In this paper we present a new variant of the Zhuang-Zi algorithm, which solves multivariate polynomial equations over a finite field by converting it into a single variable problem over a large extension field. The improvement is based on the newly developed concept of mutant in solving multivariate equations.

Let <i>X</i> be a locally compact Hausdorff space and <i>C</i> ₀(<i>X</i>) the Banach space of continuous functions on <i>X</i> vanishing at infinity. In this paper, we shall study unbounded disjointness preserving linear functionals on <i>C</i> ₀(<i>X</i>). They arise from prime ideals of <i>C</i> ₀(<i>X</i>), and we translate it into the cozero set ideal setting. In particular, every unbounded disjointness preserving linear functional of <i>c</i> ₀ can be constructed explicitly through an ultrafilter on complementary to a cozero set ideal. This ultrafilter method can be extended to produce many, but in general not all, such functionals on <i>C</i> ₀(<i>X</i>) for arbitrary <i>X</i>. We also make some remarks where <i>C</i> ₀(<i>X</i>) is replaced by a non-commutative C*-algebra.

Recent developments in multivariate polynomial solving algorithms have made algebraic cryptanalysis a plausible threat to many cryptosystems. However, theoretical complexity estimates have shown this kind of attack unfeasible for most realistic applications. In this paper we present a strategy for computing Gröbner basis that challenges those complexity estimates. It uses a flexible partial enlargement technique together with reduced row echelon forms to generate lower degree elements–mutants. This new strategy surpasses old boundaries and obligates us to think of new paradigms for estimating complexity of Gröbner basis computation. The new proposed algorithm computed a Gröbner basis of a degree 2 random system with 32 variables and 32 equations using 30 GB which was never done before by any known Gröbner bases solver.

We present the Zhuang-Zi algorithm, a new method for solving multivariate polynomial equations over a finite field. We describe the algorithm and present examples, some of which cannot be solved with the fastest known algorithms.

The notion of generating topologies, introduced by Stephani El0], is useful for studying the injective hull of an operator ideal. Using Randtke's idea (see [8, p. 90] or [12,(3.2. 1) and (3.2. 7)]), we can characterize generating topologies in terms of seminorms which satisfy some expected properties (see Lemma 3.3 and Theorems 3.4 and 3.9). By a well-known and useful idea of Grothendieck, the dual notions of generating topologies and ideal-topologies, the so-called generating bornologies, are given and studied in Sect. 4. In term of ideal-bornologies, the surjective hull of an operator ideal on Banach spaces is given (see Lemma 4.8 and Theorem 4.10). In terms of these two dual concepts, we are able to classify locally convex spaces, and to study their dual results. For instance, we show that if 9.1 is a symmetric (resp. completely symmetric) operator ideal on Banach spaces then a Banach space E is an 9~-topological

Abelian Chern-Simons theory, characterized by the so-called K matrix, has been quite successful in characterizing and classifying Abelian fractional quantum Hall effect as well as symmetry protected topological (SPT) phases, especially for bosonic SPT phases. However, there are still some puzzles in dealing with fermionic SPT (fSPT) phases. In this paper, we utilize the Abelian Chern-Simons theory to study the fSPT phases protected by arbitrary Abelian total symmetry G_f. Comparing to the bosonic SPT phases, fSPT phases with Abelian total symmetry Gf have three new features: (1) they may support gapless Majorana fermion edge modes, (2) some nontrivial bosonic SPT phases may be trivialized if G_f is a nontrivial extension of bosonic symmetry G_b by \mathbb{Z}_f^2, (3) certain intrinsic fSPT phases can only be realized in interacting fermionic system. We obtain edge theories for various fSPT phases, which can also be regarded as conformal field theories with proper symmetry anomaly. In particular, we discover the construction of Luttinger liquid edge theories with central charge n−1 for type-III bosonic SPT phases protected by (\mathbb{Z}_n)^3 symmetry and the Luttinger liquid edge theories for intrinsically interacting fSPT protected by unitary Abelian symmetry. The ideas and methods used in these examples could be generalized to derive the edge theories of fSPT phases with arbitrary unitary Abelian total symmetry G_f.

We briefly survey our recent results about the Mumford goodness of several canonical metrics on the moduli spaces of Riemann surfaces, including the Weil-Petersson metric, the Ricci metric, the Perturbed Ricci metric and the Kahler-Einstein metric. We prove the dual Nakano negativity of the Weil-Petersson metric. As applications of these results we deduce certain important results about the <i>L</i> ²-cohomology groups of the logarithmic tangent bundle over the compactified moduli spaces.

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We compute the partition function on the hemisphere of a class of two-dimensional (2,2) supersymmetric field theories including gauged linear sigma models. The result provides a general exact formula for the central charge of the D-brane placed at the boundary. It takes the form of Mellin-Barnes integral and the question of its convergence leads to the grade restriction rule concerning branes near the phase boundaries. We find expressions in various phases including the large volume formula in which a characteristic class called the Gamma class shows up. The two sphere partition function factorizes into two hemispheres glued by inverse to the annulus. The result can also be written in a form familiar in mirror symmetry, and suggests a way to find explicit mirror correspondence between branes.

In this paper, we present a new algorithm to solve algebraically the following lattice-related problems: 1) the small integer solution (SIS) problem under the condition: if the solution is bounded by an integer β in l_∞ norm, which we call a bounded SIS (BSIS) problem, and if the difference between the row dimension n and the column dimension m of the corresponding basis matrix is relatively small with respect the row dimension m; 2) the learning with errors (LWE) problems under the condition: if the errors are bounded - the errors do not span the whole prime finite field F_q but a fixed known subset of size D (D < q), which we call a learning with bounded errors (LWBE) problem. We will show that we can solve these problems with polynomial complexity.

It is well known that a vector variational inequality can be a very efficient model for use in studying vector optimization problems. By using the Ky Fan fixed point theorem and the scalarization method we will prove some existence theorems for strong solutions for generalized vector variational inequalities where discontinuous and star-pseudomonotone operators are involved. Our results can be applied to the study of the existence of solutions of vector optimal problems. Some examples are given and analyzed.

We provide a non conformal generalization of the Compère-Song-Strominger (CSS) boundary conditions for AdS3 gravity that breaks the \hat{u}(1) Kac-Moody-Virasoro symmetry to two u(1)s. The holographic dual specified by the new boundary conditions can be understood as an irrelevant deformation of a warped conformal field theory (WCFT). Upon consistent reduction to two dimensions, AdS3 gravity results in a deformed Jackiw-Teitelboim dilaton gravity model coupled to a Maxwell field. We show that near extremality the boundary conditions inherited from generalized CSS boundary conditions in three dimensions give rise to an effective action exhibiting the same symmetry breaking pattern as the complex Sachdev-Ye-Kitaev models. Besides the Schwarzian term reflecting the breaking of conformal symmetry, the effective action contains an additional term that captures the breaking of the \hat{u}(1) Kac-Moody symmetry.

Many N=(2,2) two-dimensional nonlinear sigma models with Calabi-Yau target spaces admit ultraviolet descriptions as N=(2,2) gauge theories (gauged linear sigma models). We conjecture that the two-sphere partition function of such ultraviolet gauge theories -- recently computed via localization by Benini et al. and Doroud et al. -- yields the exact Kähler potential on the quantum Kähler moduli space for Calabi-Yau threefold target spaces. In particular, this allows one to compute the genus zero Gromov-Witten invariants for any such Calabi-Yau threefold without the use of mirror symmetry. More generally, when the infrared superconformal fixed point is used to compactify string theory, this provides a direct method to compute the spacetime Kähler potential of certain moduli (e.g., vector multiplet moduli in type IIA), exactly in {\alpha}'. We compute these quantities for the quintic and for Rødland's Pfaffian Calabi-Yau threefold and find agreement with existing results in the literature. We then apply our methods to a codimension four determinantal Calabi-Yau threefold in P^7, recently given a nonabelian gauge theory description by the present authors, for which no mirror Calabi-Yau is currently known. We derive predictions for its Gromov-Witten invariants and verify that our predictions satisfy nontrivial geometric checks.

In this paper, a degree theory for finite dimensional generalized variational inequalities is built and employed to prove some results on solution existence and solution stability.

In this short note, using Siu-Yau's method [14], we give a new proof that any n-dimensional compact K ahler manifold with positive orthogonal bisectional curvature must be biholomorphic to \mathbb {P}^ n .

In 2006, Nie et al proposed an attack to break an instance of TTM cryptosystems. However, the inventor of TTM disputed this attack and he proposed two new instances of TTM to support his viewpoint. At this time, he did not give the detail of key construction --- the construction of the lock polynomials in these instances which would be used in decryption. The two instances are claimed to achieve a security of 2^{109} against Nie et al attack. In this paper, we show that these instances are both still insecure, and in fact, they do not achieve a better design in the sense that we can find a ciphertext-only attack utilizing the First Order Linearization Equations while for the previous version of TTM, only Second Order Linearization Equations can be used in the beginning stage of the previous attack. Different from previous attacks, we use an iterated linearization method to break these two instances. For any given valid ciphertext, we can find its corresponding plaintext within 2^{31} F_{2^8}-computations after performing once for any public key a computation of complexity less than 2^{44} . Our experiment result shows we have unlocked the lock polynomials after several iterations, though we do not know the detailed construction of lock polynomials.

In this paper, we show the claims in the original Kipnis-Shamir attack on the HFE cryptosystems and the improved attack by Courtois that the complexity of the attacks is polynomial in terms of the number of variables are invalid. We present computer experiments and a theoretical argument using basic algebraic geometry to explain why it is so. Furthermore we show that even with the help of the powerful new Gröbner basis algorithm like F_4, the Kipnis-Shamir attack still should be exponential but not polynomial. This again is supported by our theoretical argument.

The mutual intensity and its equivalent phase-space representations quantify an optical field's state of coherence and are important tools in the study of light propagation and dynamics, but they can only be estimated indirectly from measurements through a process called coherence retrieval, otherwise known as phase-space tomography. As practical considerations often rule out the availability of a complete set of measurements, coherence retrieval is usually a challenging high-dimensional ill-posed inverse problem. In this paper, we propose a trace-regularized optimization model for coherence retrieval and a provably convergent adaptive accelerated proximal gradient algorithm for solving the resulting problem. Applying our model and algorithm to both simulated and experimental data, we demonstrate an improvement in reconstruction quality over previous models as well as an increase in convergence speed compared to existing first-order methods.

We establish that boundary degrees of freedom associated with a generic codimension one null surface in D-dimensional pure Einstein gravity naturally admit a thermodynamical description. We expect the null surface thermodynamics to universally follow as a result of the diffeomorphism invariance of the theory, not relying on other special features of the null surface or the gravity theory. Using standard surface charge analysis and covariant phase-space method, we formulate laws of null surface thermodynamics which are local equations over an arbitrary null surface paralleling local versions of the zeroth- and first laws and the Gibbs-Duhem equation. This thermodynamical system is generally an open system and can be closed only when there is no flux of gravitons through the null surface. Our analysis extends the usual black hole thermodynamics to a universal feature of any area element on a generic null surface. We discuss the relevance of our study for the membrane paradigm and black hole microstates.

Let G=(V,E) be a locally finite connected weighted graph, and Δ be the usual graph Laplacian. In this article, we study blow-up problems for the nonlinear parabolic equation u_t = Δu + f(u) on G. The blow-up phenomenons for u_t = Δu + f(u) are discussed in terms of two cases: (i) an initial condition is given; (ii) a Dirichlet boundary condition is given. We prove that if f satisfies appropriate conditions, then the corresponding solutions will blow up in a finite time.

In this paper we will show the Mumford goodness of the metrics on the logarithmic tangent bundle of the moduli spaces of curves induced by the Ricci and the perturbed Ricci metrics. It follows from these estimates that the Ricci metric can be extended to the Deligne-Mumford compactification of the moduli spaces.

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In this paper, we propose an algebraic broadcast attack against NTRU, which recovers a single message encrypted multiple times using different NTRU public keys. Namely, when a message is broadcasted, under some reasonable assumptions, our attack can be completed in polynomial time and space. To the best of our knowledge, this is the first successful broadcast attack against NTRU.

In this paper, we will systematically study who indeed are the hard lattice cases in LLL reduction. The “hard” cases here mean for their special geometric structures, with a comparatively high “failure probability” that LLL can not solve SVP even by using a powerful relaxation factor. We define the perfect lattice as the “Beauty”, which is given by basis of vectors of the same length with the mutual angles of any two vectors to be exactly 60°. Simultaneously the “Beasts” lattice is defined as the lattice close to the Beauty lattice. There is a relatively high probability (e.g. 15.0% in 3 dimensions) that our “Beasts” bases can withstand the exact-arithmetic LLL reduction (relaxation factors δ close to 1), comparing to the probability (corresponding <0.01%) when apply same LLL on random bases from TU Darmstadt SVP Challenge. Our theoretical proof gives us a direct explanation of this phenomenon. Moreover, we give rational Beauty bases of 3 and 8 dimensions, an irrational Beauty bases of general high dimensions. We also give a general way to construct Beasts lattice bases from the Beauty ones. Experimental results show the Beasts bases derived from Beauty can withstand LLL reduction by a stable probability even for high dimensions. Our work in a way gives a simple and direct way to explain how to build a hard lattice in LLL reduction.

In CT-RSA 2019, Bauer et al. have analyzed the case when the public key is reused for the NewHope key encapsulation mechanism (KEM), a second-round candidate in the NIST Post-quantum Standard process. They proposed an elegant method to recover coefficients ranging from −6 to 4 in the secret key. We repeat their experiments but there are two fundamental problems. First, even for coefficients in [−6, 4] we cannot recover at least 262 of them in each secret key with 1024 coefficients. Second, for the coefficient outside [−6, 4], they suggested an exhaustive search. But for each secret key on average there are 10 coefficients that need to be exhaustively searched, and each of them has 6 possibilities. This makes Bauer et al.’s method highly inefficient. We propose an improved method, which with 99.22% probability recovers all the coefficients ranging from −6 to 4 in the secret key. Then, inspired by Ding et al.’s key mismatch attack, we propose an efficient strategy which with a probability of 96.88% succeeds in recovering all the coefficients in the secret key. Experiments show that our proposed method is very efficient, which completes the attack in about 137.56 ms using the NewHope parameters.