The area of proof-based verified computation (outsourced computation built atop probabilistically checkable proofs and cryptographic machinery) has lately seen renewed interest. Although recent work has made great strides in reducing the overhead of naive applications of the theory, these schemes still cannot be considered practical. A core issue is that the work for the server is immense, in general; it is practical only for hand-compiled computations that can be expressed in special forms. This paper addresses that problem. Provided one is willing to batch verification, we develop a protocol that achieves the efficiency of the best manually constructed protocols in the literature yet applies to most computations. We show that Quadratic Arithmetic Programs, a new formalism for representing computations efficiently, can yield a particularly efficient PCP that integrates easily into the core protocols, resulting in a server whose work is roughly linear in the running time of the computation. We implement this protocol in the context of a system, called Zaatar, that includes a compiler and a GPU implementation. Zaatar is almost usable for real problems-without special-purpose tailoring. We argue that many (but not all) of the next research questions in verified computation are questions in secure systems.

When a client outsources a job to a third party (e.g., the cloud), how can the client check the result, without reexecuting the computation? Recent work in proof-based verifiable computation has made significant progress on this problem by incorporating deep results from complexity theory and cryptography into built systems. However, these systems work within a stateless model: they exclude computations that interact with RAM or a disk, or for which the client does not have the full input.This paper describes Pantry, a built system that overcomes these limitations. Pantry composes proofbased verifiable computation with untrusted storage: the client expresses its computation in terms of digests that attest to state, and verifiably outsources that computation. Using Pantry, we extend verifiability to MapReduce jobs, simple database queries, and interactions with private state. Thus, Pantry takes another step toward practical proof-based verifiable computation for realistic applications.

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A long-standing open conjecture in combinatorics asserts that a Gorenstein lattice polytope with the integer decomposition property (IDP) has a unimodal (Ehrhart) h * -polynomial. This conjecture can be viewed as a strengthening of a previously disproved conjecture which stated that any Gorenstein lattice polytope has a unimodal h * -polynomial. The first counterexamples to unimodality for Gorenstein lattice polytopes were given in even dimensions greater than five by Mustaţǎ and Payne, and this was extended to all dimensions greater than five by Payne. While there exist numerous examples in support of the conjecture that IDP reflexives are h * -unimodal, its validity has not yet been considered for families of reflexive lattice simplices that closely generalize Payne's counterexamples. The main purpose of this work is to prove that the former conjecture does indeed hold for a natural generalization of Payne's examples. The second purpose of this work is to extend this investigation to a broader class of lattice simplices, for which we present new results and open problems.

Abstract. A variety of descent and major-index statistics have been defined for symmetric groups, hyperoctahedral groups, and their generalizations. Typically associated to a pair of such statistics is an Euler-Mahonian distribution, a bivariate polynomial encoding the statistics; such distributions often appear in rational bivariate generating-function identities. We use techniques from polyhedral geometry to establish new multivariate identities generalizing those giving rise to many of the known Euler-Mahonian distributions. The original bivariate identities are then specializations of these multivariate identities. As a consequence of these new techniques we obtain bijective proofs of the equivalence of the bivariate distributions for various pairs of statistics.

M. Beck et al. found that the roots of the Ehrhart polynomial of a d-dimensional lattice polytope are bounded above in norm by 1 + (d + 1)!. We provide an improved bound which is quadratic in d and applies to a larger family of polynomials.Let P be a convex polytope in R n with vertices in Z n and affine span of dimension d; we refer to such polytopes as lattice polytopes and to elements of Z n as lattice points. A remarkable theorem due to Ehrhart [5] is that the number of lattice points in the tth dilate of P , for non-negative integers t, is given by a polynomial in t of degree d called the Ehrhart polynomial of P . We denote this polynomial by L P (t), and let Ehr P (x) = t≥0 L P (t)x t denote its associated rational generating function. For more information regarding Ehrhart theory, see [2].In [1] it was shown that for a lattice polytope P of dimension d, the roots of L P (t) are bounded above in norm by 1 + (d + 1)!. However, the authors suggested that a bound that is polynomial in d should exist and questioned whether this is a property of Ehrhart polynomials in particular or of a broader class of polynomials (see Remark 4.4 on p. 26 of [1]). Our answer is the following:Theorem 1 If f is a nonzero polynomial of degree d with real-valued, non-negative coefficients when expressed with respect to the polynomial basis

An interesting open problem in Ehrhart theory is to classify those lattice polytopes having a unimodal h * -vector. Although various sufficient conditions have been found, necessary conditions remain a challenge. In this paper, we consider integrally closed reflexive simplices and discuss an operation that preserves reflexivity, integral closure, and unimodality of the h * -vector, providing one explanation for why unimodality occurs in this setting. We also discuss the failure of proving unimodality in this setting using weak Lefschetz elements.

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