An Algebraic Model For Quorum Systems
aa r X i v : . [ c s . S C ] J un An Algebraic Model For Quorum Systems
Alex Pellegrini University of Bern [email protected]
Luca Zanolini University of Bern [email protected]
Abstract
Quorum systems are a key mathematical abstraction in distributed fault-tolerant computing forcapturing trust assumptions. A quorum system is a collection of subsets of all processes, called quo-rums, with the property that each pair of quorums have a non-empty intersection. They can be foundat the core of many reliable distributed systems, such as cloud computing platforms, distributed stor-age systems and blockchains. In this paper we give a new interpretation of quorum systems, startingwith classical majority-based quorum systems and extending this to Byzantine quorum systems. Wepropose an algebraic representation of the theory underlying quorum systems making use of mul-tivariate polynomial ideals, incorporating properties of these systems, and studying their algebraicvarieties. To achieve this goal we will exploit properties of Boolean Gr¨obner bases. The nice na-ture of Boolean Gr¨obner bases allows us to avoid part of the combinatorial computations required tocheck consistency and availability of quorum systems. Our results provide a novel approach to testquorum systems properties from both algebraic and algorithmic perspectives.
Quorum systems are a key mathematical abstraction in distributed fault-tolerant computing for capturingtrust assumptions. Quorums help in reaching higher availability and fault-tolerance in distributed systems[29]. From a classical point of view, a quorum system is a collection of subsets of all processes P ,called quorums, with the property that each pair of quorums have a non-empty intersection. It is ageneralization of the concept of a majority in a democratically organized group and it is used to ensureconsistency in the context of crash failures, i.e., when processes stop executing steps [4]. Given a fail-prone system F ⊆ P , that is, a collection of subsets containing all processes that may at most failtogether in some execution, we say that Q ⊆ P is a quorum system with respect to F if each pair ofelements of Q has non empty intersection and for every element F of F there exists an element of Q that does not intersect F . However, if the failing processes can deviate in any conceivable way fromtheir algorithm, the above definition is not useful. Malkhi and Reiter [20] introduced a generalizationof classical quorum systems called Byzantine quorum systems , strengthening the definition in a way thatthe pair-wise intersection contains also some correct processes. To do this, they present the so-called dissemination quorum systems and masking quorum systems .We study the properties of quorum systems through Boolean multivariate polynomial ideals encom-passing their properties. These structures admit generating bases called Gr¨obner bases. These allow forefficient ways to compute solutions of the set of polynomials of the original ideal. We study quorumproperties by inspecting such solutions sets, called ideal varieties.Though Gr¨obner bases were originally introduced by Buchberger in polynomial rings over fields [3],many works have used Gr¨obner bases over coefficient rings that are not fields. Among them, Gr¨obnerbases of Boolean polynomial rings (Boolean Gr¨obner bases) introduced in [24, 25] have appealing prop-erties. For a comprehensive description of Boolean polynomial rings and Boolean Gr¨obner Basis thereader is referred to [26]. In this work we exploit Boolean Gr¨obner basis according to the
LEX monomial Institute of Computer Science, University of Bern, Neubr¨uckstrasse 10, 3012 CH-Bern, Switzerland.
Our results will heavily rely on the algebraic background that we will introduce in this section. Manyof the definitions and proofs can be found in [7, Section 4.3] and [27, 11, 12], we are hereby giving theresult without proofs.
Definition 1 (Ideal).
Let ( R, · , +) be a ring and ( R, +) its additive group. A subset I ⊆ R is called an ideal if • ( I, +) is a subgroup of ( R, +) ; • For every r ∈ R and f ∈ I we have that r · f ∈ I .If f , . . . , f s live in the ring K [ X , . . . , X n ] over a field K , then h f , . . . , f s i is an ideal of K [ X , . . . , X n ] .We will call h f , . . . , f s i the ideal generated by f , . . . , f s . Definition 2 (Boolean ring).
A commutative ring B with identity is called a Boolean ring if every el-ement a ∈ B is idempotent, i.e. a = a . Let B be a Boolean ring, the quotient ring B [ X , . . . , X n ] / h X − X , . . . , X n − X n i is a Boolean ring. It is called Boolean polynomial ring and denoted by B ( X , . . . , X n ) .From now on we will consider B = F , which is actually a field, and work with the polynomial ring B ( X , . . . , X n ) unless otherwise specified. Hilbert [16] proved that a polynomial ring over a Noetherianring is Noetherian. This means, in our case, that every ideal in B ( X , . . . , X n ) admits a finite basis.Notice that a polynomial in B ( X , . . . , X n ) is uniquely represented by a polynomial of B [ X , . . . , X n ] that has at most degree 1 for each variable X i . Sets of variables such as { X , . . . , X n } , { Y , . . . , Y n } and { T , . . . , T n } are abbreviated by ¯ X, ¯ Y and ¯ T , respectively. With small letters like p, q we will usuallydenote n -tuples of elements of B for some n . Let f ∈ ( ¯ X, ¯ Y ) be a polynomial and pick p ∈ B n , then f ( p, ¯ Y ) denotes a polynomial in B ( ¯ Y ) obtained by specializing ¯ X with p . Definition 3 (Sum and product of ideals). If I and J are ideals of a polynomial ring B [ X , . . . , X n ] then the sum of I and J , denoted as I + J , is the set I + J = { f + g | f ∈ I and g ∈ J } (1)and their product , denoted I · J , is defined to be the set I · J = { f · g | f ∈ I and g ∈ J } . (2) Definition 4 (Variety).
Let I ⊆ B ( ¯ X ) be an ideal. Define the variety of I as the set V ( I ) = { x ∈ B n | f ( x ) = 0 ∀ f ∈ I } . (3)An ideal I ∈ B [ X , . . . , X n ] is if the associated variety V ( I ) is a finite set, i.e. V ( I ) < ∞ . 2 heorem 5. If I and J are ideals in B [ X , . . . , X n ] , then V ( I + J ) = V ( I ) ∩ V ( J ) and V ( I · J ) = V ( I ) ∪ V ( J ) . We rely on the following theorems to support one of our main results in Section 4. During this workwe will use the set of variables denoted by ¯ X, ¯ Y , ¯ T and ¯ Z with respect to a block order ¯ T < ¯ Z < ¯ Y < ¯ X .Unless otherwise specified we use the LEX monomial ordering, i.e. ≺ = ≺ LEX on variables in each block,e.g. X n ≺ X n − ≺ · · · ≺ X . Recall that LEX is an elimination ordering on the monomials [28,Definition 11].Assume, for example, that we are working on the Boolean polynomial ring defined on blocks ofvariables ¯ X, ¯ Y and ¯ Z with block and elimination order as mentioned before. Given an ideal I and block ¯ X , from now on we will denote with π ¯ X ( V ( I )) the natural projection on the n coordinates correspondingto the ¯ X block. Example 6.
Let I ∈ B ( ¯ X, ¯ Y , ¯ Z ) with LEX order. Let v = ( v , . . . , v n ) ∈ V ( I ) , then π ¯ Y ( v ) =( v n +1 , . . . , v n ) . Definition 7 (Elimination ideal).
Given I = h f , . . . , f s i ⊆ B [ X , . . . , X n ] , the l -th elimination ideal I l is the ideal of B [ X l +1 , . . . , X n ] defined by I l = I ∩ B [ X l +1 , . . . , X n ] . (4)Important results concerning elimination ideals and related varieties are the “Elimination Theorem”and “Extension Theorem” [7, Theorem 2 and 3], respectively. We will only leverage on the latter.Loosely speaking, it says that, given a point in the variety of the elimination ideal, there exist at leastone extension of that point that lies in the variety of the original ideal. In Boolean rings we can define aspecial case of the Extension Theorem. Theorem 8 (Boolean Extension Theorem).
Let I be a finitely generated ideal in a Boolean polynomialring B ( X , . . . , X n , Y , . . . , Y n ) . For any p ∈ V ( I ∩ B ( ¯ X )) there exists q ∈ B n such that ( p, q ) ∈ V ( I ) . Moreover, Gao [12] proved a result over general finite fields F which relates the variety of an elim-ination ideal, with the corresponding projection of the variety of the original ideal. We give hereby aspecific case of the theorem, restricted to our environment. The proof that this form of the theorem holdsis straightforward. Theorem 9 ([12, Theorem 3.1]).
Let I ⊆ B ( ¯ X, ¯ Y ) be an ideal. Then π ¯ X ( V ( I )) = V ( I ∩ B ( ¯ X )) . (5)Let ≺ be a monomial ordering on B ( X , . . . , X n ) . Then we define the leading monomial LM ( f ) =max ≺ { X α | X α ∈ f } and the trailing monomial T M ( f ) = min ≺ { X α | X α ∈ f } .We give here a brief introduction to Gr¨obner basis theory and related results we will need in latersections. We will mostly exploit such results in order to design and support algorithms in Section 5. Definition 10 (Gr¨obner basis).
Let I ⊆ B [ ¯ X ] be an ideal and G = { g , . . . , g t } ⊆ B [ ¯ X ] a set ofpolynomials such that I = hGi . We say that G is a Gr¨obner basis for I if and only if ∀ f ∈ I, f = 0 , ∃ g i ∈ G s.t. LM ( g i ) | LM ( f ) . (6)Usually, a Gr¨obner basis of a set of polynomials is computed using either a variant of Buchberger’salgorithm [3] or using Faugere’s F4 [9] or F5 [10] algorithm.In some of our results we will only need to know the size of the variety of an ideal. This can be donewithout computing the variety at all. We will leverage on what follows to conclude our theses.Let I ∈ B ( ¯ X ) be an ideal. We consider the set of leading monomials of I defined as LM ( I ) = { LM ( f ) | f ∈ I } . We can define the monomial ideal generated by LM ( I ) as the polynomial ideal h LM ( I ) i ∈ B ( ¯ X ) . Define also 3 efinition 11 (Standard monomials). The set of standard monomials of any ideal I ∈ B ( ¯ X ) is denotedas follows. SM ( I ) = { X α · · · X α n n | X α · · · X α n n
6∈ h LM ( I ) i , α i ∈ B } . (7)When an ideal I has a Gr¨obner basis G , we also write the standard monomial set of I as SM ( G ) , andcall it the standard monomial set of G .The following result allows us to avoid the computation of varieties in some of our results. We givea specification related to our environment. Theorem 12 ([11, Theorem 3.2.4]).
Let I ∈ B [ ¯ X ] be a 0-dimensional ideal and G a Gr¨obner basis for I . Then | V ( I ) | = | SM ( G ) | . (8) In this section, we construct a model for representing set operations algebraically.Let n ∈ N , P = { P , . . . , P n } be any set and S ⊆ P , say S = { P j : j ∈ J } with J ⊆ { , . . . , n } a set of indexes. Define ϕ : 2 P → B n as follows ϕ ( S ) X j ∈J e j (9)where { e i } i =1 ,...,n is the canonical basis of B n and we consider the usual vector sum on ( B n , +) . With ϕ ( S ) i , we denote the i -th coordinate of the vector ϕ ( S ) . Moreover, for A ⊆ P we set ϕ ( A ) = { ϕ ( A ) | A ∈ A} . Define the inverse ϕ − : B n → P transforming a vector in B n into the associated setin P ϕ − ( X j ∈J e j )
7→ { P j : j ∈ J } . (10) Example 13.
Let n = 5 then P = { P , . . . , P } , let also S = { P , P , P } with I = { , , } then ϕ ( S ) = e + e + e = (1 , , , , . (11)Given (1 , , , , , then ϕ − ((1 , , , , { P , P , P } = S. (12) Remark 14.
We write q = P ni =1 c i e i with c i ∈ B for every i = 1 , . . . , n . We denote the support of avector q ∈ B n as Supp( q ) = { i | c i = 1 } .With this notation we can represent the powerset of P in terms of the vector space B n .In the rest of this section we define a set of Boolean multivariate polynomials that we will use todefine the polynomial ideals encompassing quorum properties. This part of the section, in other words,translates set-theoretic operations into a polynomial representation.Consider the two sets of variables ¯ X = { X , . . . , X n } and ¯ Y = { Y , . . . , Y n } and define γ ∈ B ( ¯ X, ¯ Y ) = B ( X , . . . X n , Y , . . . Y n ) as γ ( ¯ X, ¯ Y ) = n Y i =1 ( X i Y i + Y i + 1) + 1 . (13)The next lemma states that a zero of γ is an element v ∈ B n such that the support of the first halfcontains the support of the second half. 4 emma 15. Let q, p ∈ B n then Supp( q ) ⊆ Supp( p ) if and only if γ ( p, q ) = 0 . (14) Let
P, Q ⊆ P . Then Q ⊆ P if and only if γ ( ϕ ( P ) , ϕ ( Q )) = 0 . (15) Proof.
Notice that γ ( ¯ X, ¯ Y ) = 0 if and only if ( X i + 1) Y i = 0 for every i = 1 , . . . , n . This is equivalentto say ( X i , Y i ) = (0 , for every i = 1 , . . . , n . For the second part of the claim just set p = ϕ ( P ) and q = ϕ ( Q ) .In other words, we have that γ reflects the set inclusion operator.Define σ ∈ B ( ¯ X, ¯ Y ) as σ ( ¯ X, ¯ Y ) = n Y i =1 ( X i Y i + 1) . (16)Equation (16) defines a polynomial whose zeros are vectors of the form v ∈ B such that the supportsof the two components intersect. Lemma 16.
Let q, p ∈ B n then Supp( p ) ∩ Supp( q ) if and only if σ ( p, q ) = 0 . (17) Thus, given
P, Q ⊆ P . Then P ∩ Q = ∅ if and only if σ ( ϕ ( P ) , ϕ ( Q )) = 0 . (18) Proof.
By construction σ ( ¯ X, ¯ Y ) = 0 if and only if there exists i ∈ { , . . . , n } such that X i Y i = 1 which is equivalent to say that X i = Y i = 1 . For the second part of the claim just set p = ϕ ( P ) and q = ϕ ( Q ) .Similarly to the case of γ , the next corollary is a translation of Lemma 16 to sets, through the appli-cation of ϕ . In other words, σ reflects the set intersection operation.Finally, we define δ ∈ B ( ¯ X, ¯ Y , ¯ T ) as δ ( ¯ X , ¯ Y , ¯ T ) = n Y i =1 ( T i X i Y i + X i Y i + 1) . (19)Equation (19) defines a polynomial which will reflect a special set operation whose explanation andproof is the goal of the next lemma and corollary. We will need such an operation when it comes to talkabout dissemination quorum systems. Lemma 17.
Let p, q, r ∈ B n then (Supp( p ) ∩ Supp( q )) Supp( r ) if and only if δ ( p, q, r ) = 0 . (20) Thus, given
P, Q, R ⊆ P . Then ( P ∩ Q ) R if and only if δ ( ϕ ( P ) , ϕ ( Q ) , ϕ ( R )) = 0 . (21) Proof.
Assume first (Supp( p ) ∩ Supp( q )) Supp( r ) . Thus there exists an i such that p i = q i = 1 and r i = 0 . So, r i p i q i + p i q i + 1 = 0 , and it easy to show that in any other case, it has value . It follows that δ ( p, q, r ) = 0 (22)On the other hand, assume δ ( p, q, r ) = 0 . Assume (Supp( p ) ∩ Supp( q )) ⊆ Supp( r ) , meaning that forevery i such that p i = q i = 1 , also r i = 1 . Then r i p i q i + p i q i + 1 = 1 for each of those i . We obtain δ ( p, q, r ) = 1 which is a contradiction. Observe that if Supp( p ) ∩ Supp( q ) = ∅ , it means that for every i = 1 . . . n , p i = q i . Then r i p i q i + p i q i + 1 = 1 for every i = 1 , . . . , n . To prove the second part of theclaim, we set p = ϕ ( P ) , q = ϕ ( Q ) and r = ϕ ( R ) . 5he next polynomial we define is fundamental for our algebraic representation. This allows us toprecisely construct our varieties and to study them. Lemma 18.
Let Q ⊆ P . The polynomial ξ Q ∈ B ( ¯ Y ) , defined as ξ Q ( Y , . . . , Y n ) = n Y i =1 (1 + Y i + ϕ ( Q ) i ) , (23) has value at every point of B n except for ϕ ( Q ) where it assumes value . We call this the characteristicpolynomial of Q .Proof. The evaluation ξ Q ( ϕ ( Q )) = Q . Let now p ∈ B n be such that p = ϕ ( Q ) ; it exists j ∈{ , . . . , n } for which ϕ ( Q ) j = p j . Thus the factor − ( p j − ϕ ( Q ) j ) vanishes implying ξ Q ( p ) = 0 .In other words, the characteristic polynomial of a set Q is a polynomial that has as zeros vectors ofthe form v ∈ B such that ϕ − ( v ) = Q . Equivalently we can say that ξ Q has ϕ ( Q ) as unique non-zero.In the next corollary, we show, given the characteristic polynomial ξ Q of a set Q , how we can obtain thecharacteristic polynomial of Q c .Let R = { ξ S : S ⊆ P} ⊆ B ( ¯ Y ) be the set given by all the characteristic polynomials of each subset S ⊆ P . Corollary 19.
Let ξ Q ∈ R , then ξ Q c = ξ ∅ · ξ P ξ Q . (24) Proof.
Notice that ξ ∅ = Q ni =1 (1 + Y i ) while ξ P = Y · · · Y n . Let Q = { P i , . . . , P i m } , we have that ξ Q = Y i · · · Y i m · Y j i ,...,i m } (1 + Y j ) . (25)Equation 24 produces the polynomial Y j = i ,...,i m Y j · (1 + Y i ) · · · · · (1 + Y i m ) = ξ Q c . (26) Remark 20.
We have that ξ Q c ∈ B ( ¯ X ) since all of its terms are multilinear in the set of variables ¯ Y .In Appendix A we give further results and constructions using characteristic polynomials. Moreoverwith the operations we define, the set of all the characteristic polynomials becomes a Boolean ring.Let A ⊆ P and ¯ Z = { Z , . . . , Z n } be a set of variables. We denote with ξ A ¯ Z the characteristicpolynomial of A in the ¯ Z variables, defined as ξ A ¯ Z = Y A ∈A ( ξ A ( ¯ Z ) + 1) . (27) Remark 21.
It is easy to see that V ( h ξ A ( ¯ Z ) + 1 i ) = ϕ ( A ) , therefore by the product of ideals we obtain V ( h ξ A ¯ Z i ) = [ A ∈A V ( h ξ A ¯ Z + 1 i ) = { ϕ ( A ) : A ∈ A} = ϕ ( A ) . (28)6 Algebraic model for quorum systems
Let P be a set of n ∈ N processes in a system. Definition 22 (Fail-prone system). A fail-prone system F ⊆ P is a collection of subsets of P , none ofwhich is contained in another, such that some F ∈ F with F ⊆ P is called a fail-prone set and containsall processes that may at most fail together in some execution.Henceforth, the notation A ∗ for a system A ⊆ P denotes the collection of all subsets of the sets in A , that is, A ∗ = { A ′ | A ′ ⊆ A, A ∈ A} . Classical quorum systems are applicable in the context of crash failures, i.e. when processes in thesystem can only stop executing steps [14, 23].
Definition 23 (Classical quorum system).
A (classical) quorum system for a fail-prone system F is acollection of sets of processes Q ⊆ P , where each Q ∈ Q is called a quorum, such that: Consistency
The intersection of any two quorums is non empty, i.e., ∀ Q , Q ∈ Q : Q ∩ Q = ∅ . (29) Availability
For any set of processes that may fail together, there exists a disjoint quorum in Q , i.e., ∀ F ∈ F : ∃ Q ∈ Q : F ∩ Q = ∅ . (30)Our first result expresses the consistency property of a quorum system using the tools we developedin previous sections. Theorem 24.
Let Q be a quorum system and F be a fail-prone system. Consider the ideal I = h ξ Q ¯ X , ξ Q ¯ Y , σ i ⊆ B ( ¯ X, ¯ Y ) and let G be a Gr¨obner basis for I . Then, Q fulfills consistency with re-spect to F if | SM ( G ) | = |Q| . (31) Proof.
Recall that every ideal J ⊆ B ( ¯ X, ¯ Y ) is -dimensional, therefore so is I . By Theorem 12 weobtain that | SM ( G ) | = | V ( I ) | . We analyze now the set V ( I ) and we show that V ( I ) ⊆ ϕ ( Q ) × ϕ ( Q ) .First of all we have that, since I ⊆ B ( ¯ X, ¯ Y ) = F [ ¯ X, ¯ Y ] / h X + X , . . . , X n + X n , Y + Y , . . . , Y n + Y n i , then V ( I ) ⊆ B n × B n .From Remark 21 we have that V ( h ξ Q ¯ X i ) = ϕ ( Q ) × B n and V ( h ξ Q ¯ Y i ) = B n × ϕ ( Q ) . We take advantageof the sum of ideals rule to compute V ( h ξ Q ¯ X , ξ Q ¯ Y i ) = V ( h ξ Q ¯ X i + h ξ Q ¯ Y i )= V ( h ξ Q ¯ X i ) ∩ V ( h ξ Q ¯ Y i ) == ( ϕ ( Q ) × B n ) ∩ ( B n × ϕ ( Q ))= ϕ ( Q ) × ϕ ( Q ) . (32)Since h ξ Q ¯ X , ξ Q ¯ Y i ⊆ I then V ( I ) ⊆ ϕ ( Q ) × ϕ ( Q ) as we claimed. Now, from a variety point of view,adding σ to h ξ Q ¯ X , ξ Q ¯ Y i , means to filter those vectors v ∈ ϕ ( Q ) × ϕ ( Q ) that satisfy Supp ( π ¯ X ( v )) ∩ Supp ( π ¯ Y ( v )) = ∅ (33)as in Lemma 16. Through ϕ − the two projections represent two quorums which intersect. Thus, con-sistency holds if V ( I ) = ϕ ( Q ) × ϕ ( Q ) . But then | V ( I ) | = | ϕ ( Q ) × ϕ ( Q ) | = |Q| . (34)This proves the theorem. 7n other words, Theorem 24 shows that if common zeros of ξ Q ¯ X , ξ Q ¯ Y and σ cover ϕ ( Q ) × ϕ ( Q ) ,then the consistency property holds. This follows from the fact that covering ϕ ( Q ) × ϕ ( Q ) means thatevery quorum in Q has a non empty pair-wise intersection.We can state the complementary result by constructing the ideal I of Theorem 24 by substituting σ with σ + 1 . This allows us to skip the computation of the integer | SM ( G ) | . Corollary 25.
Let Q and F as in Theorem 24. Consider the ideal I = h ξ Q ¯ X , ξ Q ¯ Y , σ + 1 i and let G be aGr¨obner basis for I . Then, Q fulfills consistency with respect to F if G = { } . Corollary 25 relies only on the computation of a Gr¨obner basis of I in order to enforce consistencyof Q with respect to F . Theorem 26.
Let Q and F be a quorum system and a fail-prone system. Consider the ideal I = h ξ F ¯ X , ξ Q ¯ Y , σ + 1 i ⊆ B ( ¯ X, ¯ Y ) and G be a Gr¨obner basis for I . Then Q fulfills availability with re-spect to F if | SM ( G ∩ B ( ¯ X )) | = |F | . (35) Proof.
With the same reasoning as in Theorem 24 we can prove that V ( h ξ F ¯ X , ξ Q ¯ Y i ) = ϕ ( F ) × ϕ ( Q ) and moreover adding σ + 1 means to filter on vectors v ∈ V ( I ) such that Supp ( π ¯ X ( v )) ∩ Supp ( π ¯ Y ( v )) = ∅ . (36)Through ϕ − the two projections represent a fail-prone set and a quorum that do not intersect. For theavailability to hold we need π ¯ X ( V ( I )) = ϕ ( F ) . This is enough because we know from Theorem 9that π ¯ X ( V ( I )) = V ( I ∩ B ( ¯ X )) and from Theorem 8 that, for every p ∈ V ( I ∩ B ( ¯ X )) there exists q ∈ B n such that ( p, q ) ∈ V ( I ) . In other words, through ϕ − , q is the representation of a quorum notintersecting the fail-prone represented by p . As [19, Theorem 2.3.4] says, G ∩ B ( ¯ X ) is a Gr¨obner basisfor I ∩ B ( ¯ X ) meaning that we can apply Theorem 12 to state | SM ( G ∩ B ( ¯ X )) | = | π ¯ X ( V ( I )) | = | ϕ ( F ) | (37)proving our thesis.Theorems 24 and 26 prove that there is a relation between the consistency and availability propertiesand the size of the standard monomials of the related Gr¨obner basis. Results in this section are provedwithout ever checking the varieties of the constructed ideals, instead the Gr¨obner bases computation isrequired. We can therefore state whether a set of sets Q is a quorum system, with respect to a second setof sets F , just by inspecting the leading monomials of the related Gr¨obner bases. In the next sections weuse the same arguments in order to extend our result to other quorum systems. Malkhi and Reiter [20] introduced a generalization of classical quorum systems called Byzantine quorumsystems. These are useful in systems that may be subject to arbitrary (or Byzantine) failures, i.e., if aprocess may deviate in any conceivable way from the algorithm assigned to it [4]. They presented twokinds of quorum systems, namely dissemination quorum systems and masking quorum systems . Theformer aims at storing self-verifying (or authenticated) data in a replicated system, whereas the latter onehas the goal of storing unauthenticated data [29]. They have found many more applications in distributedprotocols.
Definition 27 (Dissemination quorum system).
A (Byzantine) dissemination quorum system for a fail-prone system F is a collection of sets of processes Q ⊆ P , where each Q ∈ Q is called a quorum, suchthat the following properties hold: Consistency
The intersection of any two quorums contains at least one process that is not fail-prone,i.e., ∀ Q , Q ∈ Q , ∀ F ∈ F : Q ∩ Q F. (38)8 vailability For any set of processes that may fail together, there exists a disjoint quorum in Q , i.e., ∀ F ∈ F : ∃ Q ∈ Q : F ∩ Q = ∅ . (39)Exploiting the properties of γ we define the Boolean polynomial γ ( ϕ ( F ) , ¯ T ) for every F ∈ F such that its zero locus are the points representing the powerset F . We can thus express the consistencyproperty in (27) like in the previous section, i.e., by defining an ideal and checking its variety’s properties. Lemma 28.
The zero locus of the polynomial λ ∈ B ( ¯ T ) , defined as λ = Y F ∈F γ ( ϕ ( F ) , ¯ T ) , (40) is the set of points of B n representing the elements of F ∗ .Proof. From Corollary 15, we have that F ′ ⊆ F if and only if γ ( ϕ ( F ) , ϕ ( F ′ )) = 0 (41)where ϕ ( F ′ ) is a point of B n representing a subset of F . It follows that the zero locus of λ is the set ofpoints of B n representing the elements of F ∗ .The consistency property of dissemination quorum systems states that the pair-wise intersections ofquorums in Q are not contained in any fail-prone set. We consider the two characteristic polynomials ξ Q ¯ X and ξ Q ¯ Y along with λ as in Lemma 28 and δ as in (19). Theorem 29.
Let Q and F be a quorum system and a fail-prone system. Let I be the ideal I = h ξ Q ¯ X , ξ Q ¯ Y , , λ, δ i ⊆ B ( ¯ X, ¯ Y , ¯ T ) and G be a Gr¨obner basis for I . We say that Q fulfills consistencywith respect to F if | SM ( G ) | = |Q| · |F ∗ | . (42) Proof.
Since I ⊆ B ( ¯ X, ¯ Y , ¯ T ) = F [ ¯ X, ¯ Y , ¯ T ] / h X + X , . . . , X n + X n , Y + Y , . . . , Y n + Y n , T + T , . . . , T n + T n i , then V ( I ) ⊆ B n × B n × B n .With the same reasoning as in Theorem 24, noticing that V ( h λ i ) = ϕ ( F ∗ ) × B n , it is possible to provethat V ( I ) ⊆ ϕ ( Q ) × ϕ ( Q ) × ϕ ( F ∗ ) . Then, adding δ to h ξ Q ¯ X , ξ Q ¯ Y , , λ i , means to filter those vectors v ∈ ϕ ( Q ) × ϕ ( Q ) × ϕ ( F ∗ ) that satisfy (Supp( p ) ∩ Supp( q )) Supp( r ) (43)as in Lemma 17. Consistency property then holds if V ( I ) = ϕ ( Q ) × ϕ ( Q ) × ϕ ( F ∗ ) . It follows that | SM ( G ) | = | V ( I ) | = |Q| · |F ∗ | Theorem follows.In other words, Theorem 29 shows that there is a relationship between common zeros of ξ Q ¯ X , ξ Q ¯ Y , λ and δ and consistency property. Asking V ( h ξ Q ¯ X , ξ Q ¯ Y , λ, δ i ) to cover ϕ ( Q ) × ϕ ( F ∗ ) means that ev-ery pair of quorums and every fail-prone are zeros of δ which represent the condition required for theconsistency property of dissemination quorum systems.Malkhi and Reiter proved that dissemination quorum systems can only exist if not too many processesfail [20]. Let us define the Q condition [18]. Definition 30 ( Q -condition). A fail prone system F satisfies the Q -condition, abbreviated as Q ( F ) ,whenever it holds, ∀ F , F , F ∈ F : P 6⊆ F ∪ F ∪ F . (44)Loosely speaking, the Q conditions ensures that there is no combination of three fail-prone sets thatcan cover the entire set of players. 9 emma 31 ([20, Theorem 5.4]). Let F be a fail-prone system. A dissemination quorum system for F exists if and only if Q ( F ) . We rephrase the Q -condition in algebraic terms. Define the polynomial ω ∈ B ( ¯ X, ¯ Y , ¯ T ) , as ω ( ¯ X, ¯ Y , ¯ T ) = n Y i =1 ( X i Y i T i + X i Y i + X i T i + Y i T i + X i + Y i + T i ) . (45) Lemma 32.
Given a, b, c ∈ B n we have that Supp ( a ) ∪ Supp ( b ) ∪ Supp ( c ) = { , . . . , n } if and only if ω ( a, b, c ) = 1 .Proof. Assume first
Supp ( a ) ∪ Supp ( b ) ∪ Supp ( c ) = { , . . . , n } , this means that for every i = 1 , . . . , n at least one of a i , b i and c i is thus the factor X i Y i T i + X i Y i + X i T i + Y i T i + X i + Y i + T i evaluates to . This implies ω ( a, b, c ) = 0 . We can prove the other way around with the reverse argumentation.Next theorem gives an algebraic way, following the same idea of previous section, to check Q condition inspecting properties of a specific ideal. Theorem 33.
Given a fail prone system F , consider the ideal I = h ξ F ¯ X , ξ F ¯ Y , ξ F ¯ T , ω i ⊆ B [ ¯ X, ¯ Y , ¯ T ] .Let G be a Gr¨obner basis for I . Then F satisfies Q ( F ) if | SM ( G ) | = | ϕ ( F ) | . (46) Proof.
Apply the same arguments as in Theorem 24.
Remark 34.
Observe that, under the threshold failure model, we can express a quorum system as Q = { ξ ∈ R | deg( T M ( ξ )) ≥ n + f +12 } where f is the number of processes that may fail together.Furthermore, a fail-prone system is F = { ξ ∈ R | deg( T M ( ξ )) ≤ f } . Definition 35 (Masking quorum system).
A (Byzantine) masking quorum system for a fail-prone sys-tem F is a collection of sets of processes Q ⊆ P , where each Q ∈ Q is called a quorum, such that thefollowing properties hold: Consistency
The intersection of any two quorums contains at least one process that is not fail-proneeven when removing from the intersection another fail-prone set, i.e., ∀ Q , Q ∈ Q , ∀ F , F ∈ F : ( Q ∩ Q ) \ F F . (47) Availability
For any set of processes that may fail together, there exists a disjoint quorum in Q , i.e., ∀ F ∈ F : ∃ Q ∈ Q : F ∩ Q = ∅ . (48)We formulate the consistency property in masking quorum system algebraically. Assuming F = { F , . . . , F m } , in the next theorem we denote F c the set { F c , . . . , F cm } . Theorem 36.
Let Q and F be a quorum system and a fail-prone system. Let I be the ideal I = h ξ Q ¯ X , ξ Q ¯ Y , ξ F c ¯ Z , λ, δ ′ i ⊆ B ( ¯ X, ¯ Y , ¯ Z, ¯ T ) , where δ ′ ( ¯ X, ¯ Y , ¯ Z, ¯ T ) = Q ni =1 ( T i X i Y i Z i + X i Y i Z i + 1) ,and G be a Gr¨obner basis for I . We say that Q fulfills consistency with respect to F if | SM ( G ) | = |Q| · |F | · |F ∗ | . (49) Proof.
Apply the same reasoning as in Theorem 29.Theorem 36 follows the same approach as Theorem 24 and Theorem 29 by showing the relationshipbetween elements of V ( h ξ Q ¯ X , ξ Q ¯ Y , ξ F c ¯ Z , λ, δ ′ i ) and the consistency property. Notice that we expressconsistency property by using the equivalence ( A ∩ B ) \ C = ( A ∩ B ) ∩ C c , with A, B and C sets.10 emark 37. Malkhi and Reiter [20] proved a similar condition as Q for masking quorum systems called Q . This is essentially the same except for quantification over fail-prone sets. In this case we say that F satisfies Q ( F ) whenever it holds ∀ F , F , F , F ∈ F : P 6⊆ F ∪ F ∪ F ∪ F . (50)We omit its algebraic construction as it is similar as the one presented for dissemination quorum systems. The present section introduces the basic algorithm for Gr¨obner basis computation introduced by Buch-berger. Such algorithm and more sophisticated ones are implemented in many symbolic computer algebrasystems like PolyBoRi [2], BooleanBG [17], Maculay2 [15] and Magma [1]. Afterwards we give an al-gorithm that makes use of Corollary 25 to characterize consistency in classical quorum systems. For thesake of completeness, we will first give some definitions that we will need to introduce the algorithms.The definitions we mention can be found in the standard literature [6].
Definition 38.
Let f , f ∈ B ( ¯ X ) . We say that f is reducible by f if LM ( f ) | LM ( f ) . The reductionof f by f is defined as red ( f , f ) := f − LM ( f ) LM ( f ) . Definition 39.
Let f ∈ B ( ¯ X ) and S ⊆ B ( ¯ X ) . The reduction of f by S is defined as red ( f , S ) := red ( red ( f , S i ) , S \ { S i } ) if it is possible to choose some S i ∈ S as a valid reductor and f otherwise. Definition 40.
Let f , f ∈ B ( ¯ X ) . The s-polynomial of f and f is defined as sp ( f , f ) := λLM ( f ) f + λLM ( f ) f where λ = LCM ( LM ( f ) , LM ( f )) . Definition 41.
Let
G ⊆ B ( ¯ X ) be a basis of I . G is a Gr¨obner basis of I if ∀ g i , g j ∈ G , red ( sp ( g i , g j ) , G ) =0 . In his Ph.D. thesis [3], Buchberger designed also two criteria to characterize when a s-polynomialreduces to zero, since these reductions do not give contributions to the computation of a Gr¨obner basis.We will call the two criteria coprime criterion and chain criterion , respectively. Theorem 42 (Coprime criterion).
Let f , f ∈ G and G ⊆ B ( ¯ X ) . The polynomial sp ( p, q ) will reduceto zero if LM ( f ) and LM ( f ) are coprime. Theorem 43 (Chain criterion).
Let f , f ∈ G and G ⊆ B ( ¯ X ) . The polynomial sp ( p, q ) will reduce tozero if ∃ g ∈ G : LM ( g ) | LM ( sp ( f , f )) and red ( sp ( f , g ) , G ) = red ( sp ( f , g ) , G ) = 0 . We give Buchberger’s algorithm which takes as input a basis G for an ideal I ⊆ B ( ¯ X ) and outputs aGr¨obner basis for I . We always assume the usage of the ordering LEX .An efficient implementation of the chain criterion was introduced by Gabauer and M¨oller [13], andrecently improved by Campos [6]. Moreover, a detailed analysis along with benchmarks, of exist-ing algorithms for computing Boolean Gr¨obner basis, is reported in [6, Section 5]. The consistencyproperty can now be tested with an algorithm that takes as input the set Q and outputs an element of { TRUE , FALSE } . Notice that we can also consider the computations of all the characteristic polynomi-als as a preprocessing step, therefore the complexity of the algorithm only relies on the Gr¨obner basiscomputation in Algorithm 5.1.Next we sketch an algorithm that uses techniques described in Theorem 26. It tests availability of aquorum system Q with respect to a fail-prone system F .Further algorithms implementing theorems in Section 4 can be devised following the structure ofAlgorithm 5.2 and Algorithm 5.3. A possible method for computing the SM ( G ) function can be im-plemented evaluating the numerator of the Hilbert series of the Stanley-Reisner ring [22]. They use acombinatorial argument on simplicial topology representing monomial ideals to study their structure. Wewant to stress that the time complexity of the proposed algorithm entirely depend on the complexity ofthe Buchberger algorithm and on the SM ( G ) algorithm.11 lgorithm 5.1 function B UCHBERGER ( G ⊆ B ( ¯ X ) ) S ← { ( g i , g j ) : g i , g j ∈ G , j > i } while S = ∅ do s ← select ( S ) S ← S \ { s } if ¬ coprime ( s , s ) ∧ ¬ chain ( s , s , G ) then r ← red ( sp ( s , s ) , G ) if r = 0 then S ← S ∪ { ( r, g ) : g ∈ G} G ← G ∪ { r } end if end if end while return G end functionAlgorithm 5.2 function C ONSISTENCY ( Q ) ξ ¯ X , ξ ¯ Y ← σ ← Q ni =1 ( X i Y i + 1) for all Q ∈ Q do ξ ¯ X ← ξ ¯ X · Q ni =1 (1 + X i + ϕ ( Q ) i ) ξ ¯ Y ← ξ ¯ Y · Q ni =1 (1 + Y i + ϕ ( Q ) i ) end for G ← B UCHBERGER ( { ξ ¯ X , ξ ¯ Y , σ + 1 } ) return G ? = { } end functionAlgorithm 5.3 function A VAILABILITY ( Q , F ) ξ F , ξ Q ← σ ← Q ni =1 ( X i Y i + 1) G ′ ← ∅ for all F ∈ F do ξ F ← ξ F · Q ni =1 (1 + X i + ϕ ( F ) i ) end for for all Q ∈ Q do ξ Q ← ξ Q · Q ni =1 (1 + Y i + ϕ ( Q ) i ) end for G ← B UCHBERGER ( { ξ F , ξ Q , σ + 1 } ) for all g ∈ G do if g ∈ B [ X , . . . , X n ] then G ′ ← G ′ ∪ { g } end if end for Standard = SM ( G ′ ) return | Standard | ? = |F | end function Conclusions and future work
In this work we took advantage of well-known algebraic techniques in order to express properties ofdifferent quorum systems. We proved that given a custom set of sets, one can, in principle, test usingGr¨obner bases whether the set fulfills the requirements of a quorum system. We leave it for futureresearch to actually evaluate the complexity of our method and to explore potential optimizations, interms of time and memory consumption. Furthermore, we strongly believe that refinements of our mainresults are possible, proving for instance that the conditions we give are not only necessary but alsosufficient. Devising an actual algorithm to implement SM ( G ) is a natural next step of this work.Traditionally, trust assumption has been symmetric, in which all processes have to adhere on a globalfail-prone structure. Damg˚ard et al. [8] introduced an asymmetric trust assumption, in which every pro-cess is allowed to trust on a personal failing structure. Cachin and Tackmann [5] introduced asymmetricByzantine quorum systems as a generalization of Byzantine quorums systems for asymmetric trust. Anasymmetric fail-prone system F consists of an array of fail-prone systems, one for every process p i inthe system. An asymmetric Byzantine quorum system Q for F is an array of quorum systems, one forevery process P i such that, in a similar way as in the symmetric case, the intersection of two quorums forany two processes contains at least one process for which both processes assume that it is not faulty and for any process P i and any set of processes that may fail together according to P i , there exists a disjointquorum for P i in its quorum system.Another approach to asymmetric trust was proposed by the Stellar blockchain. The Stellar consensusprotocol [21] powers the Stellar Lumen (XLM) cryptocurrency and introduces federated Byzantine quo-rum systems (FBQS). FBQS rely on the concept of a quorum slice , which is a subset of the processes thatcan convince one particular process of agreement. According to the formalization of Stellar, a quorumas a non-empty set Q ⊂ P that contains at least one quorum slice for each of its non-faulty members.An algebraic model of these two approaches appears interesting and feasible. The ultimate goalwill be to formulate a comprehensive model of the symmetric and asymmetric quorum-system worldswithout referring to set-system properties. We believe this will help finding new and different algorithmsfor implementing quorums in real-world distributed systems. Acknowledgments
The authors would like to express their great appreciation to Christian Cachin, Giorgia Azzurra Marsonand Alessio Meneghetti for their valuable and constructive suggestions during the planning and develop-ment of this research work.This work has been funded by the Swiss National Science Foundation (SNSF) under grant agreementNr. 200021 188443 (Advanced Consensus Protocols).
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Bulletin of the EATCS , 101:125–147, 2010. URL: http://eatcs.org/beatcs/index.php/beatcs/article/view/183 . A Operations on characteristic polynomials
We present some results on characteristic polynomials. In particular, we show how to construct charac-teristic polynomials of intersection and union of sets, and lately on sets of shape ( A ∩ B ) \ C . Lemma 44.
Let Q = { P i , . . . , P i m } ∈ P , then T M ( ξ Q ) = Y i · · · Y i m (51) Proof.
The trailing monomial is obtained by the multiplication of Y i · · · Y i m and the s in (1 − Y j ) · · · (1 − Y j n − m ) . Thus T M ( ξ Q ) = Y i · · · Y i m We start by constructing the characteristic polynomial of the intersection of two sets Q and R . Proposition 45.
Let
Q, R ∈ P and define µ = gcd( T M ( ξ Q ) , T M ( ξ R )) (52) and ν = ξ Q T M ( ξ Q ) · ξ R T M ( ξ R ) . (53) Then, ξ Q ∩ R = µ · ν (54)15 roof. Assume Q = { P i } i ∈ I and R = { P j } j ∈ J and let Q ∩ R = { P k } k ∈ K = I ∩ J . We want to prove that ξ Q ∩ R = Y k ∈ K Y k · Y l ∈{ ,...,n }\ K (1 + Y l ) = µ · ν (55)From Lemma 44 we have that T M ( Q ) = Q i ∈ I Y i and T M ( R ) = Q j ∈ J Y j and therefore µ =gcd( T M ( ξ Q ) , T M ( ξ R )) = Q k ∈ K Y k . Now ξ Q T M ( ξ Q ) = Y i ∈{ ,...,n }\ I (1 + Y i ) (56)and the same holds for ξ R T M ( ξ R ) over J . Notice that since ( { , . . . , n }\ I ) ∪ ( { , . . . , n }\ J ) = { , . . . , n }\ K and since we are working on the binary field, i.e. ( Y i ) = Y i , we can write ν = ξ Q T M ( ξ Q ) · ξ R T M ( ξ R ) = Y l ∈{ ,...,n }\ K (1 + Y l ) (57)Thus the product µ · ν gives the equality in equation (55).Furthermore, we present a construction for the union of Q and R . Proposition 46.
Let
Q, R ∈ P and define µ = T M ( ξ Q ) · T M ( ξ R ) (58) and ν = gcd( ξ Q T M ( ξ Q ) , ξ R T M ( ξ R ) ) (59) Then, ξ Q ∪ R = µ · ν (60) Proof.
Apply the same argument as in Proposition 45 bearing in mind that if A and B are two monomialsin B ( Y , . . . , Y n ) then lcm( A, B ) = A · B . Example 47.
Let n = 6 , therefore P = { P , . . . , P } , and let Q = { P , P , P , P } and R = { P , P , P } . Construct the characteristic polynomials as in Lemma 18 ξ Q = Y Y Y Y Y Y + Y Y Y Y Y + Y Y Y Y Y + Y Y Y Y (61)and ξ R = Y Y Y Y Y Y + Y Y Y Y Y + Y Y Y Y Y + Y Y Y Y Y + Y Y Y Y + Y Y Y Y + Y Y Y Y + Y Y Y (62)We obtain T M ( ξ Q ) = Y Y Y Y and T M ( ξ R ) = Y Y Y . Let us compute the characteristic polynomialof Q ∪ R . First, compute µ and ν as the following. µ = gcd( Y Y Y Y , Y Y Y ) = Y Y (63)and ν =( Y Y + Y + Y + 1)( Y Y Y + Y Y + Y Y + Y Y + Y + Y + Y + 1)= Y Y Y Y + Y Y Y + Y Y Y + Y Y Y + Y Y + Y Y + Y Y + Y Y Y ++ Y + Y Y + Y Y + Y Y + Y + Y + Y + 1 (64)16inally, ( µ · ν )( ¯ Y ) = 1 only in (0 , , , , , , i.e. ϕ − (0 , , , , ,
0) = { P , P } = Q ∩ R meaningthat µ · ν = ξ Q ∩ R = Y Y (1 + Y )(1 + Y )(1 + Y )(1 + Y ) .Now we compute the characteristic polynomial of Q ∩ R . Again, let us compute µ and ν as thefollowing. µ = Y Y Y Y Y = Y Y Y Y Y (65)and ν = gcd( Y Y + Y + Y + 1 , Y Y Y + Y Y + Y Y + Y Y + Y + Y + Y + 1) = (1 + Y ) (66)Finally, ( µ · ν )( ¯ Y ) = 1 only in (0 , , , , , .It follows that ϕ − (0 , , , , ,
1) = { P , P , P , P , P } = Q ∪ R meaning that µ · ν = ξ Q ∪ R = Y (1 + Y )(1 + Y )(1 + Y )(1 + Y )(1 + Y ) .We show how is it possible to obtain the characteristic polynomial on more complex sets as thefollowing. Proposition 48.
Let
Q, R, F ⊆ P and define µ = gcd (cid:18) ξ P T M ( ξ F ) , T M ( ξ Q ) , T M ( ξ R ) (cid:19) (67) and ν = ξ Q T M ( ξ Q ) · ξ R T M ( ξ R ) · ξ F c T M ( ξ F c ) . (68) Then, ξ ( Q ∩ R ) \ F = ξ ( Q ∩ R ) ∩ F c = µ · ν (69) Proof.
Apply the same argument as in Proposition 45 bearing in mind that
T M ( ξ F c ) = ξ P T M ( ξ F ) , with ξ F c as in Corollary 19. Example 49.
Let Q and R as in Example 47 and consider F = { P , P } . Let us compute the character-istic polynomial of ( Q ∩ R ) \ F . First, compute µ and ν as the following. µ = gcd( Y Y Y Y , Y Y Y Y , Y Y Y ) = Y (70)and ν = (1 = Y )(1 + Y )(1 + Y )(1 + Y )(1 + Y ) (71)Finally, ( µ · ν )( ¯ Y ) = 1 only in (0 , , , , , , i.e. ϕ − (0 , , , , ,
0) = { P } = ( Q ∩ R ) \ F meaningthat µ · ν = ξ ( Q ∩ R ) \ F = ξ Q ∩ R ∩ F c = Y (1 + Y )(1 + Y )(1 + Y )(1 + Y )(1 + Y ) .Consider ξ A , ξ B ∈ R . Define the following operations. Addition: ξ A ⋆ ξ B = ξ ( A ∪ B ) \ ( A ∩ B ) Multiplication: ξ A ⋄ ξ B = ξ A ∩ B . Lemma 50. ( R , ⋆, ⋄ ) is a Boolean ring.is a Boolean ring.