Constant-degree Polynomial Research Articles

Scooby: Improved multi-party homomorphic secret sharing based on FHE

In this paper we present new constructions of multi-party homomorphic secret sharing (HSS) based on a new primitive that we call homomorphic encryption with decryption to shares (HEDS). Our first scheme, which we call Scooby, is based on many popular fully homomorphic encryption (FHE) schemes with a linear decryption property. Scooby achieves an n-party HSS for general circuits with complexity O(|F|+log⁡n), as opposed to O(n2⋅|F|) for the prior best construction based on multi-key FHE. Scooby relies on a trusted setup procedure, and can be based on (ring)-LWE with a super-polynomial modulus-to-noise ratio. In our second construction, Scrappy, assuming any generic FHE plus HSS for NC1-circuits, we obtain a HEDS scheme which does not require a super-polynomial modulus. While these schemes all require FHE, in another instantiation, Shaggy, we show how it is also possible to obtain multi-party HSS without FHE, instead relying on the DCR assumption to obtain 4-party HSS for constant-degree polynomials.

Massively parallelized interpolated factored Green function method

This paper presents the first parallel implementation of the novel “Interpolated Factored Green Function” (IFGF) method introduced recently for the accelerated evaluation of discrete integral operators arising in wave scattering and other areas (Bauinger and Bruno, Jour. Computat. Phys., 2021). On the basis of the hierarchical IFGF interpolation strategy, the proposed (hybrid MPI-OpenMP) parallel implementation results in efficient data communication, and it scales up to large numbers of cores—without any hard limitations on the number of cores efficiently employed. Moreover, on any given number of cores, the proposed parallel approach preserves the O(Nlog⁡N) computing cost inherent in the sequential version of the IFGF algorithm. Unlike other approaches, the IFGF method does not utilize the Fast Fourier Transform (FFT), and it is thus better suited for efficient parallelization in distributed-memory computer systems. In particular, the IFGF method relies on a “peer-to-peer” strategy wherein, at every level, field propagation is directly enacted via “exchanges” between “peer” polynomials of constant degree, without data accumulation in large-scale “telephone-central” mathematical constructs such as those used in the Fast Multipole Method (FMM) and pure FFT-based approaches. A variety of numerical results presented in this paper illustrate the character of the proposed parallel algorithm, in particular demonstrating scaling from 1 to all 1,680 cores available in the High Performance Computing cluster used, and for problems of up to 4,096 wavelengths in acoustic size.

Testing Polynomials for Vanishing on Cartesian Products of Planar Point Sets: Collinearity Testing and Related Problems

We present subquadratic algorithms, in the algebraic decision-tree model of computation, for detecting whether there exists a triple of points, belonging to three respective sets A, B, and C of points in the plane, that satisfy a certain polynomial equation or two equations. The best known instance of such a problem is testing for the existence of a collinear triple of points in \(A\times B\times C\), a classical 3SUM-hard problem that has so far defied any attempt to obtain a subquadratic solution, whether in the (uniform) real RAM model, or in the algebraic decision-tree model. While we are still unable to solve this problem, in full generality, in subquadratic time, we obtain such a solution, in the algebraic decision-tree model, that uses roughly \(O(n^{28/15})\) constant-degree polynomial sign tests, for the special case where two of the sets lie on two respective one-dimensional curves and the third is placed arbitrarily in the plane. Our technique is fairly general, and applies to many other problems where we seek a triple that satisfies a single polynomial equation, e.g., determining whether \(A\times B\times C\) contains a triple spanning a unit-area triangle. This result extends recent work by Barba et al. (2017) and by Chan (2018), where all three sets A, B, and C are assumed to be one-dimensional. As a second application of our technique, we again have three n-point sets A, B, and C in the plane, and we want to determine whether there exists a triple \((a,b,c) \in A\times B\times C\) that simultaneously satisfies two independent real polynomial equations. For example, this is the setup when testing for collinearity in the complex plane, when each of the sets A, B, C lies on some constant-degree algebraic curve. We show that problems of this kind can be solved with roughly \(O(n^{24/13})\) constant-degree polynomial sign tests.

A #SAT Algorithm for Small Constant-Depth Circuits with PTF gates

We show that there is a better-than-brute-force algorithm that, when given a small constant-depth Boolean circuit C made up of gates that compute constant-degree Polynomial Threshold functions or PTFs (i.e., Boolean functions that compute signs of constant-degree polynomials), counts the number of satisfying assignments to C in significantly better than brute-force time. Formally, for any constants d, k, there is an \(\varepsilon > 0\) such that the zero-error randomized algorithm counts the number of satisfying assignments to a given depth-d circuit C made up of k-PTF gates such that C has at most \(n^{1+\varepsilon }\) many wires. The algorithm runs in time \(2^{n-n^{\Omega (\varepsilon )}}.\) Before our result, no algorithm for beating brute-force search was known for counting the number of satisfying assignments even for a single degree-k PTF (which is a depth-1 circuit with linearly many wires).We give two different algorithms for the case of a single PTF. The first uses a learning algorithm for learning degree-1 PTFs (or Linear Threshold Functions) using comparison queries due to Kane, Lovett and Moran (STOC 2018), and the second uses a proof of Hofmeister (COCOON 1996) for converting a degree-1 PTF to a depth-two threshold circuit with small weights. We show that both these ideas fit nicely into a memoization approach that yields the #SAT algorithms.

Orbit portraits in non-autonomous iteration

We extend the definition of an orbit portrait to the context of non-autonomous iteration, both for the combinatorial version involving collections of angles, and for the dynamic version involving external rays where combinatorial portraits can be realized by the dynamics associated with sequences of polynomials with suitably uniformly bounded degrees and coefficients. We show that, in the case of sequences of polynomials of constant degree, the portraits which arise are eventually periodic which is somewhat similar to the classical theory of polynomial iteration. However, if the degrees of the polynomials in the sequence are allowed to vary, one can obtain portraits with complementary arcs of irrational length which are fundamentally different from the classical ones.

Subquadratic Algorithms for Algebraic 3SUM

The 3SUM problem asks if an input n-set of real numbers contains a triple whose sum is zero. We qualify such a triple of degenerate because the probability of finding one in a random input is zero. We consider the 3POL problem, an algebraic generalization of 3SUM where we replace the sum function by a constant-degree polynomial in three variables. The motivations are threefold. Raz et al. gave an $$O(n^{11/6})$$ upper bound on the number of degenerate triples for the 3POL problem. We give algorithms for the corresponding problem of counting them. Gronlund and Pettie designed subquadratic algorithms for 3SUM. We prove that 3POL admits bounded-degree algebraic decision trees of depth $$O(n^{12/7+\varepsilon })$$ , and we prove that 3POL can be solved in $$O(n^2 {(\log \log n)}^{3/2} / {(\log n)}^{1/2})$$ time in the real-RAM model, generalizing their results. Finally, we shed light on the General Position Testing (GPT) problem: “Given n points in the plane, do three of them lie on a line?”, a key problem in computational geometry: we show how to solve GPT in subquadratic time when the input points lie on a small number of constant-degree polynomial curves. Many other geometric degeneracy testing problems reduce to 3POL.

The Elekes–Szabó Theorem in four dimensions

Let F ∈ C[x, y, s, t] be an irreducible constant-degree polynomial, and let A,B,C,D ⊂ C be finite sets of size n. We show that F vanishes on at most O(n8/3) points of the Cartesian product A × B × C × D, unless F has a special group-related form. A similar statement holds for A,B,C,D of unequal sizes, with a suitably modified bound on the number of zeros. This is a four-dimensional extension of our recent improved analysis of the original Elekes–Szabo theorem in three dimensions. We give three applications: an expansion bound for three-variable real polynomials that do not have a special form, a bound on the number of coplanar quadruples on a space curve that is neither planar nor quartic, and a bound on the number of four-point circles on a plane curve that has degree at least five.

Polynomials vanishing on Cartesian products: The Elekes–Szabó theorem revisited

Let $F\in\mathbb{C}[x,y,z]$ be a constant-degree polynomial,and let $A,B,C\subset\mathbb C$ be finite sets of size $n$. We show that $F$ vanishes on at most $O(n^{11/6})$ points of the Cartesian product $A\times B\times C$, unless $F$ has a special group-related form. This improves a theorem of Elekes and Szab\'o [Combinatorica, 2012], and generalizes a result of Raz, Sharir, and Solymosi [Amer. J. Math., to appear]. The same statement holds over $\mathbb{R}$, and a similar statement holds when $A, B, C$ have different sizes (with a more involved bound replacing $O(n^{11/6})$). This result provides a unified tool for improving bounds in various Erd\H os-type problems in combinatorial geometry, and we discuss several applications of this kind.

Incidences with $k$-non-degenerate sets and their applications

We study point-sphere and point-plane incidences in the three-dimensional space. In particular, for $1 0$, the number of incidences between a set of $m$ points and a $k$-non-degenerate set of $n$ spheres is \[ O(m^{3/4+\varepsilon}n^{3/4}k^{1/4}+n+mk).\] Similarly, we prove that, for every $\varepsilon>0$, the number of incidences between a set of $m$ points and a $k$-non-degenerate set of $n$ planes is \[ O(m^{4/5+\varepsilon}n^{3/5}k^{2/5} + n + mk). \] These bounds are obtained by using the polynomial partitioning technique, recently introduced by Guth and Katz. More specifically, in our proofs we use a pair of constant-degree partitioning polynomials. We also present a couple of applications of $k$-non-degenerate sets: (i) We consider an extension of the three-dimensional unit distances problem, in which we are given a set $D$ of $k$ distinct distances and ask for a three-dimensional set of $m$ points that maximizes the number of pairs of points that span a distance from $D$. By relying on $k$-non-degenerate sets of spheres, we prove an upper bound of $O(m^{236/149+\varepsilon}k^{125/149})$ for the problem (which improves the trivial bound for large values of $k$). Â Â Â (ii) We consider the maximum number of incidences between a three-dimensional set of $n$ planes (without any restrictions) and a set of $m$ points, such that no $k$ points are collinear. Our bound for $k$-non-degenerate planes immediately implies a bound of $O(n^{4/5+\varepsilon}m^{3/5}k^{2/5} + m + nk)$ for this problem, generalizing the previous bound $O(n^{4/5}m^{3/5} + n\log m)$ for the specific case where no three points are collinear (up to the $\varepsilon$ in the exponent).

Learning Read-Constant Polynomials of Constant Degree Modulo Composites

Boolean functions that have constant degree polynomial representation over a fixed finite ring form a natural and strict subclass of the complexity class ACC0. They are also precisely the functions computable efficiently by programs over fixed and finite nilpotent groups. This class is not known to be learnable in any reasonable learning model. In this paper, we provide a deterministic polynomial time algorithm for learning Boolean functions represented by polynomials of constant degree over arbitrary finite rings from membership queries, with the additional constraint that each variable in the target polynomial appears in a constant number of monomials. Our algorithm extends to superconstant but low degree polynomials and still runs in quasipolynomial time.

Author’s reply: A comment on the “Robust stability analysis of fractional order interval polynomials”, by Nusret Tan et al.

This author’s reply addresses the comment given in the note mentioned in the title. Theorem 3 given in Tan et al. (2009) [1] uses zero exclusion principle for the stability analysis of Fractional Order Interval Polynomial (FOIP). We show that the constant degree assumption is exist in the definition of zero exclusion principle. Although it has not been clearly stated in Tan et al. (2009) [1] that FOIP of Eq. (1) is a constant degree polynomial, this condition is implicit in the zero exclusion principle. Therefore, Theorem 3 is true under the constant degree assumption which is a requirement for zero exclusion principle.

Elusive Functions and Lower Bounds for Arithmetic Circuits

$\newcommand{\C}{\mathbb{C}}$ $\newcommand{\F}{\mathbb{F}}$ $\newcommand{\Image}{{\mathrm{Image}}}$ It is a basic fact of linear algebra that the image of the curve $f(x)=(x^1,x^2,x^3,\ldots,x^m)$, say over $\C$, is not contained in any $(m-1)$-dimensional affine subspace of $\C^m$. In other words, the image of $f$ is not contained in the image of any polynomial mapping $\Gamma:\C^{m-1} \rightarrow \C^m$ of degree 1 (that is, an affine mapping). Can one give an explicit example of a polynomial curve $f:\C \rightarrow \C^m$ such that the image of $f$ is not contained in the image of any polynomial mapping $\Gamma:\C^{m-1} \rightarrow \C^m$ of degree $2$ ? In this paper, we show that problems of this type are closely related to proving lower bounds for the size of general arithmetic circuits. For example, any explicit $f$ as above (with the right notion of explicitness), of degree up to $2^{m^{o(1)}}$, implies super-polynomial lower bounds for computing the permanent over$~\C$. More generally, we say that a polynomial mapping $f:\F^{n} \rightarrow \F^m$ is $(s,r)$-elusive, if for every polynomial mapping $\Gamma:\F^{s} \rightarrow \F^m$ of degree $r$, $\Image(f) \not \subset \Image(\Gamma)$. We show that for many settings of the parameters $n,m,s,r$, explicit constructions of elusive polynomial mappings imply strong (up to exponential) lower bounds for general arithmetic circuits. Finally, for every $r < \log n$, we give an explicit example of a polynomial mapping $f:\F^{n} \rightarrow \F^{n^2}$, of degree $O(r)$, that is $(s,r)$-elusive for $s = n^{1+\Omega(1/r)}$. We use this to construct for any $r$, an explicit example of an $n$-variate polynomial of total-degree $O(r)$, with coefficients in $\{0,1\}$, such that any depth-$r$ arithmetic circuit for this polynomial (over any field) is of size $\geq n^{1+\Omega(1/r)}$. In particular, for any constant $r$, this gives a constant degree polynomial such that any depth $r$ arithmetic circuit for this polynomial is of size $\geq n^{1+\Omega(1)}$. Previously, only lower bounds of the type $\Omega(n \cdot \lambda_r (n))$, where the $\lambda_r$ are extremely slowly growing functions, were known for constant-depth arithmetic circuits for polynomials of constant degree (actually, for linear functions).

Monotone separations for constant degree polynomials

We prove a separation between monotone and general arithmetic formulas for polynomials of constant degree. We give an example of a polynomial C in n variables and degree k which is computable by a homogeneous arithmetic formula of size O ( k 2 n 2 ) , but every monotone formula computing C requires size ( n / k c ) Ω ( log k ) , with c ∈ ( 0 , 1 ) . Since the upper bound is achieved by a homogeneous arithmetic formula, we also obtain a separation between monotone and homogeneous formulas, for polynomials of constant degree.

Depth-3 arithmetic circuits over fields of characteristic zero

In this paper we prove quadratic lower bounds for depth-3 arithmetic circuits over fields of characteristic zero. Such bounds are obtained for the elementary symmetric functions, the (trace of) iterated matrix multiplication, and the determinant. As corollaries we get the first nontrivial lower bounds for computing polynomials of constant degree, and a gap between the power of depth-3 arithmetic circuits and depth-4 arithmetic circuits. We also give new shorter formulae of constant depth for the elementary symmetric functions.¶The main technical contribution relates the complexity of computing a polynomial in this model to the wealth of partial derivatives it has on every affine subspace of small co-dimension. Lower bounds for related models utilize an algebraic analog of the Neciporuk lower bound on Boolean formulae.

Polynomial Time Approximation Schemes for Dense Instances of [formula omitted][formula omitted]-Hard Problems

We present a unified framework for designing polynomial time approximation schemes (PTASs) for “dense” instances of many NP-hard optimization problems, including maximum cut, graph bisection, graph separation, minimumk-way cut with and without specified terminals, and maximum 3-satisfiability. By dense graphs we mean graphs with minimum degreeΩ(n), although our algorithms solve most of these problems so long as the average degree isΩ(n). Denseness for nongraph problems is defined similarly. The unified framework begins with the idea ofexhaustive sampling:picking a small random set of vertices, guessing where they go on the optimum solution, and then using their placement to determine the placement of everything else. The approach then develops into a PTAS for approximating certainsmoothinteger programs, where the objective function and the constraints are “dense” polynomials of constant degree.

When bears are blue and bulls are red

Analysts deal with real numerical data. Often the data is collected over time. Because numbers are abstract, analysts often use graphs to make numerical data more meaningful. Real data typically fluctuates in some random fashion. However, patterns often exist in the data. An example would be the seasonal fluctuations in commodities. Analysts use a variety of techniques to help them find patterns that have the potential to improve their skill in predicting future trends. This paper suggests some ways to make some of these patterns more obvious.We are here presenting techniques to enhance traditional methods of providing information in graphs. By taking advantage of current computer technology to provide graphic images and to draw graphs, new and exciting visual images are possible. In addition, animation strategies allow us to simulate the passage of time. Finally, we add a whimsical touch.By animating the graphs of mathematical functions we illuminate aspects of the graphing process which might otherwise go unnoticed. The focus of this paper is on curve fitting. APL and J provide the matrix divide function for finding the coefficients of the constant, linear, quadratic, cubic or nth degree polynomial which best fit a collection of data.Each of these polynomials also has a polynomial derivative. These derivatives are useful for three reasons. Select a point on the x-axis. The y-value of the polynomial is the height of the polynomial for that x-value. But the y-value of the derived polynomial for the chosen x-value provides information about the original polynomial in three ways. First the y-value for the chosen x-value is the slope of the original polynomial at that point. The second useful aspect of the derivative is that positive and negative values indicate when the original polynomial is increasing or decreasing. A derivative is one degree less than the original polynomial so for example, a cubic polynomial has a quadratic polynomial derivative.Applications will demonstrate how the slope of the tangent together with the use of images and color can enhance the reader's ability to assimilate data. These enhancements give additional meaning to graphs and add life to the data.