids

  • Renata Kawa ORCID logo EMAIL logo and Mieczysław Kula ORCID logo

Abstract

In this article, all multipartite access structures obtained from uniform integer polymatroids were investigated using the method developed by Farràs, Martí-Farré, and Padró. They are matroid ports, i.e., they satisfy the necessary condition to be ideal. Moreover, each uniform integer polymatroid defines some ideal access structures. Some objects in this family can be useful for the applications of secret sharing. The method presented in this article is universal and can be continued with other classes of polymatroids in further similar studies. Here, we are especially interested in hierarchy of participants determined by the access structure, and we distinguish two main classes: they are compartmented and hierarchical access structures. The main results obtained for access structures determined by uniform integer polymatroids and a monotone increasing family Δ can be summarized as follows. If the increment sequence of the polymatroid is non-constant, then the access structure is connected. If Δ does not contain any singletons or the height of the polymatroid is maximal and its increment sequence is not constant starting from the second element, then the access structure is compartmented. If Δ is generated by a singleton or the increment sequence of the polymatroid is constant starting from the second element, then the obtained access structures are hierarchical. They are proven to be ideal, and their hierarchical orders are completely determined. Moreover, if the increment sequence of the polymatroid is constant and Δ > 1 , then the hierarchical order is not antisymmetric, i.e., some different blocks are equivalent. The hierarchical order of access structures obtained from uniform integer polymatroids is always flat, that is, every hierarchy chain has at most two elements.

MSC 2010: 94A62

1 Introduction

A secret sharing scheme is a method of sharing a secret among a finite set of participants in such a way that only certain specified subsets of participants can compute the secret data. Secret sharing was originally introduced by Blakley [1] and Shamir [2] independently in 1979 as a solution for safeguarding cryptographic keys, but nowadays, it is used in many cryptographic protocols. The reader is referred to the studies of Beimel [3] and Padró [4] for a general introduction to secret sharing.

Let P be a finite set of participants, and let p 0 P be a special participant called the dealer. Given a secret, the dealer computes the shares and distributes them secretly to the participants, so that each participant receives only his/her share. It is required that only certain authorized subsets of P can recover the secret by pooling their shares together. It is easily seen that the family Γ of all authorized sets, called an access structure, is monotone increasing, which means that any superset of an authorized subset is also authorized. To avoid borderline cases, we assume that Γ and P Γ . If no unauthorized set has any information about the secret, regardless of the computational power available, then the secret sharing scheme is said to be perfect. Such a scheme can be considered as unconditionally secure.

Ito et al. [5] and Benaloh and Leichter [6] independently proved, in a constructive way, that every monotone increasing family of subsets of P admits a perfect secret sharing scheme. Therefore, every monotone increasing family of subsets of P is referred to as an access structure. Obviously, every access structure is uniquely determined by the family of its minimal sets. An access structure is said to be connected if every participant in P is a member of a minimal authorized set.

In a perfect secret sharing scheme, the length of every share is at least the length of the secret. The secret sharing schemes such that all shares have the same length as the secret are said to be ideal, and their access structures are called ideal as well. More formal definitions can be found in the articles of Beimel [3] and Padró [4]. Shamir’s threshold schemes [2] are the best known examples of ideal secret sharing schemes. The secret sharing schemes constructed for a given access structure in the articles of Ito et al. [5] and Benaloh and Leichter [6] are very far from being ideal because the length of the shares grows exponentially with the number of participants.

An access structure is said to be multipartite if the set of participants is divided into several blocks that are pairwise disjoint and participants in individual blocks are equivalent (precise definition can be found in Section 2.1). The study of multipartite access structures was initiated by Kothari [7], who posed the open problem of constructing ideal hierarchical secret sharing schemes, and by Simmons [8], who introduced the multilevel and compartmented access structures. This approach, developed by many authors, provides a very effective tool for describing structures in a compact way, by using a few conditions that are independent of the total number of participants (cf. [913]) and others.

The characterization of ideal access structures is one of the main open problems in the secret sharing theory. This problem seems to be extremely difficult, and only some particular results are known. In many articles, the authors consider some specific classes of access structures with prescribed properties and try to check whether these structures are ideal. Most of the results obtained are based on the connections between ideal secret sharing schemes and matroids discovered by Brickell [14] and Brickell and Davenport [15]. Later, the use of polymatroids proposed by Farràs et al. in [10] provided a new tool for studying ideal multipartite access structures. In particular, they proved that each access structure determined by a polymatroid with a ground set J and a suitable family of subsets of J is a matroid port with a ground set P { p 0 } . A concise review of the results contained in the literature can be found in the articles [1012].

Since ideal access structures are known to be matroid ports, it seems quite natural to look for ideal access structures among matroid ports. Given a specific class of polymatroids, one can take all multipartite access structures determined by these polymatroids and investigate their properties. The ideality can be established on the base of properties of particular polymatroids. In this article, the study is restricted to uniform integer polymatroids. This choice is motivated by the fact that each such polymatroid defines a family of ideal access structure (cf. Remark 2.5). But the method presented here is universal and can be continued with other classes of polymatroids in further similar studies (cf. [16]).

Here, we deal with multipartite access structures Γ = Γ ( Π , Z , Δ ) in a set of participants divided into a partition Π determined by uniform integer polymatroids Z and monotone increasing families Δ . We examine hierarchical order among the participants induced by the obtained access structure. A short introduction to matroids and polymatroids and their relation to access structures are presented in Section 2.2. In particular, we recall the result of Farràs et al. that every polymatroid with the ground set J and a monotone increasing family of subsets of J , which is compatible with the polymatroid, determine a unique access structure, which is a matroid port. The details are described in Definition [10]. In Section 2.3, some relations between uniform integer polymatroids Z = ( J , h , g ) and monotone increasing families Δ P ( J ) { } are presented. We prove several technical properties that are useful in the next sections.

Section 3 is devoted to the study of necessary condition for an access structure obtained from a uniform integer polymatroid to be hierarchical. Under some special conditions, it is proved that the existence of comparable blocks in the access structure Γ ( Π , Z , Δ ) implies that the increment sequence of the polymatroid is (almost) constant. Another result of this section (Corollary 3.10) states that if the height of Z is greater than 1 or g is not constant, then different blocks in Π are not equivalent.

The main results obtained for access structures determined by uniform polymatroids and a monotone increasing family Δ are contained in Sections 4 and 5. If the increment sequence of the polymatroid is non-constant, then the access structure is connected (Theorem 4.1). This theorem combined with Corollary 3.10 shows that in general, the Π -partite access structures determined by uniform integer polymatroids are well constructed, i.e., all participants are important and the basic partition Π cannot be improved. The exceptions are generated by polymatroids with extreme height (1 or m ) and constant increment sequence.

Theorems 4.2 and 4.3 show that large majority of access structures determined by uniform integer polymatroids are compartmented.

Exceptions occur for polymatroids with a maximum height. If Δ is generated by a singleton or the increment sequence is constant starting from the second element, then the obtained access structures are hierarchical (Theorems 4.8, 4.9, and 4.11). In these cases, the hierarchical orders are completely determined. Moreover, if the polymatroid height is two, then complete description of hierarchical structures is also given (Theorem 4.6).

Moreover, we prove in Theorem 4.12 that the maximal length of chains in such hierarchical access structures is equal to 1. This fact seems quite surprising, because for other polymatroids, one can construct hierarchical access structures with chains of arbitrary length. For instance, such constructions can be found in [1113,16] and others.

As was mentioned earlier, every uniform integer polymatroid determines some ideal access structures, but the question is whether all access structures determined by uniform integer polymatroids are ideal. A direction, which is worth considering and may result in obtaining the answer, is using the fact that a sufficient condition (for access structures to be ideal) can be obtained by proving that the one point extension of a given uniform integer polymatroid is representable (cf. [10, Corollary 6.7]). This method has been applied in Section 5 to the proof that all the structures described in Theorems 4.6, 4.8, and 4.11 are ideal. It is worth noting that the class of access structures obtained from uniform integer polymatroids contains some interesting families of objects that can be useful for the applications of secret sharing.

Another interesting example is the family of uniform access structures characterized by Farrás et al. in [12, Section VI] (cf. Remark 4.10). It consists of multipartite access structures that are invariant under any permutation of blocks of participants. In other words, all participants have the same rights, although they are not hierarchically equivalent. A different situation occurs in compartmented access structures, where there is a set of distinguished participants, whose representatives must be present in all authorized sets. Such a case is described in Theorem 5.2.

This article is intended to initiate research on the access structures obtained from uniform integer polymatroids, but it does not exhaust the topic and leaves space for further study. Some remarks on the new research possibilities can be found in Section 6. Appendix contains a classification of all access structures with four parts obtained from uniform integer polymatroids.

2 Preliminaries

The aim of this section is to provide the necessary definitions and results regarding multipartite access structures and polymatroids. In general, we are using the same or similar notations and definitions as in the articles [11] and [12]. The family of all subsets of a set X is denoted by P ( X ) (the power set). Similarly, P k ( X ) denotes the collection of all of k -element subsets of X . Let N 0 and N denote the set of all non-negative integers and positive integers, respectively. Let J be a finite set. For two vectors u ¯ = ( u x ) x J and v ¯ = ( v x ) x J N 0 J , we write u ¯ v ¯ if u x v x for all x J . Moreover, u ¯ < v ¯ denotes u ¯ v ¯ and u ¯ v ¯ . Given a vector v ¯ = ( v x ) x J , we define the support supp ( v ¯ ) = { x J : v x 0 } and the modulus v ¯ = x J v x . Furthermore, we write v ¯ X = ( v x ) x J , where X J and

v x = v x if x X , 0 if x X .

In particular, v ¯ = ( 0 ) x J . Let us observe that v ¯ = v ¯ X is equivalent to supp ( v ¯ ) X . For every z J , we define the vector e ¯ ( z ) N 0 J such that e ¯ ( z ) = ( e x ( z ) ) x J with e z ( z ) = 1 and e x ( z ) = 0 for all x z . For undefined notions, see the articles [10] and [17].

2.1 Multipartite access structures

Let Γ be an access structure on a set of participants P . A participant p P is said to be hierarchically superior or equivalent to a participant q P (written q p ), if A { p } Γ for all subsets A P { p , q } with A { q } Γ . If p q and q p , then the participants p and q are called hierarchically equivalent.

By a partition ( Π -partition) of the set of participants P , we mean a family Π = ( P x ) x J of pairwise disjoint and nonempty subsets of P , called blocks, such that P = x J P x . An access structure Γ is said to be multipartite ( Π -partite) if all participants in every block P x are pairwise hierarchically equivalent. Thus, we are allowed to define a hierarchy in Π . Namely, P x is said to be hierarchically superior or equivalent to P y (written P y P x ) if there are p P y and q P x such that p q . In other words, it can be said that P y is hierarchically inferior or equivalent to P x . By transitivity, we have p q for all p P y and q P x whenever P y P x . The relation both in P and in Π is reflexive and transitive but not antisymmetric in general, so it is a preorder. If P x P y and P y P x , then the blocks P x and P y are called hierarchically equivalent. Moreover, if P x P y or P y P x , then the blocks P x and P y are called hierarchically comparable; otherwise, they are called hierarchically independent. Finally, if P x P y and the blocks are not hierarchically equivalent, then we write P x P y .

Let us recall that an access structure is said to be connected if every participant in P is a member of a minimal authorized set. A participant who does not belong to any minimal authorized set is called redundant. It is easy to see that every participant is hierarchically superior or equivalent to any redundant participant. In particular, all redundant participants are hierarchically equivalent. A block of participants that contains a redundant participant will also be called redundant.

A Π -partite access structure is said to be hierarchical if there are blocks P x and P y in Π such that P x P y . Otherwise, the access structure is referred to as compartmented.

A hierarchical access structure such that the relation is antisymmetric and every pair of blocks is hierarchically comparable is referred to as totally hierarchical. A complete characterization of ideal totally hierarchical access structure was presented by Farràs and Padró [11]. It is worth pointing out that the phrase “compartmented access structure” used here is very general and covers several notions with the same name appearing in the literature.

Given a partition Π = ( P x ) x J of P and a subset A P , we define the vector π ( A ) = ( v x ) x J , where v x = A P x . If Γ is a Π -partite access structure, then all participants in every subset P x are pairwise hierarchically equivalent, so if A Γ , B P and π ( A ) = π ( B ) , then B Γ . We put π ( Γ ) = { π ( A ) N 0 J : A Γ } and

P ( Π ) = { π ( A ) N 0 J : A P } = { v ¯ N 0 J : v ¯ π ( P ) } .

Obviously, if A B P , then π ( A ) π ( B ) . Moreover, if u ¯ π ( Γ ) and u ¯ v ¯ π ( P ) , then v ¯ π ( Γ ) . Indeed, there is A Γ such that u ¯ = π ( A ) . The set A can be extended to a set B P such that v ¯ = π ( B ) . Hence, B Γ and consequently, v ¯ π ( Γ ) . This shows that π ( Γ ) P ( Π ) is a set of vectors monotone increasing with respect to . On the other hand, every monotone increasing set Γ P ( Π ) determines the Π -partite access structure Γ = { A P : π ( A ) Γ } . This shows that there is a one-to-one correspondence between the family of Π -partite access structures defined on P and the family of monotone increasing subsets of P ( Π ) . Therefore, we use the same notation Γ for both the access structure and its vector representation. With this convention, we define supp ( Γ ) = { supp ( v ¯ ) P ( J ) : v ¯ Γ } .

The hierarchy among blocks in Π can be characterized in vector terms as follows: P y P x if and only if

(1) v ¯ e ¯ ( y ) + e ¯ ( x ) Γ for all v ¯ Γ with v y 1 and v x < P x .

To show that P y P x , it is enough to check if the aforementioned condition is satisfied for all vectors v min Γ . A block P x in Π is redundant if and only if v x = 0 for every v ¯ min Γ .

2.2 Polymatroids and access structures

Let J be a nonempty finite set and let P ( J ) denote the power set of J . A polymatroid Z is a pair ( J , h ) where h is a mapping h : P ( J ) R satisfying

  • h ( ) = 0 ;

  • h is monotone increasing: if X Y J , then h ( X ) h ( Y ) ;

  • h is submodular: if X , Y J , then h ( X Y ) + h ( X Y ) h ( X ) + h ( Y ) .

The mapping h is called the rank function of a polymatroid. If all values of the rank function are integer, then the polymatroid is called integer. An integer polymatroid ( J , h ) such that h ( X ) X for all X J is called a matroid. All polymatroids considered in this article are assumed to be integer, so we omit the term “integer” when dealing with polymatroid.

Let Z = ( J , h ) be a polymatroid and let x J such that h ( { x } ) = 1 . The set { X P ( J { x } ) : h ( X { x } ) = h ( X ) } is called a polymatroid port or more precisely, the port of polymatroid Z at the point x. One can show that every polymatroid port is a monotone increasing family of some subsets of J { x } , which does not contain .

The following examples of polymatroids play a special role in studying ideal access structures. Let V be a vector space of finite dimension, and let V = ( V x ) x J be a family of subspaces of V . One can show that the mapping h : P ( J ) N 0 defined by h ( X ) = dim ( x X V x ) for X P ( J ) is the rank function of the polymatroid Z = ( J , h ) . The polymatroids that can be defined in this way are said to be representable. If dim V x 1 for all x J , then we obtain a matroid, which is called representable as well. The family V is referred to as a vector space representation of the polymatroid (matroid).

Let Z = ( J , h ) be a polymatroid. For J = J { x 0 } with a certain x 0 J and a monotone increasing family Δ P ( J ) { } , we define the function h : P ( J ) N 0 by h ( X ) = h ( X ) for all X P ( J ) and

h ( X { x 0 } ) = h ( X ) if X Δ , h ( X ) + 1 if X P ( J ) Δ .

If h is monotone increasing and submodular, then Δ is said to be compatible with Z and Z = ( J , h ) is a polymatroid, which is called the one point extension of Z induced by Δ . It is easy to see that h ( x 0 ) = 1 and Δ is the polymatroid port of Z at the point x 0 . The next result, which is a consequence of [18, Proposition 2.3] (cf. also [11, Proposition 5.2]), is very useful in the investigation of access structures induced by polymatroids.

Lemma 2.1

([18] Csirmaz, [10]) A monotone increasing family Δ P ( J ) { } is compatible with the integer polymatroid Z = ( J , h ) if and only if the following conditions are satisfied:

  1. If Y X J and Y Δ while X Δ , then h ( Y ) < h ( X ) .

  2. If X , Y Δ and X Y Δ , then h ( X Y ) + h ( X Y ) < h ( X ) + h ( Y ) .

The following notation will be used very often throughout this article. Let Z = ( J , h ) be a polymatroid and let X J . We define the following set:

(2) ( Z , X ) = { v ¯ N 0 J : supp ( v ¯ ) X , v ¯ = h ( X ) , Y X v ¯ Y h ( Y ) } .

The notation ( Z , X ) was introduced in [10]. Here, we will use its simplified form ( X ) . It is easy to see that

(3) if Y X J and h ( Y ) = h ( X ) , then ( Y ) ( X ) .

On the other hand, ( Y ) ( X ) = whenever h ( Y ) h ( X ) .

The connection between matroids and ideal access structures was discovered by Brickell and Davenport [15]. They proved that if Γ P ( P ) is the access structure of an ideal secret sharing scheme on a set of participants P with a dealer p 0 P , then there is a matroid S with the ground set P { p 0 } such that Γ is the port of S at the point p 0 .

The converse is not true. For example, the ports of the Vamos matroid are not ideal access structures (cf. [19]). But the linear construction of ideal secret sharing schemes proposed by Brickell [14] shows that every port of a representable matroid is an ideal access structure.

Following [12, Definition 2.3], we define the main notion of this article.

Definition 2.2

Let Π = ( P x ) x J be a partition of a set P of participants. Consider a polymatroid Z = ( J , h ) with h ( { x } ) P x for every x J , and a monotone increasing family Δ P ( J ) { } , which is compatible with Z . We define a Π -partite access structure Γ ( Π , Z , Δ ) in the following way: a vector u ¯ P ( Π ) is in Γ ( Π , Z , Δ ) if and only if there exist a subset X Δ and a vector v ¯ ( X ) such that v ¯ u ¯ . The family Γ ( Π , Z , Δ ) will be called the Π -partite access structure determined by the polymatroid Z and the monotone increasing family Δ .

Let Π = ( P x ) x J be a partition of a set P of participants. Farràs et al. [10, Theorem 5.3] proved that a Π -partite access structure Γ on P is a matroid port if and only if Γ = Γ ( Π , Z , Δ ) for some polymatroid Z with ground set J and monotone increasing family Δ P ( J ) { } compatible with Z . Access structures that are matroid ports are called κ -ideal in the literature. Moreover, if there is a (linearly) representable one point extension Z of Z and Δ is a polymatroid port of Z , then Γ is an ideal acces structure.

Example 2.3

Let us consider J = { 0 , 1 , 2 , 3 } and the function h : P ( J ) N 0 defined by:

h ( X ) = 0 if X = 0 ; 1 if X = 1 ; 2 if X 2 .

It is easy to check that Z = ( J , h ) is a polymatroid and Δ = { { 1 , 2 } , { 1 , 3 } , { 2 , 3 } , { 1 , 2 , 3 } } is its port at 0. Moreover, Z is a one point extension of Z = Z J , where J = { 1 , 2 , 3 } . Thus, Δ is compatible with Z . Hence, we obtain ( { 1 , 2 } ) = { ( 1 , 1 , 0 ) } , ( { 1 , 3 } ) = { ( 1 , 0 , 1 ) } , ( { 2 , 3 } ) = { ( 0 , 1 , 1 ) } and ( { 1 , 2 , 3 } ) = { ( 1 , 1 , 0 ) , ( 1 , 0 , 1 ) , ( 0 , 1 , 1 ) } . From the aforementioned definition, we have u ¯ Γ ( Π , Z , Δ ) if and only if u ¯ π ( P ) = ( P 1 , P 2 , P 3 ) and supp ( u ¯ ) 2 .

2.3 Uniform polymatroids

We begin this subsection with the definition of uniform polymatroids that play a major role in this article. To shorten notation, we set I m = { 0 , 1 , , m } .

Definition 2.4

An integer polymatroid Z = ( J , h ) is called uniform if

X = Y h ( X ) = h ( Y ) for all X , Y J .

Let m J . We define h i = h ( X ) for every i I m with X J , X = i . It is obvious that the sequence ( h i ) i I m determines the rank function of the polymatroid. For this sequence, we define the increment sequence g = ( g i ) i I m by g i = h i + 1 h i for i = 0 , , m 1 , and additionally, g m = 0 . It is easy to see that g is nonincreasing sequence of non-negative integers.

On the other hand, every nonincreasing sequence g = ( g i ) i I m of non-negative integers with g m = 0 , determines h j by:

(4) h j = i = 0 j 1 g i for all j = 1 , , m and h 0 = 0 .

This sequence ( h j ) j I m actually defines an integer polymatroid.

We define the height of a polymatroid as the number of nonzero elements in g . A polymatroid is said to be of a maximal height if g m 1 > 0 . Note that g 0 = 0 h 1 = = h m = 0 and g 1 = 0 h 1 = = h m = g 0 . Hence, according to the assumption that we consider only polymatroids such that their rank functions do not have all values equal to 0, from now on, we assume that the height of each uniform polymatroid is greater than zero. To avoid repetition in the further part of this article, a uniform polymatroid will be denoted by Z = ( J , h , g ) , where g = ( g i ) i I m , g 0 > g m = 0 is a nonincreasing sequence of non-negative integers and h : P ( J ) N 0 is the rank function such that h ( X ) = h k = i = 0 k 1 g i for every X P ( X ) with k = X .

Remark 2.5

We shall show that every uniform polymatroid determines at least one ideal access structure. Indeed, uniform polymatroids are known to be representable (cf. [9, Theorem 6]). Let K be a finite field, and let ( V x ) x J be a K -vector space representation of a uniform polymatroid Z = ( J , h , g ) . Then, V x are the subspaces of the vector space K h m and dim V x = h 1 = g 0 for every x J . For any X J , we define V X = x X V x . Given a nonzero vector β K h m , the family Δ = { X J : β V X } P ( J ) is a monotone increasing family of subsets of J and Δ is compatible with the polymatroid Z . It is easily seen that ( V x ) x J { x 0 } , where x 0 J and V x 0 = span ( { β } ) is a vector space representation of the one point extension of Z induced by Δ . This shows that the access structure Γ ( Π , Z , Δ ) is ideal. Varying the representation of Z and the vector β , we can control to some extent the selection of Δ that allows us to obtain different ideal access structures. This idea will be used in Section 5 in proofs that the structures considered there are linearly representable.

In order to continue our studies, we need some elementary properties of vectors in ( X ) that are proved in several technical lemmas. In the remainder of this subsection, we assume that Z = ( J , h , g ) is a uniform polymatroid and X J . Let us recall that ( X ) is defined by equation (2).

Lemma 2.6

If 1 k = X and w ¯ ( X ) , then:

  1. For every x X we have w x g k 1 .

  2. If w x = g k 1 for some x X , then w ¯ w x e ¯ ( x ) ( X { x } ) .

Proof

  1. Let us note that w ¯ X = h ( X ) = h k and w ¯ X { x } h ( X { x } ) = h k 1 ; hence,

    w x = w ¯ X w ¯ X { x } h k h k 1 = g k 1 .

  2. If we set v ¯ w ¯ w x e ¯ ( x ) , then we have supp ( v ¯ ) X { x } and

    v ¯ = h k g k 1 = h k 1 = h ( X { x } ) .

Lemma 2.7

Let x , y X , x y , and w ¯ ( X ) such that w x = g 0 , w y 0 . If v ¯ ( supp ( v ¯ ) ) and v ¯ w ¯ e ¯ ( y ) + e ¯ ( x ) , then y supp ( v ¯ ) .

Proof

Let w ¯ w ¯ e ¯ ( y ) + e ¯ ( x ) and Y supp ( v ¯ ) . It is clear that v ¯ ( Y ) implies v x h 1 = g 0 and v ¯ = h ( Y ) . Moreover, Y X and w ¯ Y h ( Y ) . Suppose that y Y . If x Y , then we have

h ( Y ) = v ¯ w x + ( w y 1 ) + w ¯ Y { x , y } = w ¯ Y 1 h ( Y ) 1 ,

which is a contradiction.

Similarly, if x Y , then we have

h ( Y ) = v ¯ ( w y 1 ) + w ¯ Y { y } = w ¯ Y 1 h ( Y ) 1 ,

which is a contradiction. This completes the proof.□

Lemma 2.8

Let y X and x J X and w ¯ ( X ) such that w y = g 0 . If k X , g k > 0 , and v ¯ ( supp ( v ¯ ) ) such that v ¯ w ¯ e ¯ ( y ) + e ¯ ( x ) , then y supp ( v ¯ ) . Moreover, if g 0 > 1 , then x , y supp ( v ¯ ) , i.e., supp ( v ¯ ) X { y } .

Proof

Let w ¯ w ¯ e ¯ ( y ) + e ¯ ( x ) . Clearly, supp ( v ¯ ) supp ( w ¯ ) X { x } . Let Y X supp ( v ¯ ) , and let l Y . Suppose that y supp ( v ¯ ) . If x supp ( v ¯ ) , then supp ( v ¯ ) = Y { x } , and we have l k and

h l + 1 = v ¯ w ¯ Y { y } + 1 + ( g 0 1 ) = w ¯ Y { y } + g 0 = w ¯ Y h l .

Hence, 0 < g k g l = h l + 1 h l 0 , which is a contradiction.

If x supp ( v ¯ ) , then supp ( v ¯ ) = Y , and we have

h l = v ¯ w ¯ Y { y } + ( g 0 1 ) = w ¯ Y { y } + g 0 1 = w ¯ Y 1 = h l 1 ,

which is a contradiction. Thus, we have proved that supp ( v ¯ ) ( Y { y } ) { x } .

Now, we assume g 0 > 1 and suppose supp ( v ¯ ) = ( Y { y } ) { x } .

h l = v ¯ w ¯ Y { y } + 1 = w ¯ Y { y } + 1 = w ¯ Y { y } + g 0 ( g 0 1 ) = w ¯ Y ( g 0 1 ) < h l ,

as g 0 1 > 0 , which is a contradiction. This shows supp ( v ¯ ) = Y { y } X { y } , which completes the proof.□

Lemma 2.9

Let x , y X , x y , and w ¯ ( X ) . If w y > 0 , then either w ¯ w ¯ e ¯ ( y ) + e ¯ ( x ) ( X ) , or there exists a set Y X { y } , x Y , such that v ¯ w ¯ Y ( Y ) . Furthermore, v ¯ w ¯ and v ¯ w ¯ .

Proof

Note that supp ( w ¯ ) X and w ¯ X = w ¯ X = h ( X ) = h X . Let us consider the case w ¯ ( X ) , that is, there is a set Y X that w ¯ Y h ( Y ) + 1 . Let us choose a minimum set Y for this property. It is easy to see that x Y and y Y . Setting the notation l Y , we obtain

h l + 1 w ¯ Y = ( w x + 1 ) + w ¯ Y { x } = w ¯ Y + 1 h l + 1 ,

and consequently, w ¯ Y = h l . Thus, for v ¯ w ¯ Y , we have v ¯ ( Y ) . It is clear that v ¯ w ¯ and v ¯ w ¯ , which completes the proof.□

Now, we introduce a notion of a vertex vector. Let J be a finite set and m J and let g = ( g i ) i I m be the increment sequence of a uniform polymatroid Z = ( J , h , g ) . Given X J and a bijection σ : X { 0 , 1 , , k 1 } , where k = X , we define the vector w ¯ = ( w x ) x J by:

w ¯ = x X g σ ( x ) e ¯ ( x ) ,

which is referred to as a vertex vector with basic set X. Note that in general, we have supp ( w ¯ ) X , but supp ( w ¯ ) = X whenever g k 1 > 0 . Vertex vectors are the vertices of the convex polytope

T = { w ¯ N 0 J : w ¯ X h ( X ) for every X J } ,

determined by the polymatroid ( J , h ) .

Lemma 2.10

For every vertex vector w ¯ , we have w ¯ ( supp ( w ¯ ) ) .

Proof

Let w ¯ be any vertex vector and k supp ( w ¯ ) . Let us take a subset Y supp ( w ¯ ) and set l Y k . The sequence g being nonincreasing implies

w ¯ Y = x Y w x = x Y g σ ( x ) i = 0 l 1 g i = h l = h ( Y ) .

Here, we use the fact that the sum of l arbitrary elements of a nonincreasing sequence does not exceed the sum of the l initial entries of the sequence. In particular, we obtain w ¯ supp ( w ¯ ) = i = 0 k 1 g i = h k = h ( supp ( w ¯ ) ) , which shows that w ¯ ( supp ( w ¯ ) ) .□

Remark 2.11

Note that if Z is a uniform polymatroid, then the set ( X ) is always nonempty since it contains vertex vectors with basic set X . In extreme cases when X = or the range function of the polymatroid has all values equal to 0, the family ( X ) contains only the zero vector. Moreover, it is easy to check that if w ¯ ( X ) for some X J , then w ¯ ( supp ( w ¯ ) ) .

Deciding if a monotone increasing family is compatible with a given polymatroid is not an easy task. The Csirmaz lemma seems to be the most general tool for solving this problem. For example, it is easy to check that if the increment sequence of a polymatroid with ground set J is strictly decreasing, then every proper monotone increasing family of subsets of J is compatible with the polymatroid. At the end of this section, we present several facts related to the compatibility of monotone increasing families and polymatroids, which are used in proofs in subsequent sections.

Lemma 2.12

Let Z = ( J , h , g ) be a uniform polymatroid and let a monotone increasing family Δ P ( J ) { } be compatible with Z .

  1. If g k = 0 for some 1 k J , then all subsets of the set J with at least k elements belong to Δ .

  2. If Δ contains a minimal set with k elements, then g k 1 > 0 .

Proof

(1) By assumption, we have g i = 0 for all i = k , , m . Let us consider X J , l X k . Then, we have

h ( J ) h ( X ) = h J h X = i = l m 1 g i = 0 .

This implies h ( X ) = h ( J ) , and by the Csirmaz lemma, we obtain X Δ .

(2) Assume that X J is a minimal set in Δ , X = k . Then, for every Y X with Y = k 1 , we have Y Δ , so by the Csirmaz lemma h Y < h X . Hence,

g k 1 = h k h k 1 = h X h Y > 0 .

Lemma 2.13

If Δ P ( J ) { } is a monotone increasing family such that min Δ = { X } for some X J , then Δ is compatible with a uniform polymatroid Z = ( J , h , g ) if and only if g m 1 > 0 .

Proof

Assume Δ is compatible with Z . If x X , then J { x } Δ , so by Csirmaz lemma h ( J { x } ) < h ( J ) , thus g m 1 = h ( J ) h ( J { x } ) > 0 .

Now, we shall show that the conditions of the Csirmaz lemma are met whenever g m 1 > 0 . Let us note that h i h i 1 = g i 1 > 0 for all i = 1 , , m , so the sequence h 0 , h 1 , , h m is strictly increasing. Let us take such sets Y W J , that Y Δ and W Δ . Of course, Y < W , so we have h ( Y ) < h ( W ) ; thus, condition (1) is satisfied.

Now, let us consider W , Y Δ . Then, X W and X Y since min Δ = { X } , so W Y Δ . This shows that the second condition of the Csirmaz lemma is also satisfied.□

Let us recall a result of Farràs et al., which can be restated as follows.

Lemma 2.14

([12], Lemma 6.1) For a positive integer k I m , the monotone increasing family Δ such that min Δ = P k ( J ) is compatible with a uniform polymatroid Z = ( J , h , g ) if and only if g k 1 > g k .

Further results concerned with compatibility can be found in Section 4.

3 Access structures determined by uniform polymatroids

This section is devoted to the study of necessary conditions for an access structure obtained from a uniform polymatroid to be hierarchical. It is proved in Propositions 3.4, 3.5, 3.7, and 3.8, and Corollary 3.9 that under some special conditions, the existence of comparable blocks in the access structure Γ ( Π , Z , Δ ) implies g 1 = g m 1 i.e., the increment sequence of the polymatroid is (almost) constant. Another result of this section (Corollary 3.10) states that if the height of Z is greater than 1 or g is not constant, then different blocks in Π are not equivalent. This means that the relation is antisymmetric in this case.

From now on, we make the assumptions: J is a finite set with m J 2 , g = ( g i ) i I m being the increment sequence of a uniform polymatroid Z with ground set J and Γ = Γ ( Π , Z , Δ ) . Moreover, 0 < g 0 < P x for all x J ; hence, height of Z is greater than or equal to 1. We define μ ( Δ ) = min { X : X Δ } . Note that μ ( Δ ) 1 , as Δ .

Example 3.1

Let us consider a uniform polymatroid Z = ( J , h , g ) such that the height of Z equals 1, i.e., g 0 > g 1 = 0 , and a monotone increasing family Δ of subsets of J is compatible with Z . Applying Lemma 2.12 (1) yields Δ = P ( J ) { } . According to equation (4), we have h ( X ) = g 0 for all nonempty subsets X of J . Hence, ( X ) ( J ) for every X J (cf. equation (3)), and consequently, X Δ ( X ) = ( J ) . Let Γ = Γ ( Π , Z , Δ ) . This implies that w ¯ min Γ if and only if w ¯ = g 0 or equivalently w ¯ Γ if and only if w ¯ g 0 . This shows that the threshold access structure is the only type of access structures determined by uniform polymatroids with height equal to 1. In particular, all blocks (and participants) are hierarchically equivalent.

Let us collect several simple observations, which are very helpful in many proofs.

Lemma 3.2

  1. ( X ) Γ for all X Δ .

  2. supp ( Γ ) = Δ .

  3. If w ¯ min Γ , then w ¯ ( supp ( w ¯ ) ) and supp ( w ¯ ) Δ .

  4. If w ¯ Γ , then there exists v ¯ min Γ such that v ¯ w ¯ , v ¯ ( supp ( v ¯ ) ) and supp ( v ¯ ) Δ .

  5. If w ¯ is a vertex vector and supp ( w ¯ ) Δ , then w ¯ Γ .

Proof

  1. This follows directly from Definition 2.2. (2) Let us consider Y supp ( Γ ) . Then, there exists w ¯ Γ such that supp ( w ¯ ) = Y . Let us consider two cases:

    1. w ¯ min Γ . Then, there exists X Δ such that w ¯ ( X ) , so Y X . If Y = X , then Y Δ . If Y X , then also Y Δ . Indeed, let us note that w ¯ Y h ( Y ) , w ¯ X = h ( X ) , and w ¯ Y = w ¯ X , where the later equality follows from the fact supp ( w ¯ ) = Y X . Moreover, if Y Δ , then by the Csirmaz lemma, we would obtain

      h ( X ) = w ¯ X = w ¯ Y h ( Y ) < h ( X ) ,

      which is a contradiction.

    2. w ¯ Γ and w ¯ min Γ . Then, there is v ¯ min Γ such that v ¯ w ¯ . From Case (i), we obtain supp ( v ¯ ) Δ . Let us note that supp ( v ¯ ) supp ( w ¯ ) . Moreover, Δ is a monotone increasing family, so Y = supp ( w ¯ ) Δ .

    Now, we shall show the converse inclusion. Let us take X Δ . As we already have observed in Remark 2.11, the family ( X ) cannot be empty, so there is a certain vector w ¯ ( X ) . By (1), we obtain w ¯ Γ , so supp ( w ¯ ) supp ( Γ ) . The family supp ( Γ ) is monotone increasing and supp ( w ¯ ) X , so X supp ( Γ ) .

  2. If w ¯ min Γ , then w ¯ ( X ) for some X Δ . Remark 2.11 implies w ¯ ( supp ( w ¯ ) ) . Moreover, supp ( w ¯ ) supp ( Γ ) ; hence, and by (2), we obtain supp ( w ¯ ) Δ .

  3. It follows from (3) immediately.

  4. If w ¯ is a vertex vector, then we have w ¯ ( supp ( w ¯ ) ) by Lemma 2.10. By assumption and part (1) of this lemma, we obtain w ¯ Γ .□

Lemma 3.3

If g 1 = g n 1 > 0 for some 2 n m and if X , Y min Δ as well as X Y n , then X = Y or both sets are singletons. Moreover, if g 0 = g 1 , then X = Y even if both X and Y are singletons.

Proof

For n = 2 , the claim is obvious. Let us assume n 3 . It is enough to consider the case X Y . Suppose that at least one of these sets, for example X , has at least two elements. Let us fix x X and consider the set

Y = Y when X Y ; Y { x } when X Y = .

Note that X Y = X Y n and W X Y . In addition, W is a proper subset of X , which is a minimum set in Δ , so it does not belong to Δ . Hence, according to the Csirmaz lemma, we obtain

h ( W ) + h ( X Y ) < h ( X ) + h ( Y ) .

On the other hand, the assumption g 1 = g n 1 implies h l = g 0 + ( l 1 ) g 1 for every 1 l n . From this, we obtain

h W + h ( X + Y W ) < h X + h Y ,

g 0 + ( W 1 ) g 1 + g 0 + ( X + Y W 1 ) g 1 < g 0 + ( X 1 ) g 1 + g 0 + ( Y 1 ) g 1 .

The aforementioned expression simplifies to 0 < 0 , which is a contradiction. This shows that if X and Y are different, then they cannot have more than one element.

Let us assume g 0 = g 1 and X = Y = 1 . Let us suppose X Y . Then, X Y = , so by the Csirmaz lemma we have

h ( X Y ) + h ( X Y ) < h ( X ) + h ( Y ) ,

and consequently, h 2 < 2 h 1 or equivalently g 0 + g 1 < 2 g 0 , which is a contradiction.□

Proposition 3.4

If X min Δ , then for all x , y X , x y , the blocks P x and P y are hierarchically independent in the access structure Γ = Γ ( Π , Z , Δ ) .

Proof

Let X min Δ and let x and y be two different elements in X . Suppose P y P x , and consider a vertex vector w ¯ with basic set X and w x = g 0 . Setting k X and applying Lemma 2.12 (2), we have g k 1 > 0 so supp ( w ¯ ) = X , in particular, w y > 0 , and by Lemma 3.2 (5), we obtain w ¯ Γ . Thus, w ¯ = w ¯ e ¯ ( y ) + e ¯ ( x ) Γ . By Lemma 3.2 (4), there is v ¯ min Γ such that v ¯ w ¯ and v ¯ ( supp ( v ¯ ) ) Γ , so applying Lemma 2.7, we have y supp ( v ¯ ) X , which contradicts the fact that X min Δ .□

Proposition 3.5

If X min Δ , 1 k X m 1 and g k > 0 , then for every y X , the block P y is not hierarchically inferior or equivalent to any block P x P y in the access structure Γ = Γ ( Π , Z , Δ ) .

Proof

Let y X and let us suppose that P y P x for some x J . By Proposition 3.4, we have x J X . Let us consider a vertex vector w ¯ with basic set X and w y = g 0 . Obviously, w ¯ Γ by Lemma 3.2 (5). Then, the vector w ¯ w ¯ e ¯ ( y ) + e ¯ ( x ) also belongs to Γ .

By Lemma 3.2 (4), there exists a minimal authorized vector v ¯ such that v ¯ w ¯ , v ¯ ( supp ( v ¯ ) ) and supp ( v ¯ ) Δ . If g 0 > 1 , then Lemma 2.8 implies supp ( v ¯ ) X { y } , but this contradicts the assumption X min Δ .

If g 0 = 1 , then g 0 = g 1 = g k , and by Lemma 2.8, we have supp ( v ¯ ) ( X { y } ) { x } . For Y min Δ such that Y supp ( v ¯ ) , we have X Y X { x } , so X Y k + 1 . Applying Lemma 3.3 yields X = Y but y X and y Y , which is a contradiction.□

Lemma 3.6

Let us assume that X min Δ with k X , and there are x , y J , x y such that X { x , y } 3 and the blocks P y and P x are hierarchically comparable in the access structure Γ . Furthermore, we assume that g 1 = g k and g l > 0 for certain 1 l < m . If X { x , y } or g 0 = g 1 , then g 1 = g l .

Proof

If g 1 = 1 , then the claim is obvious.

Assume that g 1 > 1 and assume that this is not the case. Let l be the least positive integer that does not satisfy the claim. That means, g 1 = g l 1 > g l > 0 . Obviously, k + 1 l m 1 . This implies k m 2 . Without loss of generality, we can assume that P y P x . By Proposition 3.5, we have y X . Let now Y J be a set with l + 1 elements that contains X { x , y } . Moreover, let us take an element z X { x , y } .

Let us consider a vertex vector w ¯ with basic set Y and w x = g 0 and w z = g l . Obviously, supp ( w ¯ ) = Y , as g l > 0 . Under the aforementioned assumptions, w t = g 1 for all t Y { x , z } ; in particular, we have w y = g 1 . For every 0 < j l , we have

(5) h j = i = 0 j 1 g i = g 0 + ( j 1 ) g 1 .

Let us note that Y Δ as X Y . Hence, w ¯ Γ by Lemma 3.2 (5). Moreover, h l + 1 = w ¯ = g 0 + ( l 1 ) g 1 + g l . Since P y P x , we have w ¯ w ¯ e ¯ ( y ) + e ¯ ( x ) Γ . Let us note supp ( w ¯ ) = Y . Now by Lemma 3.2 (4), there exists a minimal authorized vector v ¯ such that v ¯ w ¯ , v ¯ ( supp ( v ¯ ) ) and W supp ( v ¯ ) Δ . Lemma 2.7 implies y W , i.e., W Y { y } , so W l . Let Z min Δ such that Z W .

Thus, X Z X W Y { y } , so X Z l and applying Lemma 3.3 yields X = Z or both X and W are singletons. If X { x , y } , then x , z X , so X is not a singleton; thus, X = Z . If g 0 = g 1 , then X = Z . Thus, in both cases, we have z X = Z W , so applying Lemma 2.6 (1), we obtain g W 1 v z . Note also that v z w z = w z = g l < g l 1 g W 1 , a contradiction that proves that g 1 = g l .□

Proposition 3.7

Let us assume that the height of Z equals n 3 . If there are X min Δ such that 1 X n 2 and x , y J X such that the blocks P x and P y are hierarchically comparable in the access structure Γ , then g 0 = g 1 = = g n 1 > g n = 0 .

Proof

If g 0 = 1 , then let us observe that

1 = h 1 = g 0 g 1 g n 1 1 .

Hence, g 0 = g 1 = = g n 1 > g n = 0 .

Thus, we assume g 0 2 . If the blocks P x and P y are hierarchically comparable, then one can assume without loss of generality that P y P x . Let us consider a vertex vector w ¯ with basic set X { y } such that w y = g 0 . Obviously, by Lemma 3.2 (5), we have w ¯ Γ . Then, the vector w ¯ w ¯ e ¯ ( y ) + e ¯ ( x ) belongs to Γ . By Lemma 3.2 (4), there exists a minimal authorized vector v ¯ such that v ¯ w ¯ , v ¯ ( supp ( v ¯ ) ) and supp ( v ¯ ) Δ . By Lemma 2.8, we have supp ( v ¯ ) X , but X is minimal in Δ , so supp ( v ¯ ) = X . Thus, we have

h k = v ¯ w ¯ X = w ¯ X = i = 1 k g i = h k + 1 g 0 ,

so g 0 h k + 1 h k = g k . The sequence g is nonincreasing, so g 0 = g 1 = = g k . Thus, we have shown that g 1 = = g k . To complete the proof, it is enough to apply Lemma 3.6, assuming l = n 1 .□

Proposition 3.8

Let us assume that the height of Z equals n 3 . If there are X min Δ with 2 X n 1 and x X and y J X such that the blocks P x and P y are hierarchically comparable in the access structure Γ , then g 1 = = g n 1 > g n = 0 .

Proof

If the blocks P x and P y are hierarchically comparable, then it follows from Proposition 3.5 that P y P x . Let us consider a vertex vector w ¯ with basic set X { y } such that w x = g 0 and w y = g 1 . Obviously, w ¯ Γ by Lemma 3.2 (5). Then, also the vector w ¯ w ¯ e ¯ ( y ) + e ¯ ( x ) belongs to Γ and supp ( w ¯ ) X { y } . Hence, by Lemma 3.2 (4), there is a minimal authorized vector v ¯ , such that v ¯ w ¯ , v ¯ ( supp ( v ¯ ) ) , and Y supp ( v ¯ ) Δ . Let us observe Y supp ( w ¯ ) X { y } .

By Lemma 2.7, we have y Y that shows Y X , but X is minimal in Δ , so Y = X . Thus, we have

h k = v ¯ g 0 + i = 2 k g i = h k + 1 g 1 ,

where k X . Hence, g 1 h k + 1 h k = g k and g 1 = g k as the sequence g is nonincreasing. To complete the proof, it is enough to apply Lemma 3.6, assuming l = n 1 .□

Corollary 3.9

Let n be the height of Z . If there are x , y J such that P x and P y are hierarchically comparable and 3 X { x , y } n for a certain X min Δ , then g 1 = g m 1 . Moreover, if X { x , y } = , then g 0 = g 1 = g m 1 .

Proof

Assuming with no loss of generality that P y P x , we obtain that { x , y } is not contained in X , by Proposition 3.4, so X n 1 . Applying Proposition 3.5 yields y X . If x X , then 2 X n 1 , and applying Proposition 3.8 yields g 1 = g n 1 > g n = 0 . If x X , in particular X = 1 , then applying Proposition 3.7 yields g 0 = g 1 = g n 1 > g n = 0 .

Suppose, contrary to our claim, that n < m . Then, there is a subset Z J such that Z = n + 1