M. Karchmer and A. Wigderson. On span programs. In Proc. of the 8th IEEE Structure in Complexity Theory, pages 102--111, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Consider the special case where V = K for some e and S i = K d i for some d 1 ; dn , where every inner product is the respective standard inner product, and where M is a matrix multiplication M : K d1 , x 7 M Delta x. In this case, M; is called a monotone span program [11]. Clearly, by fixing orthogonal bases of V and S 1 ; Sn , respectively, one can always have this simplified and more familiar view. However, as this simplification might (and indeed would in Section 5.1) destroy the naturalness of additional structures in V or S, we keep this more general ....

...., respectively, one can always have this simplified and more familiar view. However, as this simplification might (and indeed would in Section 5.1) destroy the naturalness of additional structures in V or S, we keep this more general view. Nevertheless, because of this reduction, it follows from [11] that the access structure Gamma and the privacy structure Delta of a linear secret sharing scheme (M; are given by Gamma = fQ P j 9 2 S : supp( Q; M and Delta = Gamma , respectively, where M : S V is the conjugate of M (i.e. such that h; MxiS = hM ; xi V for all 2 S ....

M. Karchmer and A. Wigderson. On span programs. In 8th Annual Conference on Structure in Complexity Theory (SCTC '93). IEEE, 1993.

....computation of a nondeterministic Branching Program by means of randomizing group products. We use a new technique, relying on linear algebraic machinery, to simplify and improve the efficiency of the result from [15] and to extend it to presumably wider log space language classes such as mod p L [11, 16] and to log space counting classes, such as #L. These extensions provide efficient PSM solutions for natural linear algebraic and graph theoretic problems (cf. 23, 2, 1, 11] not covered by previous solutions. But perhaps most of the added value is in the simple and efficient solutions obtained ....

....define the counting logspace classes #L (resp. diffL) to include any integer valued function f for which there exists a nondeterministic logspace TM such that on every input x the number of accepting computation paths (resp. accepting minus rejecting) on x is equal to f(x) The reader may refer to [11, 16, 1] for further discussion of the structure and importance of these log space classes. We use the terms protocol and function to denote parameterized families of protocols and functions. RANK SUM(A1 ; A2 ; An) Input: a Theta a matrices over GF (q) A1 ; A2 ; An . Computes: ....

M. Karchmer and A. Wigderson. On span programs. In Proc. of 8th Structure in Complexity, pages 102--111, 1993.

....realizing non threshold access structures, were introduced by Ito, Saito, and Nishizeki [42] where it was shown that every monotone access structure can be (inefficiently) realized by a secret sharing scheme. More efficient schemes for specific types of access structures were presented, e.g. in [11, 51, 19, 43]. We refer the reader to [50, 53] for extensive surveys on secret sharing literature. # A preliminary version of this paper appeared in the proceedings of the 16th Annu. IEEE Conf. on Computational Complexity, pages 188 202, 2001. Similarly to almost all of the vast literature on ....

....the shares are obtained by applying a linear mapping to the secret and several independent random field elements. Linear schemes may be equivalently defined by requiring that each authorized set reconstructs the secret by applying a linear function to its shares [8] For example, the schemes of [49, 15, 42, 11, 51, 19, 14, 43, 32] are all linear. The share size in linear schemes over F realizing a monotone function f is proportional to the monotone span program size of f over F . Span programs are a linear algebraic model of computation introduced in [43] In fact, there is a one to one correspondence between linear ....

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M. Karchmer and A. Wigderson. On span programs. In Proc. of the 8th Annu. IEEE Structure in Complexity Theory, pages 102--111, 1993.

....networks: It was shown in [30] that any switching and rectifier network (in particular, any nondeterministic branching program) for a large variety of symmetric functions must have size nff(n) where ff(n) is a function which slowly grows to infinity. A similar result was proven in [18] for Phi branching programs. The proofs are based upon a purely combinatorial characterization of the network size in terms of particular instances of the MINIMUM COVER problem. Let C n be the set of those functions f n for which the size (f n ) of the minimal solution to the corresponding ....

....n be the set of those functions for which any restriction ae assigning n=2 variables to zero can be extended to another restriction ae by assigning to zero (n=2 Gamma log log n) additional variables in such a way that the induced function has A log log n . To see C n C n , recall from [30, 18] that every covering set ffi i;ffl (A) has its associated variable x i such that restricting this variable to 0 kills ffi i;ffl (A) Now, for any collection of o(nff(n) covering sets we simply assign n=2 most frequently represented x i s to 0, and this leaves us with a collection in which every ....

M. Karchmer and A. Wigderson. On span programs. In Proceedings of the 8th Structure in Complexity Theory Annual Conference, pages 102--111, 1993.

....addition is always denoted by and multiplication by (or nothing) However, it should always be clear from the context, which addition or multiplication is meant. 3. 2 Span Programs over Rings and Linear Secret Sharing Monotone span programs over (finite) fields were introduced in [32] and turned out to be in a oneto one correspondence to linear secret sharing schemes (over finite fields) This notion was extended in [16] to monotone span programs over (possibly infinite) rings, and it was shown that integer span programs, i.e. span programs over Z, have a similar ....

M. Karchmer and A. Wigderson. On span programs. In Proc. of 8th Conference on Structure in Complexity Theory, pp. 102-111, 1993.

....bounded depth circuits with modular gates [18, 11, 2] for switching and rectifier networks ( nondeterministic branching programs) 19] # Supported by the grant # 93 011 16015 of the Russian Foundation for Fundamental Research . for # branching programs (see [7] for definitions) [6]. The reader willing to learn more about these and related results or about the general perspective of the field is referred to the survey paper [3] Concrete approximation models appeared in the literature can be naturally subdivided into two large groups. Models from the first group use inputs ....

....from [16, 17, 14, 1, 12, 15, 18, 11, 2] We will call the method based on models of this kind the pure approximation method. Other models use as error tests specially designed functionals, every functional being attached to a single input. These models were studied, and sometimes actually used in [9, 19, 4, 5, 6, 8]. See [13] for an extended survey; following this source, we will call the corresponding method the fusion method. The same word fusion will be also used for functionals and models. The most interesting question is, of course, to which extent the approximation method might be useful in proving ....

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M. Karchmer and A. Wigderson. On span programs. In Proceedings of the 8th Structure in Complexity Theory Annual Conference, pages 102--111, 1993.

....result proved here yields quadratic lower bounds for the size of monotone span programs, improving on the largest previously known bounds for explicit functions. The bound is asymptotically tight for the function corresponding to a class of 4 cliques. 1 Introduction Karchmer and Wigderson [14] introduced span programs as a linear algebraic model of computation. A span program for a Boolean function is presented as a matrix over some field with rows labeled by literals of the variables, and the size of the program is the number of rows. The span program accepts an assignment if and only ....

....if the all ones row is a linear combination of the rows whose labels are consistent with the assignment. Definitions are given in Section 2. The class of functions with polynomial size span programs is equivalent to the class of functions with polynomial size counting branching programs [8] [14]. Span program size is a lower bound on the size of symmetric branching programs [14] The model of symmetric branching programs is essentially the same as that of (undirected) contact schemes (for definitions, see [14] Lower bounds for span programs also imply lower bounds for formula size. ....

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M. Karchmer and A. Wigderson. On Span Programs. In Proceedings of the 8th Annual Structure in Complexity Theory, (1993) 102--111.

....Research in Computer Science, Centre of the Danish National Research Foundation. time. The containment SL RL is the only non trivial one in the line above and follows directly from the randomized Logspace algorithm for USTCON of [AKL 79] It is also known that SL SC [Nis92] SL L L [KW93] and SL DSPACE(log n) NSW92] After the surprising proofs that NL is closed under complement were found [Imm88, Sze88] Borodin et al. [BCD 89] asked whether the same is true for SL. They could prove only the weaker statement, namely that SL co Gamma RL, and left SL = co Gamma SL ....

Karchmer and Wigderson. On span programs. In Annual Conference on Structure in Complexity Theory, 1993.

....groups, algebraic number theory. 1 Introduction A black box secret sharing scheme for the threshold access structure T t,n is one which works over any finite Abelian group G. Briefly, such a scheme di#ers from an ordinary linear secret sharing scheme (over, say, a given finite field; see e.g. [Bla79, Sha79, Bri89, BI92, BL88, KW93, Gal95, Bei96, Dij97, CDM00]) in that distribution matrix and reconstruction vectors are defined over Z and are designed independently of the group G from which the secret and the shares may be sampled. In other words, the dealer computes the shares for the n players as Z linear combinations of the secret group element of ....

.... extensions of Z over which there exists a pair of su#ciently large Vandermonde matrices with co prime determinants and show how this allows us to construct, for arbitrary given T t,n , a black box secret sharing scheme with expansion factor O(log n) Using a result of Karchmer and Wigderson [KW93], we show that this is minimal. It is not hard to find an exceptional set of size p in this ring. To see that the maximal size of such a set is p, let K be a number field of degree m, and let denote its ring of algebraic integers. For an arbitrary non trivial ideal I of , it is easy to ....

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M. Karchmer and A. Wigderson. On span programs. In: Proc. Structures in Complexity Theory '93, IEEE Computer Society Press, pp. 102--111, 1993.

....protocols: their generalization of [RBO] runs in time wyx:z e , where w is the size of the smallest monotone formula consisting of [ QQpJ gates which rejects the adversary structure 0 . 1. 3 Monotone span programs Span programs were introduced as a model of computation in [KW93] They were first used as a tool for multiparty computation in [CDM98] In this section we define the concepts related to monotone span programs relevant to this report. Definition 3 A monotone span program (MSP) over a set is a triple r W where is a finite field, is a , ....

M. Karchmer and A. Wigderson. On span programs. In Proceedings of the Eighth Annual Structure in Complexity Theory Conference, pages 102--111, San Diego, California, 18--21 May 1993. IEEE Computer Society Press.

....compute the secret. Linear secret sharing schemes were first introduced by Brickell [9] who considered only ideal linear schemes, i.e. with dim E i = 1, for any C i 2 C[fAg. General linear secret sharing schemes were introduced by Simmons [24] Jackson and Martin [16] and Karchmer and Wigderson [18] under other names, such as geometric secret sharing schemes ormonotone span programs. In an ideal linear secret sharing scheme with dim E 0 = 1, we can consider that the surjective linear mappings i are non zero vectors in the dual space E . In that case, a subset X ae C is qualified if and ....

M. Karchmer, A. Wigderson, On Span Programs, in Proc. of the 8th Annual IEEE Symposium on Structure in Complexity, pp. 102--111, 1993.

....S 0 2 ; S 0 3 ) and to DB 111 the subcube C 111 = S 1 1 ; S 1 2 ; S 1 3 ) as in the elementary cube scheme. We would like the answers of each of the two databases to include the 4 answer bits of the 4 databases it emulates. To achieve this 5 Using the secret sharing scheme proposed in [14], this can be generalized to any function f with a span program (over GF (2) of size S(n) 6 goal, DB 000 sends the single bit b 000 (which it is able to compute from the query it received) along with 3 bit long strings, each of which contains the answer bit of one of the other databases it ....

M. Karchmer and A. Wigderson. On span programs. In Proc. of 8th IEEE Structure in Complexity Theory, pages 102--111, 1993.

....realizing non threshold access structures, were introduced by Ito, Saito, and Nishizeki [41] where it was shown that every monotone access structure can be (inefficiently) realized by a secret sharing scheme. More efficient schemes for specific types of access structures were presented, e.g. in [11, 50, 18, 42]. We refer the reader to [49, 52] for This paper was accepted for publication in the proceedings of the 16th Annu. IEEE Conf. on Computational Complexity, 2001. extensive surveys on secret sharing literature. 1 Originally motivated by the problem of secure information storage, ....

....the shares are obtained by applying a linear mapping to the secret and several independent random field elements. Linear schemes may be equivalently defined by requiring that each authorized set reconstructs the secret by applying a linear function to its shares [8] For example, the schemes of [48, 14, 41, 11, 50, 18, 42, 31] are all linear. The share size in linear schemes over F realizing a monotone function f is proportional to the monotone span program size of f over F . Span programs are a linear algebraic model of computation introduced in [42] In fact, there is a one to one correspondence between linear ....

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M. Karchmer and A. Wigderson. On span programs. In Proc. of the 8th Annu. IEEE Structure in Complexity Theory, pages 102--111, 1993.

....M, then the dual access structure Gamma is computed by a monotone span program M of the same size. We will strengthen this result by proving that such an M not only exists, but can be efficiently computed from M. 1 Introduction Monotone span programs, introduced by Karchmer and Wigderson in [KW93], are a model of computation, based on linear algebra, for computing monotone functions. Since there is a natural one to one correspondence between monotone functions f0; 1g n f0; 1g and access structures over the set P = f1; ng, every access structure Gamma can be represented, we ....

Maurizio Karchmer and Avi Wigderson. On span programs. In 8th Annual Conference on Structure in Complexity Theory (SCTC '93), pages 102--111, San Diego, CA, USA, May 1993. IEEE Computer Society Press.

....by bounded depth circuits over f; MOD p g, if p is a prime and r is not a power of p. A very exciting area of research is trying to find new techniques that might lead to superlinear lower bounds for general Boolean circuits [76, 47, 50, 98] 1.3.1. Span programs Karchmer and Wigderson [49] introduced span programs as a linear algebraic model of computation. A span program for a Boolean function is presented as a matrix over some field with rows labeled by literals of the variables, and the size of the program is the number of rows. The span program accepts an assignment if and only ....

....span programs have only positive literals (non negated variables) as labels of the rows. They compute only monotone functions, even though the computation uses non monotone linear algebraic operations. It is known that every function with a polynomial size span program is in NC (this follows from [19, 23, 49, 58]) but no monotone analog of this result is known. In this model, it is not known how to prove large lower bounds for explicit functions even in the monotone case. The Omega Gamma m 2 = log m) lower bound implied by [25] 8 for monotone span program size is the strongest previously known ....

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M. Karchmer and A. Wigderson, "On span programs", In Proceedings of the 8th Annual Structure in Complexity Theory, 1993, pp. 102-111.

....1. Introduction In the context of complexity lower bounds, the approximation method usually refers to the method originated by Razborov in [58] and [60] for proving monotone lower bounds. The approach was continued by [59] and [67] and several others including [6] 68] 9] 69] 72] 78] [40] for general lower bounds, and further used in monotone lower bounds such as [2] 79] and [10] Other complexity lower bounds that can be generally classified as being based on nonapproximability by low degree or sparse polynomials, or other basis functions include many of the lower bounds on ....

....that of stability and corresponding methods for nonapproximability and lower bounds can be found in [21] Finally, our analytic framework is suitable primarily for questions that can be decomposed into linear approximation questions. Many of the lower bounds based on [58] and [60] such as [40] [2] 79] and [10] use distinctly non linear approximation methods. While it is an open question whether these, too, can be treated using purely linear approximation methods, we discuss the current points of difference in Section 3. 1.3. General results. Below, we give an informal description of ....

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M. Karchmer, A. Wigderson, "On span programs," 8 th Ann. IEEE conf. on Struct. in Compl. Theory, pp. 102-111, 1993.

.... natural It was shown in [30] that any switching and rectifier network (in particular, any nondeterministic branching program) for a large variety of symmetric functions must have size Omega Gamma nff(n) where ff(n) is a function which slowly grows to infinity. A similar result was proven in [18] for Phi branching programs. The proofs are based upon a purely combinatorial characterization of the network size in terms of particular instances of the MINIMUM COVER problem. Let C n be the set of those functions f n for which the size (f n ) of the minimal solution to the corresponding ....

....C n be the set of those functions for which any restriction ae assigning n=2 variables to zero can be extended to another restriction ae 0 by assigning to zero (n=2 Gamma log log n) additional variables in such a way that the induced function has A log log n . To see C n C n , recall from [30, 18] that every covering set ffi i;ffl (A) has its associated variable x i such that restricting this variable to 0 kills ffi i;ffl (A) Now, for any collection of o(nff(n) covering sets we simply assign n=2 most frequently represented x i s to 0, and this leaves us with a collection in which every ....

M. Karchmer and A. Wigderson. On span programs. In Proceedings of the 8th Structure in Complexity Theory Annual Conference, pages 102--111, 1993.

....reach unambiguity and symmetry. Both RUSPACE(logn) and the symmetric logspace class SymSPACE(logn) possess complete problems and share nearly the same structural upper bounds, which seem to distinguish them from NSPACE(logn) they are contained in parity logspace, DSPACE(o(log 2 n) and SC 2 ([12,16,4,17,2]) Open questions here are: what is the relationship between SymSPACE(logn) and RUSPACE(logn) Can the inclusion of SymSPACE(logn) in randomized logspace ( 1] be extended to RUSPACE(logn) If so, the deterministic space bound of O(log 2 n= log log n) for RUSPACE(logn) could be improved to ....

M. Karchmer and A. Wigderson. On span programs. In Proc. of the 8th IEEE Structure in Complexity Theory Conference, pages 102--111, 1993.

....with one sided error by a randomized Logspace machine running in polynomial time. The containment SL RL is the only non trivial one in the line above and follows directly from the randomized Logspace algorithm for USTCON of [AKL 79] It is also known that SL SC [Nis92] SL L L [KW93] and SL DSPACE(log 1:5 n) NSW92] After the surprising proofs that NL is closed under complement were found [Imm88, Sze88] Borodin et al. [BCD 89] asked whether the same is true for SL. They could prove only the weaker statement, namely that SL coRL, and left SL = coSL as an open ....

Karchmer and Wigderson. On span programs. In Annual Conference on Structure in Complexity Theory, 1993.

....to Rutgers University during summer 1999. implicitly acknowledged in [22, pp. 518 519] Interestingly, Mario Szegedy has pointed out to us (personal communication) that an algebraic structure proposed by Tutte [28] when combined with more recent results about span programs and counting classes [14], gives a #L algorithm for planarity testing. It is listed as an open question by Ja Ja and Simon [13] if planarity is in NL, although the subsequent discovery that NL is closed under complementation [11, 27] allows one to verify that one of the algorithms of [12, 13] can in fact be implemented ....

....question if their algorithm can be implemented in SL, but in this paper we observe that the algorithm of Ramachandran and Reif can be implemented in SL. We also show that the planarity problem is hard for L under projection reducibility. Recall that L # SL # NL # AC 1 SL # #L. See [14]. L (respectively SL, NL) denotes deterministic (respectively symmetric, nondeterministic) logarithmic space, AC 1 denotes problems solvable by polynomial size AND OR circuits of logarithmic depth, where the gates are allowed to have any number of inputs. The class #L consists of problems ....

M. Karchmer and A. Wigderson. On span programs. In Proceedings of the 8th Conference on Structure in Complexity Theory, pages 102--111. IEEE Computer Society Press, 1993.

....up of size. All our simulations (as well as Valiant Vazirani s) are non uniform (or at least randomized) A very interesting question that remains open is whether one can remove the nonuniformity from any of these results. The only result in this direction we are aware of is the observation in [KW] that SL PhiL, where SL stands for symmetric logspace (the class of languages logspace reducible to undirected s t connectivity) 2 Preliminaries and Results 2.1 The Isolation Lemma Let E be a finite set, and let w : E R be an arbitrary (weight) function. Extend w to subsets of E by w(S) ....

M. Karchmer and A. Wigderson, "On span programs," In Proceedings of the 8th Annual Symposium on Structure in Complexity Theory, (1993), pp. 102-111.

....where the adversary is restricted to corrupting one of these sets dishonest minority is clearly a special case. Our results in this paper extend to the general scenario, following ideas developed in [CDM99] First, by replacing Shamir secret sharing by monotone span program (MSP) secret sharing [KW93] in our VSS, we immediately obtain WSS protocols secure against any Q 2 adversary [HM97] with communication and computation polynomial in the monotone span program complexity of the adversary [CDM99] A Q 2 adversary is an adversary who is capable of corrupting only subsets of players in a ....

M. Karchmer and A. Wigderson. On span programs. In Proc. of Structure in Complexity, pp. 383--395, 1993.

....the secret, the collection of such qualified sets is specified by an access structure, which naturally corresponds to a monotone Boolean function hM : f0; 1g m f0; 1g (i.e. S [m] is qualified iff hM ( S ) 1) See [4, 14] for a formal definition. 4 Using the secret sharing scheme of [15], this can be generalized to any function h with a span program of size S(n) at least one player. Then there exists a protocol for disclosing s subject to the condition h, whose total communication complexity is S(n) 1. Proof. The players first conditionally disclose a shared random bit r, ....

.... w j , we obtain a monotone Boolean formula HM of size S(n) computing a monotone function hM (y1 ; yn ; w1 ; wn ) Note that h is a projection of hM , since h(y1 ; yn ) hM (y1 ; yn ; y 1 ; y n ) Using the generalized secret sharing scheme of [4] or [15]) in which the total share size can be bounded by the size of a monotone Boolean formula representing the access structure, it follows from Claim 1 that the players can disclose the bit r subject to the condition h using S(n) communication. Finally, a single player holding s simultaneously ....

M. Karchmer and A. Wigderson. On span programs. In Proc. of 8th IEEE Structure in Complexity Theory, pages 102--111, 1993.

....the exact counting logspace hierarchy collapses to L C=L . It collapses all the way to C=L if and only if C=L is closed under complement. We further show that NC 1 (C= L) L C=L , and that this class consists of exactly those languages with logspace uniform span programs over the rationals (cf. [KW93]) We show that testing feasibility of a system of linear equations is complete for this hierarchy. Another complete problem for this class is computing the rank of a matrix (or even determining the low order bit of the rank) In contrast, verifying that a matrix has a particular rank is complete ....

....to FSLE. Given (A; i; b) let S = fj n j bit i of j is equal to bg. Then (A; i; b) is in Comp.RANK if and only if W j2S (A; j) 2 Ver.RANK. The result now follows by Lemma 2.11 and Theorem 2.6. 2 2. 4 Span programs The span program model of computation was introduced by Karchmer and Wigderson [KW93]. A span program on n Boolean variables x 1 ; x n consists of a target vector b in some vector space V , together with a collection of 2n subspaces U z V , for each literal z 2 fx 1 ; x 1 ; x n ; x n g (each subspace is represented by a possibly redundant generating set) The ....

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M. Karchmer and A. Wigderson, On Span Programs. In Proceedings of the 8th Conference on Structure in Complexity Theory, IEEE Computer Society Press, 1993, 102--111.

....3.5. Corollary 3.6 For the problem of computing majority or mod a;b p nc for any a, every branching program has size Omega i n log n log log n j and every oblivious branching program of width w p n has length Omega i n log n log w j . Razborov [Raz90] and Karchmer and Wigderson [KW93] proved unconditional size lower bounds of Omega0 n log log log n) for computing majority on nondeterministic extensions of the branching program model. Razborov proved it for rectifierswitching networks; Karchmer and Wigderson proved it for span programs. The 26 results are based on a ....

Mauricio Karchmer and Avi Wigderson. On span programs. In Proceedings, Structure in Complexity Theory, Eighth Annual Conference, pages 102--111, San Diego, CA, May 1993. IEEE.

....of the complexity of the propositional calculus. This direction of research has been getting attention recently with the hope that the connection to well developed fields of mathematics like algebra can be helpful in proving lower bounds. Span programs as a model of computation were introduced in [13]. A span program is a device for defining boolean functions, where the function is defined to be 1 iff a fixed vector can be expressed as a linear combination of vectors chosen by the input. Span programs polynomially simulate branching programs (for finite fields they are equivalent to counting ....

....u 2i Gamma1 = u 2i = 0, which is impossible for a minterm, a contradiction. Hence f has no small monotone dependency programs. 3.2. Closure properties and some variants of the definitions. First we prove that our models are closed under restrictions. For span programs this was noticed already in [13], using a more complicated argument. Lemma 3.3. If g is a restriction of a function f , then mDP (g) mDP (f) for an arbitrary field K; similarly for span and polynomial programs and also for the non monotone versions. Proof. Suppose that x i is assigned a constant, hence some rows (or ....

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M. Karchmer and A. Wigderson. On span programs. In Proc. of the 8th Structure in Complexity Theory, pages 102--111. IEEE, 1993.

....Supported in part by NSF grant CCR 9509603. y Supported in part by NSF grant CCR 9509603. Recently, surprising results have indicated that symmetric space bounded computation is weaker than nondeterminism. In particular, symmetric logspace has been shown to be contained in parity logspace [14], in SC 2 [20] and in DSPACE(log 4=3 n) 2] None of these upper bounds is known to hold in the nondeterministic case. If we consider these questions for space bounded unambiguous classes, we are confronted with the fact that there are several ways to define notions of unambiguity that ....

M. Karchmer and A. Wigderson. On span programs. In Proc. of the 8th IEEE Structure in Complexity Theory Conference, pages 102--111, 1993.

....considered in several papers (cf. BD90, BDGV92, BC92] Schemes suggested in [BL88] for structures represented by monotone formulas turn out to be important for our quorum systems. The most general access structures for which efficient secret sharing schemes are known is that of span programs [KW93]. All our schemes fall into this category. Krawczyk [Kra94] suggested the notion of computational secret sharing which we adopt for our purposes. Pseudo random functions: Our constructions employ pseudo random functions (cf. GGM86, Lub] for two purposes: encrypting the database and generating ....

....availability possible for such a load, namely F p (Paths) exp( Gamma Omega Gamma p n) The smallest quorums in the Paths system have cardinality O( p n) 6.1. 2 The Scheme Paths SSS The scheme is based on the construction of Rudich for s Gamma t connectivity that was generalized in [KW93] for span programs. The system elements in the Paths system are the edges of the grid, however we first assign intermediate values to the vertices, from which we compute the shares. The basic secret unit s is a single bit. The secret is first randomly split into four bits l, r, t and b, such that ....

M. Karchmer and A. Wigderson. On span programs. In Proc. Structures in Complexity Theory, pages 102--111, 1993.

....protocols: their generalization of [RB89] runs in time m O(log log m) where m is the size of the smallest monotone formula consisting of majority accepting gates which rejects the adversary structure A. 1. 3 Monotone span programs Span programs were introduced as a model of computation in [KW93] They were rst used as a tool for multiparty computation by Cramer, Damg rd and Maurer [CDM98] In this section we de ne the concepts related to monotone span programs relevant to this paper. De nition 3 A monotone span program (MSP) over a set P is a triple (K; M; where K is a nite eld, M ....

M. Karchmer and A. Wigderson. On span programs. In Proceedings of the Eighth Annual Structure in Complexity Theory Conference, pages 102111, San Diego, California, 1821 May 1993. IEEE Computer Society Press.

....fan in case. All our simulations (as well as Valiant Vazirani s) are non uniform (or at least randomized) A very interesting question that remains open is whether one can remove the nonuniformity from any of these results. The only result in this direction we are aware of is the observation in [KW] that SL PhiL, where SL stands for symmetric logspace (the class of languages logspace reducible to undirected s t connectivity) 2 Preliminaries and Results 2.1 The Isolation Lemma Let E be a finite set, and let w : E R be an arbitrary (weight) function. Extend w to subsets of E by w(S) ....

M. Karchmer and A. Wigderson, "On span programs," In Proceedings of the 8th Annual Symposium on Structure in Complexity Theory, (1993), pp. 102-111.

....nodes results in the same power as OR nodes for polynomial size programs . Allowing both AND nodes and OR nodes enables polynomial size programs to recognize alternating logspace, which is equal to P. By allowing parity nodes, polynomial programs recognize PhiL, a logspace analogue to PhiP [KW93]. Meinel [Me89] explores the range of all possibilities and concludes that allowing nodes of other binary Boolean functions does not give classes different from L, NL, P, or PhiL. It is easy to see that the proof of [Im88] yields the same result in the non uniform case: Given a polynomial size ....

M. Karchmer and A. Wigderson. On span programs. Proceedings of the 8th Structure in Complexity Theory, (1993), pp. 102--111.

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M. Karchmer and A. Wigderson. On span programs. In Proc. of the 8th IEEE Structure in Complexity Theory, pages 102--111, 1993.

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M. Karchmer and A. Wigderson. On Span Programs In Proc. of the 8th IEEE Structure in Complexity Theory, pages 102--111, 1993.

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M. Karchmer and A. Wigderson. On span programs. In Proc. of the 8th IEEE Structure in Complexity Theory, pages 102--111, 1993.

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M. Karchmer and A. Wigderson. On Span Programs. In The Eighth Annual Structure in Complexity Theory, pages 102--111, 1993.

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M. Karchmer, A. Wigderson. On span programs. In: Proceedings of the 8th Annual Symposium on Structure in Complexity Theory, (1993).

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M. Karchmer and A. Wigderson. On span programs. In Proc. 8th Ann. Symp. Structure in complexity Theory, IEEE 1993, pp. 102-111.

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M. Karchmer and A. Wigderson. On span programs. Proceedings of the Eighth Annual Structure in Complexity Theory Conference (San Diego, CA, 1993.

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M. Karchmer and A. Wigderson. On span programs. In Proceedings of the Eighth Annual Structure in Complexity Theory Conference, pages 102--111, San Diego, California, 18--21 May 1993. IEEE Computer Society Press.

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M. Karchmer and A. Wigderson. On span programs. In Annual Conference on Structure in Complexity Theory, 1993.

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M. Karchmer and A. Wigderson. On span programs. In Proc. of the 8th Structure in Complexity Theory, pages 102--111, 1993.

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M. Karchmer and A. Wigderson. On span programs. In Proceedings of the Eighth Annual Structure in Complexity Theory Conference, pages 102111, San Diego, California, 1821 May 1993. IEEE Computer Society Press.

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Maurizio Karchmer and Avi Wigderson. On span programs. In 8th Annual Conference on Structure in Complexity Theory (SCTC '93). IEEE, 1993.

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M. Karchmer and A. Wigderson. On span programs. In Proceedings of the Eighth Annual Structure in Complexity Theory Conference, pages 102--111, San Diego, California, 18-- 21 May 1993. IEEE Computer Society Press.

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M. Karchmer and A. Wigderson. On span programs. In Proc. of 8th Conference on Structure in Complexity Theory, pp. 102-111, 1993.

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M. Karchmer and A. Wigderson. On span programs. In Proceedings of the 8th Annual IEEE Structure in Complexity Theory, pp. 102-111, 1993.

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M. Karchmer and A. Wigderson. On span programs. In Proceedings of the Eighth Annual Structure in Complexity Theory Conference, pages 102-111, San Diego, California, 18-21 May 1993. IEEE Computer Society Press.

No context found.

M. Karchmer and A. Wigderson. On span programs. In Proceedings of the 8th Conference on Structure in Complexity Theory, pages 102--111. IEEE Computer Society Press, 1993.

No context found.

M. Karchmer and A. Wigderson. On span programs. In Proceedings of the Eighth Annual Structure in Complexity Theory Conference, pages 102--111. IEEE Computer Society Press, 1993.

No context found.

M. Karchmer and A. Wigderson. On span programs. In 8th Annual Conference on Structure in Complexity Theory (SCTC '93), pages 102--111, 1993.

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