Narrow Results

For a lattice $L$, we associate a graph called the annihilator intersection graph of $L$, denoted by $\mathrm{\text{AIG}}(L).$ The vertex set of $\mathrm{\text{AIG}}(L)$ is the set of all nonzero zero-divisors of $L$ and any two distinct vertices $u$ and $v$ are adjacent in $\mathrm{\text{AIG}}(L)$ if and only if $\mathrm{\text{Ann}}(u)\cap \mathrm{\text{Ann}}(v)\ne \{0\}$. It has shown that the $\mathrm{\text{AIG}}(L)$ is disconnected if and only if the number of atoms in $L$ is two. If $\mathrm{\text{AIG}}(L)$ is connected, then we determine the diameter and the girth of $\mathrm{\text{AIG}}(L).$ We characterize all lattices whose annihilator intersection graph is planar. Further, we obtain the clique number and chromatic number of $\mathrm{\text{AIG}}(L)$ when $L$ is a finite Boolean lattice. We show that the domination number of $\mathrm{\text{AIG}}(L)$ is not exceeding two. Finally, we obtain a condition under which the annihilator intersection graph is identical with the zero-divisor graph and the annihilator ideal graph of lattices.

As the progress of quantum computers, it is desired to propose many more efficient cryptographic constructions with post-quantum security. In the literatures, almost all cryptographic schemes and protocols can be explained and constructed modularly from certain cryptographic primitives, among which an Identity-Based Hash Proof System (IB-HPS) is one of the most basic and important primitives. Therefore, we can utilize IB-HPSs with post-quantum security to present several types of post-quantum secure schemes and protocols. Up until now, all known IB-HPSs with post-quantum security are instantiated based on latticed-based assumptions. However, all these lattice-based IB-HPSs are either in the random oracle model or not efficient enough in the standard model. Hence, it should be of great significance to construct more efficient IB-HPSs from lattices in the standard model.

In this paper, we propose a new smooth IB-HPS with anonymity based on the Learning with Errors (LWE) assumption in the standard model. This new construction is mainly inspired by a classical identity-based encryption scheme based on LWE due to Agreawal et al. in Eurocrypt 2010. And our innovation is to employ the algorithm SampleGaussian introduced by Gentry et al. and the property of random lattice to simulate the identity secret key with respect to the challenge identity. Compared with other existing IB-HPSs in the standard model, our master public key is quite compact. As a result, our construction has much lower overheads on computation and storage.

Multikey fully homomorphic encryption (MFHE) allows homomorphic operations between ciphertexts encrypted under different keys. In applications for secure multiparty computation (MPC) protocols, MFHE can be more advantageous than usual fully homomorphic encryption (FHE) since users do not need to agree with a common public key before the computation when using MFHE. In EUROCRYPT 2016, Mukherjee and Wichs constructed a secure MPC protocol in only two rounds via MFHE which deals with a common random/reference string (CRS) in key generation. After then, Brakerski et al. replaced the role of CRS with the distributed setup for CRS calculation to form a four round secure MPC protocol. Thus, recent improvements in round complexity of MPC protocols have been made using MFHE.

In this paper, we go further to obtain round-efficient and secure MPC protocols. The underlying MFHE schemes in previous works still involve the common value, CRS, it seems to weaken the power of using MFHE to allow users to independently generate their own keys. Therefore, we resolve the issue by constructing an MFHE scheme without CRS based on LWE assumption, and then we obtain a secure MPC protocol against semi-malicious security in three rounds. We also define a new security notion “multikey-CPA security” to prove that a multikey ciphertext cannot be decrypted unless all the secret keys are gathered and our scheme is multikey-CPA secure.

We study the behavior of solutions of the Helmholtz equation $(-{\mathrm{\Delta }}_{\mathrm{\text{disc}},h}-E)u_h=f_h$ on a periodic lattice as the mesh size $h$ tends to 0. Projecting to the eigenspace of a characteristic root ${\lambda }_h(\xi )$ and using a gauge transformation associated with the Dirac point, we show that the gauge transformed solution $u_h$ converges to that for the equation $(P(D_x)-E)v=g$ for a continuous model on $R^d$, where ${\lambda }_h(\xi )\to P(\xi )$. For the case of the hexagonal and related lattices, in a suitable energy region, it converges to that for the Dirac equation. For the case of the square lattice, triangular lattice, hexagonal lattice (in another energy region) and subdivision of a square lattice, one can add a scalar potential, and the solution of the lattice Schrödinger equation $(-{\mathrm{\Delta }}_{\mathrm{\text{disc}},h}+V_{\mathrm{\text{disc}},h}-E)u_h=f_h$ converges to that of the continuum Schrödinger equation $(P(D_x)+V(x)-E)u=f$.

The automorphism group of a K3 surface with Picard number two is either the infinite cyclic group or the infinite dihedral group, if it is infinite. In this paper, we determine some conditions for a K3 surface of Picard number two to have the infinite dihedral automorphism group.

The mathematical formalisms and computer enumerations of two polymeric models: trails and silhouettes are presented. Trails have been quite extensively studied, but their equivalence class, silhouettes are less familiar. By drawing parallelism from n→0 magnetic analog, the tricritical properties of silhouettes are studied both analytically and numerically. The tricriticality of trails, which is inaccessible by a renormalization group analysis in ε=4−D dimensions, can therefore only be investigated numerically. The tricritical exponents thus obtained indicate that trails belong to a distinct universality class from silhouettes.

A class of numerical algorithms for solving the classical equations of motion in lattice gauge field theories which exactly fulfill the constraints imposed by the unitarity of the local group elements, by the local color charge conservation (Gauss law) and by the total energy conservation is constructed. The performance of these constrained algorithms is comparatively discussed.

Site and bond percolation thresholds are calculated for the face centered cubic, body centered cubic and diamond lattices in four, five and six dimensions. The results are used to study the behavior of percolation thresholds as a functions of dimension. It is shown that the predictions from a recently proposed invariant for percolation thresholds are not satisfactory for these lattices.

The universal law for percolation thresholds proposed by Galam and Mauger (GM) is found to apply also to dynamical situations. This law depends solely on two variables, the space dimension d and a coordinance number q. For regular lattices, q reduces to the usual coordination number while for anisotropic lattices it is an effective coordination number. For dynamical percolation we conjecture that the law is still valid if we use the number q₂ of second nearest neighbors instead of q. This conjecture is checked for the dynamic epidemic model which considers the percolation phenomenon in a mobile disordered system. The agreement is good.

A calculation of site-bond percolation thresholds in many lattices in two to five dimensions is presented. The line of threshold values has been parametrized in the literature, but we show here that there are strong deviations from the known approximate equations. We propose an alternative parametrization that lies much closer to the numerical values.

The continuum World Line Formalism permits a transparent discussion of bosonic and fermionic determinants in some background field. For general, nontrivial backgrounds numerical evaluations must be envisaged. In this work we implement this formalism on the lattice by using statistically generated random walk world line loops. We illustrate the method by applying it to special cases and discuss the results in comparison with known analytic solutions in continuum.

Using the recently proposed generalization to an arbitrary number of colors of the strong coupling approach to lattice gauge theories,¹ we compute the chiral condensate of massless QCD in the 't Hooft limit.

The six quark state(uuddss) called H dibaryon(J^P = 0⁺,S = -2) has been calculated to study its existence and stability. The simulations are performed in quenched QCD on 8³ × 24 and 16³ × 48 anisotropic lattices with Symanzik improved gauge action and Clover fermion action. The gauge coupling is β = 2.0 and aspect ratio ξ = a_s/a_t = 3.0. Preliminary results indicate that mass of H dibaryon is 2134(100)Mev on 8³ × 24 lattice and 2167(59)Mev on 16³ × 48 respectively. It seems that the radius of H dibaryon is very large and the finite size effect is very obvious.

Though being weakly interacting, QED can support bound states. In principle, this can be expected for the weak interactions in the Higgs sector as well. In fact, it has been argued long ago that there should be a duality between bound states and the elementary particles in this sector, at least in leading order in an expansion in the Higgs quantum fluctuations around its expectation value. Whether this remains true beyond the leading order is being investigated using lattice simulations, and support is found. This provides a natural interpretation of peaks in cross-sections as bound states. This would imply that (possibly very broad) resonances of Higgs and W and Z bound states could exist within the Standard Model.

I review lattice QCD calculations of the strong coupling and quark masses.

Quantized Yang–Mills fields lie at the heart of our understanding of the strong nuclear force. To understand the theory at low energies, we must work in the strong coupling regime. The primary technique for this is the lattice. While basically an ultraviolet regulator, the lattice avoids the use of a perturbative expansion. I discuss the historical circumstances that drove us to this approach, which has had immense success, convincingly demonstrating quark confinement and obtaining crucial properties of the strong interactions from first principles.

We review the determination of ${\alpha }_{\mathrm{\text{s}}}$ that follows from comparing at short distances the QCD static energy at three loops and resummation of the next-to-next-to leading logarithms with its determination in 2+1-flavor lattice QCD provided by the HotQCD collaboration. The result that we obtain is ${\alpha }_{\mathrm{\text{s}}}$(1.5 GeV) = 0.336${}_{-0.008}^{+0.012}$, corresponding to ${\alpha }_{\mathrm{\text{s}}}$(M_Z) = 0.1166${}_{-0.0008}^{+0.0012}$. We outline future possible developments.

The Super Tau-Charm Facility (STCF) is a proposed dual-ring electron–positron collider in China, designed to operate at a center-of-mass energy (CME) range of 2–7GeV with symmetric beam energies. The collider aims for a luminosity exceeding $5\times 10^{34}{\mathrm{\text{cm}}}^{-2}{\mathrm{\text{s}}}^{-1}$ at the beam energy of 2.0GeV. This work presents the preliminary lattice design for the STCF collider ring, specifically optimized for the 2.0GeV beam energy. To improve the nonlinear dynamics performance, the optimization of lattice is carried out by adjusting the strengths of sextupole and octupole magnets following the tuning of phase advances.

Using a one-loop renormalization group improvement for the effective potential in the Higgs model of electrodynamics with electrically and magnetically charged scalar fields, we argue for the existence of a triple (critical) point in the phase diagram (), where λ_run is the renormalized running self-interaction constant of the Higgs scalar monopoles and g_run is their running magnetic charge. This triple point is a boundary point of three first-order phase transitions in the dual sector of the Higgs scalar electrodynamics: The "Coulomb" and two confinement phases meet together at this critical point. Considering the arguments for the one-loop approximation validity in the region of parameters around the triple point A we have obtained the following triple point values of the running couplings: , which are independent of the electric charge influence and two-loop corrections to with high accuracy of deviations. At the triple point the mass of monopoles is equal to zero. The corresponding critical value of the electric fine structure constant turns out to be by the Dirac relation. This value is close to the , which in a U(1) lattice gauge theory corresponds to the phase transition between the "Coulomb" and confinement phases. In our theory for α ≥ α_crit there are two phases for the confinement of the electrically charged particles. The results of the present paper are very encouraging for the antigrand unification theory which was developed previously as a realistic alternative to SUSY GUT's. The paper is also devoted to the discussion of this problem.

Using a two-loop approximation for β functions, we have considered the corresponding renormalization group improved effective potential in the dual Abelian Higgs model (DAHM) of scalar monopoles and calculated the phase transition (critical) couplings in U(1) and SU(N) regularized gauge theories. In contrast to our previous result α_crit≈0.17, obtained in the one-loop approximation with the DAHM effective potential (see Ref. 20), the critical value of the electric fine structure constant in the two-loop approximation, calculated in the present paper, is equal to α_crit≈0.208 and coincides with the lattice result for compact QED¹⁰: . Following the 't Hooft's idea of the "Abelization" of monopole vacuum in the Yang–Mills theories, we have obtained an estimation of the SU(N) triple point coupling constants, which is . This relation was used for the description of the Planck scale values of the inverse running constants (i= 1, 2, 3 correspond to U(1), SU(2) and SU(3) groups), according to the ideas of the multiple point model.¹⁶