Number Of Monomials Research Articles

Weighted Sum-of-Squares Lower Bounds for Univariate Polynomials Imply VP≠VNP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{VP} \ eq \ ext{VNP}$$\\end{document}

For a polynomial f,a weighted sum-of-squares representation (SOS) has the form f=∑i∈[s]cifi2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f = \\sum_{i\\in [s]} c_i f_i^2$$\\end{document}, where the weightsci\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c_i$$\\end{document} are field elements. The size of the representation is the number of monomials that appear across the fi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f_i$$\\end{document}'s.Its minimum across all such decompositions is called the support-sum S(f) of f.For a univariate polynomial f of degree d of full support,a lower bound for the support-sum is S(f)≥d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S(f) \\ge \\sqrt d$$\\end{document}.We show that the existence of an explicit univariate polynomial fwith support-sum just slightly larger than the lower bound, that is, S(f)≥d0.5+ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S(f) \\ge d^{0.5+\\varepsilon}$$\\end{document}, for some ε>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon > 0$$\\end{document},implies that VP≠\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ e$$\\end{document}VNP,the major open problem in algebraic complexity.In fact, our proof works for some subconstant functions ε(d)>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon(d) > 0$$\\end{document} as well.We also consider the sum-of-cubes representation (SOC) of polynomials. We show that an explicit hard polynomialimplies both blackbox-PIT is in P, and VP≠\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ eq$$\\end{document}VNP.

Multilinear sets with two monomials and cardinality constraints

Binary polynomial optimization is equivalent to the problem of minimizing a linear function over the intersection of the multilinear set with a polyhedron. Many families of valid inequalities for the multilinear set are available in the literature, though giving a polyhedral characterization of the convex hull is not tractable in general as binary polynomial optimization is NP-hard. In this paper, we study the cardinality constrained multilinear set in the special case when the number of monomials is exactly two. We give an extended formulation, with two more auxiliary variables and exponentially many inequalities, of the convex hull of solutions of the standard linearization of this problem. We also show that the separation problem can be solved efficiently.

Combined weight and density bounds on the polynomial threshold function representation of Boolean functions

In an earlier report it was shown that an arbitrary n-variable Boolean function f can be represented as a polynomial threshold function (PTF) with 0.75×2n or less number of monomials. In this report, we derive an upper bound on the absolute value of the (integer) weights of a PTF that represents f and still obeys the aforementioned density bound. To our knowledge this provides the best combined bound on the PTF density (number of monomials) and PTF weight (sum of the coefficient magnitudes) of general Boolean functions. For the special case of bent functions, it is found that any n-variable bent function can be represented with integer coefficients less than or equal to 2n with density no more than 0.75×2n, and for the case of m-sparse Boolean functions that are almost constant except for small (m≪2n) number of variable assignments, it is shown that they can be represented with small weight PTFs with density at most m+2n−1. In addition, tight PTF weight bounds with conformance to the density bound of 0.75×2n are numerically obtained for the general Boolean functions up to 6 variables.

RevSCA-2.0: SCA-Based Formal Verification of Nontrivial Multipliers Using Reverse Engineering and Local Vanishing Removal

The formal verification of integer multipliers is one of the important but challenging problems in the verification community. Recently, the methods based on symbolic computer algebra (SCA) have shown very good results in comparison to all other existing proof techniques. However, when it comes to verification of huge and structurally complex multipliers, they completely fail as an explosion happens in the number of monomials. The reason for this explosion is the generation of redundant monomials known as vanishing monomials. This article introduces the SCA-based approach RevSCA-2.0 that combines reverse engineering and local vanishing removal to verify large and nontrivial multipliers. For our approach, we first come up with a theory for the origin of vanishing monomials, i.e., we prove that the gates/nodes where both outputs of half adders (HAs) converge are the origins of vanishing monomials. Then, we propose a dedicated reverse engineering technique to identify atomic blocks including HAs. The identified HAs are the basis for detecting converging cones and locally removing vanishing monomials, which finally results in a vanishing-free global backward rewriting. The efficiency of RevSCA-2.0 is demonstrated using an extensive set of multipliers with up to several million gates.

COMBINING AND FILTERING FUNCTIONS IN THE FRAMEWORK OF NONLINEAR-FEEDBACK SHIFT REGISTER

Strong cryptography of stream ciphers is determined according to the ability of the generated pseudorandom sequence to resist analytical attacks. One of the main components of the pseudorandom stream cipher sequence generating algorithm is Boolean functions for combining and filtering. The paper considers the possibility of applying nonlinear-feedback shift registers that generate a maximum length sequence as a combining or filtering function. The main indicators of cryptographic strength of such functions as: balance, the prohibitions presence, correlation immunity and nonlinearity are examined in this work. The study analyzes and demonstrates correlation immunity and nonlinearity experimental values for all nonlinear feedback shift registers that generate a maximum length sequence, for register sizes up to 6 cells inclusively, and register sizes up to 9 cells inclusively with algebraic degree of the polynomial under 2. The possibility of optimizing the process of selecting Boolean functions according to the criteria of maximum correlation immunity and nonlinearity with various algebraic degrees and minimization of the number of monomials in the polynomial is studied.

The number of unimodular roots of some reciprocal polynomials

We introduce a sequence $P_{2n}$ of monic reciprocal polynomials with integer coefficients having the central coefficients fixed. We prove that the ratio between number of nonunimodular roots of $P_{2n}$ and its degree $d$ has a limit when $d$ tends to infinity. We present an algorithm for calculation the limit and a numerical method for its approximation. If $P_{2n}$ is the sum of a fixed number of monomials we determine the central coefficients such that the ratio has the minimal limit. We generalise the limit of the ratio for multivariate polynomials. Some examples suggest a theorem for polynomials in two variables which is analogous to Boyd's limit formula for Mahler measure.

OPPORTUNITIES TO MINIMIZE HARDWARE AND SOFTWARE COSTS FOR IMPLEMENTING BOOLEAN FUNCTIONS IN STREAM CIPHERS

Currently, nonlinear Boolean functions are actively investigated worldwide. However, many questions remain unanswered. The theory of nonlinear Boolean functions that are suitable for use in cryptographically strong algorithms is significantly incomplete. Despite the existence of numerous publications on these themes, many issues related to the interconnection of design characteristics affecting the generator’s performance and its cryptographic characteristics still remain unsolved. The possibility of generating a special type of sequence, called de Bruijn sequence, at minimal hardware and software costs to implement nonlinear Boolean functions in stream encryption systems, is the main subject of this work. The paper presents the possible structure boundaries (algebraic degree of a Boolean function, the number of monomials in a function) of iterative de Bruijn sequence bitrate generators for various generated sequence characteristics, such as linear complexity and autocorrelation function. The profile of the linear complexity of the studied sequences is close to the expected value of the linear complexity, as well as for a truly random sequence.

Lower bounds for Mahler measure that depend on the number of monomials

We prove a new lower bound for the Mahler measure of a polynomial in one and in several variables that depends on the complex coefficients and the number of monomials. In one variable, our result generalizes a classical inequality of Mahler. In [Formula: see text] variables, our result depends on [Formula: see text] as an ordered group, and in general, our lower bound depends on the choice of ordering.

Root separation for trinomials

We give a separation bound for the complex roots of a trinomial f∈Z[X]. The logarithm of the inverse of our separation bound is polynomial in the size of the sparse encoding of f; in particular, it is polynomial in log⁡(deg⁡f). It is known that no such bound is possible for 4-nomials (polynomials with 4 monomials). For trinomials, the classical results (which are based on the degree of f rather than the number of monomials) give separation bounds that are exponentially worse.As an algorithmic application, we show that the number of real roots of a trinomial f can be computed in time polynomial in the size of the sparse encoding of f. The same problem is open for 4-nomials.

Using jointly geometry and algebra to determine RC-constructibility

In most cases in geometry, applying analytic or algebraic tools on coordinates helps to solve some difficult problems. For instance, proving that a geometrical construction problem is solvable using ruler and compass is often impossible within a synthetic geometry framework. But in an analytic geometry framework, it is a direct application of Galois theory after performing triangularizations. However, these algebraic tools lead to a large amount of computation. Their implementation in modern Computer Algebra Systems (CAS) are still too time consuming to provide an answer in a reasonable time. In addition, they require a lot of memory space which can grow exponentially with the size of the problem. Fortunately, some geometrical properties can be used to setup the algebraic systems so that they can be more efficiently computed. These properties turn polynomials into new ones so as to reduce both the degrees and the number of monomials. The present paper promotes this approach by considering two corpora of geometric construction problems, namely Wernick's and Connely's lists. These lists contain about 280 problems. The purpose is to determine their status i.e. whether they are constructible or not with ruler and compass. Some of these problems had unknown status that will be settled in this paper. More generally, the status of all problems of these corpora are fully automatically given by an approach combining geometry and algebra.

Mahler measures of polynomials that are sums of a bounded number of monomials

We study Laurent polynomials in any number of variables that are sums of at most [Formula: see text] monomials. We first show that the Mahler measure of such a polynomial is at least [Formula: see text], where [Formula: see text] is the height of the polynomial. Next, restricting to such polynomials having integer coefficients, we show that the set of logarithmic Mahler measures of the elements of this restricted set is a closed subset of the nonnegative real line, with [Formula: see text] being an isolated point of the set. In the final section, we discuss the extent to which such an integer polynomial of Mahler measure [Formula: see text] is determined by its [Formula: see text] coefficients.

Polynomial threshold functions and Boolean threshold circuits

We initiate a comprehensive study of the complexity of computing Boolean functions by polynomial threshold functions (PTFs) on general Boolean domains (as opposed to domains such as {0,1}n or {−1,1}n that enforce multilinearity). A typical example of such a general Boolean domain, for the purpose of our results, is {1,2}n. We are mainly interested in the length (the number of monomials) of PTFs, with their degree and weight being of secondary interest.First we motivate the study of PTFs over the {1,2}n domain by showing their close relation to depth two threshold circuits. In particular we show that PTFs of polynomial length and polynomial degree compute exactly the functions computed by polynomial size THR∘MAJ circuits. We note that known lower bounds for THR∘MAJ circuits extend to the likely strictly stronger model of PTFs (with no degree restriction). We also show that a “max-plus” version of PTFs is related to AC0∘THR circuits.We exploit this connection to gain a better understanding of threshold circuits. In particular, we show that (super-logarithmic) lower bounds for 3-player randomized communication protocols with unbounded error would yield (super-polynomial) size lower bounds for THR∘THR circuits.Finally, having thus motivated the model, we initiate structural studies of PTFs. These include relationships between weight and degree of PTFs, and a degree lower bound for PTFs of constant length.

Computing Walsh coefficients from the algebraic normal form of a Boolean function

We study the relationship between the Walsh Transform of a Boolean function and its Algebraic Normal Form(ANF), and present algorithms that compute the Walsh coefficients at a small set of points in terms of certain parameters derived from the ANF of a Boolean function. In the first part of this paper, based on the previous result by Gupta and Sarkar, we investigate the formula in Gupta-Sarkar's algorithm in a novel iterative method and obtain a recurrence relation for the Walsh Transform of a Boolean function. The second part is devoted to applying this formula to algorithms to evaluate it. Experimental result shows that for the specified classes of Boolean functions, our algorithms can perform better than Gupta-Sarkar's algorithm. For example, the proposed algorithm "ComputeWalsh" is able to compute the Walsh coefficients of the functions for which the complexity of Gupta-Sarkar's algorithm is impractical. Besides, for functions acting on high number of variables (m>30) and having low number of monomials, the proposed algorithms are advantageous over the Fast Walsh Transform which is a standard method of computing the Walsh Transform with a complexity of O(m2 m ) operations.

Parallel Gaussian elimination for XL family over GF(2)

ABSTRACTThe security of many cryptographic systems is based on the difficulty of solving systems of quadratic multivariate polynomial equations. XL family (which stands for eXtended Linearization) is a traditional type of algorithm for solving systems of multivariate polynomial equations over finite fields. The main idea of these algorithms is to extend the initial system by monomials multiplications and use linear algebra techniques to solve the system. The overall complexity of XL family is essentially determined by the workload of Gaussian elimination step. In this paper, we use the structured Gaussian elimination and parallel fast Gaussian elimination to reduce the complexity of XL family over GF(2). Because of the sparsity of the extended system, structured Gaussian elimination is applied to reduce the dimensions of the extended system. Thus, a great reduction of the storage can be obtained. By slightly modifying the logic of ordinarily Gaussian elimination, we parallelly compute the reduced system. Theory analysis indicates that the complexity of our improved XL, XL_SP, is (λT)2, which is far lower than (7/64)T2.8, the complexity of XL, where T is the number of monomials and λ is the ratio of the dimensions of the reduced system to the extended system. Further experiments to Hidden Field Equation cryptosystems show that about 90% reduction of the extended system is achieved by using structured Gaussian elimination, which means λ is about 1/10 in our experiments. Copyright © 2013 John Wiley & Sons, Ltd.

A new Lenstra-type algorithm for quasiconvex polynomial integer minimization with complexity [formula omitted

We study the integer minimization of a quasiconvex polynomial with quasiconvex polynomial constraints. We propose a new algorithm that is an improvement upon the best known algorithm due to Heinz [S. Heinz, Complexity of integer quasiconvex polynomial optimization, Journal of Complexity 21(4) (2005) 543–556]. This improvement is achieved by applying a new modern Lenstra-type algorithm, finding optimal ellipsoid roundings, and considering sparse encodings of polynomials. For the bounded case, our algorithm attains a time-complexity of s(rlMd)O(1)22nlog2(n)+O(n) when M is a bound on the number of monomials in each polynomial and r is the binary encoding length of a bound on the feasible region. In the general case, slO(1)dO(n)22nlog2(n)+O(n). In each we assume d≥2 is a bound on the total degree of the polynomials and l bounds the maximum binary encoding size of the input.

Algebraic attacks on a class of stream ciphers with unknown output function

Attacks on linear feedback shift register (LFSR) based cryptosystems typically assume that all the system details except the initial state of the LFSR are known. We address the problem assuming that the nonlinear output function is also unknown and frame the problem as one of a multivariate interpolation. The solution to this problem yields a system that produces an output identical to that of the original system with some other initial state. The attack needs to observe M bits of data and has complexity O(M ?) where $${M = \sum_{i=0}^{d} C(n, i)}$$ is the number of monomials of degree at most d in n variables, n being the state size and d the degree of the output function. When the output function has annihilators of degree e < d then with O(D) bits of data one can reconstruct parts of the keystream where $${D = \sum_{i=0}^{e} C(n, i)}$$ .

Gcd of multivariate polynomials via Newton polytopes

We study geometric criteria to determine coprimality between multivariate polynomials. Our main contribution is the development of a polynomial-time algorithm (on the number of monomials) that detects coprimality of multivariate polynomials using Newton polytopes. We also show how to construct the gcd of two bivariate polynomials using their Newton polygons.

A small decrease in the degree of a polynomial with a given sign function can exponentially increase its weight and length

A Boolean function f: {−1, +1}n → {−1, +1} is called the sign function of an integer-valued polynomial p(x) if f(x) = sgn(p(x)) for all x ∈ {−1, +1}n. In this case, the polynomial p(x) is called a perceptron for the Boolean function f. The weight of a perceptron is the sum of absolute values of the coefficients of p. We prove that, for a given function, a small change in the degree of a perceptron can strongly affect the value of the required weight. More precisely, for each d = 1, 2, ..., n − 1, we explicitly construct a function f: {−1, +1}n → {−1, +1} that requires a weight of the form exp{Θ(n)} when it is represented by a degree d perceptron, and that can be represented by a degree d + 1 perceptron with weight equal to only O(n2). The lower bound exp{Θ(n)} for the degree d also holds for the size of the depth 2 Boolean circuit with a majority function at the top and arbitrary gates of input degree d at the bottom. This gap in the weight values is exponentially larger than those that have been previously found. A similar result is proved for the perceptron length, i.e., for the number of monomials contained in it.

Betti number bounds for fewnomial hypersurfaces via stratified Morse theory

We use stratified Morse theory for a manifold with corners to give a new bound for the sum of the Betti numbers of a fewnomial hypersurface in R N. In the book Fewnomials (6), A. Khovanskii gives bounds on the Betti numbers of varieties X in R n or in the positive orthant R n . These varieties include algebraic varieties, varieties defined by polynomial functions in the variables and exponentials of the variables, and also by even more general functions. Here, we consider only algebraic varieties. For these, Khovanskii bounds the sum b∗(X) of Betti numbers of X by a function that depends on n, the codimension of X, and the total number of monomials appearing (with non zero coefficients) in polynomial equations definingX. For real algebraic varieties X, the problem of bounding b∗(X) has a long history. O. A. Oleinik (8) and J. Milnor (7) used Morse theory to estimate the number of critical points of a suitable Morse function to obtain a bound. R. Thom (12) used Smith Theory to bound the mod-2 Betti numbers. This Smith-Thom bound has the form b∗(X) ≤ b∗(XC), where XC is the complex variety defined by the same polynomials as X. For instance, if X ⊂ R n is defined by polynomials of degree at most d, then Milnor's bound is b∗(X) ≤ d(2d − 1) n . If X ⊂ (R {0}) n is a smooth hypersurface defined by a polynomial with Newton polytope P , then the Smith-Thom bound is b∗(X) ≤ b∗(XC) = n! ¢ Vol(P ), where Vol(P ) is the usual volume of P. We refer to (10) for an informative history of the subject, and to the book (1) for more details. These bounds are given in term of degree or volume of a Newton polytope which are numerical deformation invariants of the complex variety XC. In contrast, the topology of a real algebraic variety depends on the coefficients of its defini ng equations, and in partic- ular, on the number of monomials involved in these equations. For instance, Descartes's rule of signs implies that the number of positive roots of a real univariate polynomial is less than its number of monomials, but the number of complex roots is equal to its degree. Khovanskii's bound can be seen as a generalization of this Descartes bound. It is smaller than the previous bounds when the defining equations have few monomials compared to their degrees. While it has always been clear that Khovanskii's bounds are unrealistically large, it appears very challenging to sharpen them. Some progress has been made recently for the number of non degenerate positive solutions to a system of n polynomials in n variables (3). Here, we bound b∗(X), when X is a fewnomial hypersurface. Suppose that X ⊂ R n is a smooth hypersurface defined by a Laurent polynomial with

Approximability of Minimum AND-Circuits

Given a set of monomials, the Minimum AND-Circuit problem asks for a circuit that computes these monomials using AND-gates of fan-in two and being of minimum size. We prove that the problem is not polynomial-time approximable within a factor of less than 1.0051 unless $\mathsf{P}=\mathsf{NP}$, even if the monomials are restricted to be of degree at most three. For the latter case, we devise several efficient approximation algorithms, yielding an approximation ratio of 1.278. For the general problem, we achieve an approximation ratio of d−3/2, where d is the degree of the largest monomial. In addition, we prove that the problem is fixed parameter tractable with the number of monomials as parameter. Finally, we discuss generalizations of the Minimum AND-Circuit problem and relations to addition chains and grammar-based compression.