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MultilinearNC1 ≠ MultilinearNC2
"... An arithmetic circuit or formula is multilinear if the polynomial computed at each of its wires is multilinear. We give an explicit example for a polynomial f(x 1 , ..., x n ), with coe#cients in n). ..."
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An arithmetic circuit or formula is multilinear if the polynomial computed at each of its wires is multilinear. We give an explicit example for a polynomial f(x 1 , ..., x n ), with coe#cients in n).
Multilinear NC1 6= Multilinear NC2
 In FOCS
, 2004
"... An arithmetic circuit or formula is multilinear if the polynomial computed at each of its wires is multilinear. We give an explicit example for a polynomial f(x1;:::; xn), with coe±cients in f0; 1g, such that over any ¯eld: 1. f can be computed by a polynomialsize multilinear circuit of depth O(log ..."
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Cited by 1 (0 self)
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(log2 n). 2. Any multilinear formula for f is of size n(logn). This gives a superpolynomial gap between multilinear circuit and formula size, and separates multilinear NC1 circuits from multilinear NC2 circuits. 1
Checking Computations in Polylogarithmic Time
, 1991
"... . Motivated by Manuel Blum's concept of instance checking, we consider new, very fast and generic mechanisms of checking computations. Our results exploit recent advances in interactive proof protocols [LFKN92], [Sha92], and especially the MIP = NEXP protocol from [BFL91]. We show that every no ..."
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Cited by 274 (11 self)
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. Motivated by Manuel Blum's concept of instance checking, we consider new, very fast and generic mechanisms of checking computations. Our results exploit recent advances in interactive proof protocols [LFKN92], [Sha92], and especially the MIP = NEXP protocol from [BFL91]. We show that every nondeterministic computational task S(x; y), defined as a polynomial time relation between the instance x, representing the input and output combined, and the witness y can be modified to a task S 0 such that: (i) the same instances remain accepted; (ii) each instance/witness pair becomes checkable in polylogarithmic Monte Carlo time; and (iii) a witness satisfying S 0 can be computed in polynomial time from a witness satisfying S. Here the instance and the description of S have to be provided in errorcorrecting code (since the checker will not notice slight changes). A modification of the MIP proof was required to achieve polynomial time in (iii); the earlier technique yields N O(log log N)...
Candidate indistinguishability obfuscation and functional encryption for all circuits
 In FOCS
, 2013
"... In this work, we study indistinguishability obfuscation and functional encryption for general circuits: Indistinguishability obfuscation requires that given any two equivalent circuits C0 and C1 of similar size, the obfuscations of C0 and C1 should be computationally indistinguishable. In functional ..."
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Cited by 169 (37 self)
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. The security of this construction is based on a new algebraic hardness assumption. The candidate and assumption use a simplified variant of multilinear maps, which we call Multilinear Jigsaw Puzzles. • We show how to use indistinguishability obfuscation for NC 1 together with Fully Homomorphic Encryption (with
Exactly marginal operators and duality in fourdimensional N=1 supersymmetric gauge theory,” Nucl
 Phys. B
, 1995
"... We show that manifolds of fixed points, which are generated by exactly marginal operators, are common in N=1 supersymmetric gauge theory. We present a unified and simple prescription for identifying these operators, using tools similar to those employed in twodimensional N=2 supersymmetry. In parti ..."
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Cited by 240 (7 self)
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We show that manifolds of fixed points, which are generated by exactly marginal operators, are common in N=1 supersymmetric gauge theory. We present a unified and simple prescription for identifying these operators, using tools similar to those employed in twodimensional N=2 supersymmetry. In particular we rely on the work of Shifman and Vainshtein relating the βfunction of the gauge coupling to the anomalous dimensions of the matter fields. Finite N=1 models, which have marginal operators at zero coupling, are easily identified using our approach. The method can also be employed to find manifolds of fixed points which do not include the free theory; these are seen in certain models with product gauge groups and in many nonrenormalizable effective theories. For a number of our models, Sduality may have interesting implications. Using the fact that relevant perturbations often cause one manifold of fixed points to flow to another, we propose a specific mechanism through which the N=1 duality discovered by Seiberg could be associated with the duality of finite N=2 models. (Submitted to Nuclear Physics B)
Separation of Multilinear Circuit and Formula Size
 Theory of Computing
, 2006
"... Abstract: An arithmetic circuit or formula is multilinear if the polynomial computed at each of its wires is multilinear. We give an explicit polynomial f (x1,...,xn) with coefficients in {0,1} such that over any field: 1. f can be computed by a polynomialsize multilinear circuit of depth O(log 2 n ..."
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Cited by 30 (7 self)
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n). 2. Any multilinear formula for f is of size n Ω(logn). This gives a superpolynomial gap between multilinear circuit and formula size, and separates multilinear NC1 circuits from multilinear NC2 circuits. ACM Classification: F.2.2, F.1.3, F.1.2, G.2.0
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