Bound Of Degree Research Articles

Solving Degree Bounds for Iterated Polynomial Systems

For Arithmetization-Oriented ciphers and hash functions Gröbner basis attacks are generally considered as the most competitive attack vector. Unfortunately, the complexity of Gröbner basis algorithms is only understood for special cases, and it is needless to say that these cases do not apply to most cryptographic polynomial systems. Therefore, cryptographers have to resort to experiments, extrapolations and hypotheses to assess the security of their designs. One established measure to quantify the complexity of linear algebra-based Gröbner basis algorithms is the so-called solving degree. Caminata & Gorla revealed that under a certain genericity condition on a polynomial system the solving degree is always upper bounded by the Castelnuovo-Mumford regularity and henceforth by the Macaulay bound, which only takes the degrees and number of variables of the input polynomials into account. In this paper we extend their framework to iterated polynomial systems, the standard polynomial model for symmetric ciphers and hash functions. In particular, we prove solving degree bounds for various attacks on MiMC, Feistel-MiMC, Feistel-MiMC-Hash, Hades and GMiMC. Our bounds fall in line with the hypothesized complexity of Gröbner basis attacks on these designs, and to the best of our knowledge this is the first time that a mathematical proof for these complexities is provided. Moreover, by studying polynomials with degree falls we can prove lower bounds on the Castelnuovo-Mumford regularity for attacks on MiMC, Feistel-MiMC and Feistel-MiMCHash provided that only a few solutions of the corresponding iterated polynomial system originate from the base field. Hence, regularity-based solving degree estimations can never surpass a certain threshold, a desirable property for cryptographic polynomial systems.

Sharp Degree Bounds for Fake Weighted Projective Spaces

We give sharp upper bounds on the anticanonical degree of fake weighted projective spaces, only depending on the dimension and the Gorenstein index.

Irregularity of Graphs Respecting Degree Bounds

Albertson defined the irregularity of a graph $G$ as $$irr(G)=\sum\limits_{uv\in E(G)}|d_G(u)-d_G(v)|.$$ For a graph $G$ with $n$ vertices, $m$ edges, maximum degree $\Delta$, and $d=\left\lfloor \frac{\Delta m}{\Delta n-m}\right\rfloor$, we show $$irr(G)\leq d(d+1)n+\frac{1}{\Delta}\left(\Delta^2-(2d+1)\Delta-d^2-d\right)m.$$

Sharp Effective Finite-Field Nullstellensatz

The (weak) Nullstellensatz over finite fields says that if are n-variate degree-d polynomials with no common zero over a finite field then there are polynomials such that . Green and Tao [14, Proposition 9.1] used a regularity lemma to obtain an effective proof, showing that the degrees of the polynomials Ri can be bounded independently of n, though with an Ackermann-type dependence on the other parameters m, d, and . In this paper we use the polynomial method to give a proof with a degree bound of . We also show that the dependence on each of the parameters is the best possible up to an absolute constant. We further include a generalization, offered by Pete L. Clark, from finite fields to arbitrary subsets in arbitrary fields, provided the polynomials Pi take finitely many values on said subset.

Beam Shaping for Massive MIMO Systems With One-Bit Spatial $\Sigma -\Delta$ ADCs

One-bit quantization opens up opportunities for massive MIMO implementation using cheap and power-efficient radio-frequency front-ends, but unfortunately brings severe amplitude distortion. The spatial <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Sigma -\Delta$</tex-math></inline-formula> structure is one important manner that can recover the amplitude information from one-bit quantized signals. In this paper, we demonstrate that the upper bound of the spatial degree of quantization noise reduction (DQNR) is 3 for the considered spatial <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Sigma -\Delta$</tex-math></inline-formula> structure. However, the upper bound is hard to be achieved via the zero forcing (ZF) or the maximum ratio combination (MRC) schemes, and the achieved spatial DQNR is even not greater than 1 in the worst case. Thus the distortion suppression is inefficient. To address this issue, we propose a random beam shaping (RBS) scheme and prove it can achieve the spatial DQNR of 3. Through the random beam shaping, the beams are smeared into the whole beamspace. Meanwhile, the random beam shaping results in the power leakage of the beams and it may limit the distortion suppression when considering the additive noise. To reduce the power leakage, we further propose an accurate beam shaping (ABS) scheme, in which the vector for the beam shaping is optimized through the alternative minimization algorithm and the beams are concentrated around the region with low power of quantization noise. The simulation results illustrate that the proposed beam shaping schemes can more efficiently exploit the large-scale antenna array to recover the amplitude information.

Two variants of Hilbert’s Nullstellensatz

ABSTRACT We prove two variations of Hilbert’s Nullstellensatz in the univariate case: (1) If h and are polynomials in and there is a positive integer m and a positive constant C such that then there are n polynomials such that with the degree bound (2) If and h are polynomials in and there is a positive integer m and a positive constant C such that then there are n polynomials such that with the degree bound

Perceptions of Physics Teachers of Adopting Invention in Science Labs (ISL) Framwork in Ma'an Governorate

This study aimed at revealing the physics teachers’ perceptions in Ma’an of adopting Invention in Science Labs (ISL) Framework as an effective teaching strategy, and predicting the intention of implementing ISL from their perceptions. The study used the descriptive methodology. The sample consisted of (11) male teachers and (13) female teachers frpm the Directorate of Education of Ma’an. The tools of the study were: a questionnaire consisted of (32) items based on Unified Theory of Acceptance and Use of Technology (UTAUT) which mofyfied according to the study requirements, and a prosure describes the ISL. The findings indicated that there were statistical significance differences at (α≤0.05) between teachers’ perceptions and the crucial score of the previous upper bound of categorize degrees, which indicate that teachers’ perception were high in expected performance and social influence, and at average on: expected effort, available facilities and intention to use. Findings related to gender showed no statistical significance differences for each of teachers’ perceptions degrees at the five domains. Results of correlation showed the existence of a positive relations between all the domains. Results of regression showed that entention to use can be predicted by the teachers perceptions of the four domsins. According to the study results the recommendation were to conduct more training workshops about using ISL and conducting further studies in a large sample for the intention of using ISL.

Bounds for degrees of syzygies of polynomials defining a grade two ideal

We make explicit the exponential bound on the degrees of the polynomials appearing in the Effective Quillen-Suslin Theorem, and apply it jointly with the Hilbert-Burch Theorem to show that the syzygy module of a sequence of m polynomials in n variables defining a complete intersection ideal of grade two is free, and that a basis of it can be computed with bounded degrees. In the known cases, these bounds improve previous results.

Two completely independent spanning trees of claw-free graphs

If a graph G contains two spanning trees T1,T2 such that for each two distinct vertices x,y of G, the (x,y)-path in each Ti has no common edge and no common vertex except for the two ends, then T1,T2 are called two completely independent spanning trees (CISTs) of G,i∈{1,2}. If the subgraph induced by the neighbor set of a vertex u in a graph G is connected, then u is an eligible vertex of G. An hourglass is a graph with degree 4,2,2,2,2 and (P6)2 denotes the square graph of a path P6 on six vertices. Araki (2014) [1] proved that if a graph G satisfies the Dirac's condition, which is also sufficient condition for hamiltonian graphs, then G contains two CISTs. Ryjáček proposed that a claw-free graph G is hamiltonian if and only if its Ryjáček's closure, denoted by cl(G), is hamiltonian. In this paper, we prove that if G is a {claw, hourglass, (P6)2}-free graph with δ(G)≥4, then G contains two CISTs if and only if cl(G) has two CISTs. The bound of the minimum degree in our result is best possible.

Sparse polynomial interpolation based on derivatives

We propose two new interpolation algorithms for sparse multivariate polynomials represented by a straight-line program (SLP). Both of our algorithms work over any finite fields Fq with large characteristic.The first algorithm is randomized of the Monte Carlo type. Its bit complexity is linear in the number of non-zero terms T and the number of variables n of f. Let D be the partial degree bound of f. If q is in O((nTD)(1)), our algorithm has better complexity than other existing algorithms.The second algorithm is deterministic; it has better complexity than any existing deterministic algorithms in the field with large characteristic. Its bit complexity is quadratic in n,T,log⁡D.

Representation of real solvable Lie algebras having 2-dimensional derived ideal and geometry of coadjoint orbits of corresponding Lie groups

Let Lie ([Formula: see text]) be the class of all [Formula: see text]-dimensional real solvable Lie algebras having [Formula: see text]-dimensional derived ideals. In 2020, Le et al. gave a classification of all non-2-step nilpotent Lie algebras of Lie ([Formula: see text], 2). We propose in this paper to study representations of these Lie algebras as well as their corresponding connected and simply connected Lie groups. That is, for each algebra, we give an upper bound of the minimal degree of a faithful representation. Then, we give a geometrical description of coadjoint orbits of corresponding groups. Moreover, we show that the characteristic property of the family of maximal dimensional coadjoint orbits of an MD-group studied by Shum et al. is still true for the Lie groups considered here. Namely, we prove that, for each considered group, the family of the maximal dimensional coadjoint orbits forms a measurable foliation in the sense of Connes. The topological classification of these foliations is also provided.

Local Energy Bounds and Strong Locality in Chiral CFT

A family of quantum fields is said to be strongly local if it generates a local net of von Neumann algebras. There are few methods of showing directly strong locality of a quantum field. Among them, linear energy bounds are the most widely used, yet a chiral conformal field of conformal weight $d>2$ cannot admit linear energy bounds. In this paper we give a new direct method to prove strong locality in two-dimensional conformal field theory. We prove that if a chiral conformal field satisfies an energy bound of degree $d-1$, then it also satisfies a certain local version of the energy bound, and this in turn implies strong locality. A central role in our proof is played by diffeomorphism symmetry. As a concrete application, we show that the vertex operator algebra given by a unitary vacuum representation of the $\mathcal{W}_3$-algebra is strongly local. For central charge $c > 2$, this yields a new conformal net. We further prove that these nets do not satisfy strong additivity, and hence are not completely rational.

Examples of Surfaces with Canonical Map of Maximal Degree

It was shown by Beauville that if the canonical map $\varphi_{|K_M|}$ of a complex smooth projective surface $M$ is generically finite, then $\operatorname{deg}(\varphi_{|K_M|}) \leq 36$. The first example of a surface with canonical degree $36$ was found by the second author. In this article, we show that for any surface which is a degree four Galois étale cover of a fake projective plane $X$ with the largest possible automorphism group $\operatorname{Aut}(X) = C_7:C_3$ (the unique non-abelian group of order $21$), the base locus of the canonical map is finite, and we verify that $35$ of these surfaces have maximal canonical degree $36$. We also classify all smooth degree four Galois étale covers of fake projective planes, which give possible candidates for surfaces of canonical degree $36$. Finally, we also confirm in this paper the optimal upper bound of the canonical degree of smooth threefolds of general type with sufficiently large geometric genus, related to earlier work of Hacon and Cai.

Three effective preconditioners for double saddle point problem

In this paper, we mainly propose three preconditioners for solving double saddle point problems, which arise from some practical problems. Firstly, the solvability of this kind of problem is investigated under suitable assumption. Next, we prove that all the eigenvalues of the three preconditioned matrices are $ 1 $. Furthermore, we analyze the eigenvector distribution and the upper bound of the minimum polynomial degree of the corresponding preconditioned matrix. Finally, numerical experiments are carried to show the effectiveness of the proposed preconditioners.

Oracle-supported drawing of the Gröbner escalier

The aim of this note is to discuss the following quite queer problem: to compute the Grobner basis of an ideal I w.r.t. a term-ordering ≺ without knowing neither the ideal nor the term-ordering but only a degree bound of the required Grobner basis, being allowed to pose a finite number of queries to an oracle which, given a term τ ∈ T, returns its canonical form Can(τ, I, ≺) w.r.t. the unknown ideal I and term-ordering ≺. This problem was suggested to us by the desire to definitely dispose of a very weak paper wrongly claiming a cryptographic application of (non commutative) Grobner bases. The commutative reformulation is instead a non-obvious challenge and we consider it an helpful tool for understanding and visually describe the structure of the Grobner escalier of an ideal; moreover it allows to describe (and compute) the corner set, an helpful tool for computing Macaulay decomposition of a (non-necessarily 0-dimensional) algebra.

Fast algorithms at low temperatures via Markov chains†

Efficient algorithms for approximate counting and sampling in spin systems typically apply in the so‐called high‐temperature regime, where the interaction between neighboring spins is “weak.” Instead, recent work of Jenssen, Keevash, and Perkins yields polynomial‐time algorithms in the low‐temperature regime on bounded‐degree (bipartite) expander graphs using polymer models and the cluster expansion. In order to speed up these algorithms (so the exponent in the run time does not depend on the degree bound) we present a Markov chain for polymer models and show that it is rapidly mixing under exponential decay of polymer weights. This yields, for example, an ‐time sampling algorithm for the low‐temperature ferromagnetic Potts model on bounded‐degree expander graphs. Combining our results for the hard‐core and Potts models with Markov chain comparison tools, we obtain polynomial mixing time for Glauber dynamics restricted to appropriate portions of the state space.

On Structural Parameterizations of the Bounded-Degree Vertex Deletion Problem

We study the parameterized complexity of the Bounded-Degree Vertex Deletion problem (BDD), where the aim is to find a maximum induced subgraph whose maximum degree is below a given degree bound. Our focus lies on parameters that measure the structural properties of the input instance. We first show that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, pathwidth, treedepth, and even the size of a minimum vertex deletion set into graphs of pathwidth and treedepth at most three. We thereby resolve an open question stated in Betzler, Bredereck, Niedermeier and Uhlmann (2012) concerning the complexity of BDD parameterized by the feedback vertex set number. On the positive side, we obtain fixed-parameter algorithms for the problem with respect to the decompositional parameter treecut width and a novel problem-specific parameter called the core fracture number.

Projective normality of torus quotients of flag varieties

Let G=SLn(C) and T be a maximal torus in G. We show that the quotient T\\\\G/Pα1∩Pα2 is projectively normal with respect to the descent of a suitable line bundle, where Pαi is the maximal parabolic subgroup in G associated to the simple root αi, i=1,2. We give a degree bound of the generators of the homogeneous coordinate ring of T\\\\(G3,6)Tss(L2ϖ3). If G=Spin7, we give a degree bound of the generators of the homogeneous coordinate ring of T\\\\(G/Pα2)Tss(L2ϖ2) whereas we prove that the quotient T\\\\(G/Pα3)Tss(L4ϖ3) is projectively normal with respect to the descent of the line bundles L4ϖ3.

Distributed algorithm for AP association with random arrivals and departures of users

Here, the authors study the novel problem of optimising access point (AP) association by maximising the network throughput, subject to the degree bound of AP. The formulated problem is a combinatorial optimisation. They resort to the Markov Chain approximation technique to design a distributed algorithm. They first approximate their optimal objective via Log-Sum-Exp function. Thereafter, they construct a special class of Markov Chain with steady-state distribution specify to their problem to yield a distributed solution. Furthermore, they extend the static problem setting to a dynamic environment where the users can randomly leave or join the system. Their proposed algorithm has provable performance, achieving an approximation gap of ( 1 / η ) log ⁡ | F | . It is simple and can be implemented in a distributed manner. Their extensive simulation results show that the proposed algorithm can converge very fast, and achieve a close-to-optimal performance with a guaranteed loss bound.

Bounds on maximum concurrent flow in random bipartite graphs

Algorithms that exploit the duality between maximum concurrent flow and sparse cuts in a graph can effectively discover hierarchical community structures in social networks. We analyze the maximum concurrent flow on random bipartite graphs and compare the results to two bounds based on graph structure.. The minimum degree bound is based on a sparse cut that removes a node of minimum degree. The other is the “gridlock” bound, where all edges are saturated with flow. We show that the gridlock bound is more constraining when the minimum degree of the graph is sufficiently large, and provide a way to calculate this bound on graphs of diameter three, which occurs with high probability on larger random bipartite graphs.