In this article, we define a “ recursive local” algorithm in order to construct two reccurent numerical sequences of positive prime numbers (𝑈2𝑛) and (𝑉2𝑛), ((𝑈2𝑛) function of (𝑉2𝑛)), such that for any integer n≥ 2, their sum is 2n. To build these , we use a third sequence of prime numbers (𝑊2𝑛) defined for any integer n≥ 3 by : 𝑊2𝑛 = Sup(p∈IP : p ≤ 2n-3), where IP is the infinite set of positive prime numbers. The Goldbach conjecture has been verified for all even integers 2n between 4 and 4.1018. . In the Table of Goldbach sequence terms given in paragraph § 10, we reach values of the order of 2n= 101000 . Thus, thanks to this algorithm of “ascent and descent”, we can validate the strong Euler-Goldbach conjecture.

In this paper, we consider the cutoff Boltzmann equation near Maxwellian, we proved the global existence and uniqueness for the cutoff Boltzmann equation in polynomial weighted space for all $\gamma \in (-3, 1]$. We also proved initially polynomial decay for the large velocity in $L^2$ space will induce polynomial decay rate, while initially exponential decay will induce exponential rate for the convergence. Our proof is based on newly established inequalities for the cutoff Boltzmann equation and semigroup techniques. Moreover, by generalizing the $L_x^\infty L^1_v \cap L^\infty_{x, v}$ approach, we prove the global existence and uniqueness of a mild solution to the Boltzmann equation with bounded polynomial weighted $L^\infty_{x, v}$ norm under some small condition on the initial $L^1_x L^\infty_v$ norm and entropy so that this initial data allows large amplitude oscillations.

We study an asymptotic estimate on the number of negative eigenvalues of the Schr\"odinger operators on unbounded fractal spaces which admit a cellular decomposition. We first give some sufficient conditions for Weyl-type asymptotic formula to hold. Second, we verify these conditions for the infinite blowup of Sierpi\'nski gasket and unbounded generalized Sierpi\'nski carpets. Final, we demonstrate how the result can be applied to the infinite blowup of certain fractals associated with iterated function systems with overlaps, including those defining the classical infinite Bernoulli convolution with golden ratio.

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With the recent rise of Metaverse, online multiplayer VR applications are becoming increasingly prevalent worldwide. Allowing users to move easily in virtual environments is crucial for high-quality experiences in such collaborative VR applications. This paper focuses on redirected walking technology (RDW) to allow users to move beyond the confines of the limited physical environments (PE). The existing RDW methods lack the scheme to coordinate multiple users in different PEs, and thus have the issue of triggering too many resets for all the users. We propose a novel multi-user RDW method that is able to significantly reduce the overall reset number and give users a better immersive experience by providing a more continuous exploration. Our key idea is to first find out the ”bottleneck” user that may cause all users to be reset and estimate the time to reset, and then redirect all the users to favorable poses during that maximized bottleneck time to ensure the subsequent resets can be postponed as much as possible. More particularly, we develop methods to estimate the time of possibly encountering obstacles and the reachable area for a specific pose to enable the prediction of the next reset caused by any user. Our experiments and user study found that our method outperforms existing RDW methods in online VR applications.

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We study the non-klt locus of singularities of pairs. We show that given a pair (X, B) and a projective morphism X → Z with connected fibres such that −(KX +B) is nef over Z, the non-klt locus of (X, B) has at most two connected components near each fibre of X → Z. This was conjectured by Hacon and Han. In a different direction we answer a question of Mark Gross on connectedness of the non-klt loci of certain pairs. This is motivated by constructions in Mirror Symmetry.

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We introduce the notion of symplectic flatness for connections and fiber bundles over symplectic manifolds. Given an $A_\infty$-algebra, we present a flatness condition that enables the twisting of the differential complex associated with the $A_\infty$-algebra. The symplectic flatness condition arises from twisting the $A_\infty$-algebra of differential forms constructed by Tsai, Tseng and Yau. When the symplectic manifold is equipped with a compatible metric, the symplectic flat connections represent a special subclass of Yang-Mills connections. We further study the cohomologies of the twisted differential complex and give a simple vanishing theorem for them.

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Data-based detection and quantification of causation in complex, nonlinear dynamical systems is of paramount importance to science, engineering, and beyond. Inspired by the widely used methodology in recent years, the cross-map-based techniques, we develop a general framework to advance towards a comprehensive understanding of dynamical causal mechanisms, which is consistent with the natural interpretation of causality. In particular, instead of measuring the smoothness of the cross-map as conventionally implemented, we define causation through measuring the scaling law for the continuity of the investigated dynamical system directly. The uncovered scaling law enables accurate, reliable, and efficient detection of causation and assessment of its strength in general complex dynamical systems, outperforming those existing representative methods. The continuity scaling-based framework is rigorously established and demonstrated using datasets from model complex systems and the real world.

Finite Quot schemes were used by Bertram, Johnson, and the first author to study Le Potier’s strange duality conjecture on del Pezzo surfaces when one of the moduli spaces is the Hilbert scheme of points. In order to rigorously enumerate the finite Quot scheme, we study the moduli space of limit stable pairs in which the target has rank one on a smooth complex projective surface. We obtain an embedding of this moduli space into a smooth space that induces a perfect obstruction theory. This obstruction theory yields a virtual fundamental class that can be computed explicitly. Because the moduli space coincides with the Quot scheme when they have dimension 0, this makes the desired count of the finite Quot scheme in Bertram et al. rigorous. As another application, we obtain a universality result for tautological integrals on the moduli space of stable pairs.

Recently Chen and Gao~\cite{ChenGao2017} proposed a new quantum algorithm for Boolean polynomial system solving, motivated by the cryptanalysis of some post-quantum cryptosystems. The key idea of their approach is to apply a Quantum Linear System (QLS) algorithm to a Macaulay linear system over $\CC$, which is derived from the Boolean polynomial system. The efficiency of their algorithm depends on the condition number of the Macaulay matrix. In this paper, we give a strong lower bound on the condition number as a function of the Hamming weight of the Boolean solution, and show that in many (if not all) cases a Grover-based exhaustive search algorithm outperforms their algorithm. Then, we improve upon Chen and Gao's algorithm by introducing the Boolean Macaulay linear system over $\CC$ by reducing the original Macaulay linear system. This improved algorithm could potentially significantly outperform the brute-force algorithm, when the Hamming weight of the solution is logarithmic in the number of Boolean variables. Furthermore, we provide a simple and more elementary proof of correctness for our improved algorithm using a reduction employing the Valiant-Vazirani affine hashing method, and also extend the result to polynomial systems over $\FF_q$ improving on subsequent work by Chen, Gao and Yuan \cite{ChenGao2018}. We also suggest a new approach for extracting the solution of the Boolean polynomial system via a generalization of the quantum coupon collector problem \cite{arunachalam2020QuantumCouponCollector}.

Key exchange protocols from the learning with errors (LWE) problem share many similarities with the Diffie–Hellman–Merkle (DHM) protocol, which plays a central role in securing our Internet. Therefore, there has been a long time effort in designing authenticated key exchange directly from LWE to mirror the advantages of DHM-based protocols. In this paper, we revisit signal leakage attacks and show that the severity of these attacks against LWE-based (authenticated) key exchange is still underestimated. In particular, by converting the problem of launching a signal leakage attack into a coding problem, we can significantly reduce the needed number of queries to reveal the secret key. Specifically, for DXL-KE we reduce the queries from 1,266 to only 29, while for DBS-KE, we need only 748 queries, a great improvement over the previous 1,074,434 queries. Moreover, our new view of signals as binary codes enables recognizing vulnerable schemes more easily. As such we completely recover the secret key of a password-based authenticated key exchange scheme by Dabra et al. with only 757 queries and partially reveal the secret used in a two-factor authentication by Wang et al. with only one query. The experimental evaluation supports our theoretical analysis and demonstrates the efficiency and effectiveness of our attacks. Our results caution against underestimating the power of signal leakage attacks as they are applicable even in settings with a very restricted number of interactions between adversary and victim.

Lattice problem such as NTRU problem and LWE problem is widely used as the security base of post-quantum cryptosystems. And currently doing lattice reduction by BKZ algorithm is the most efficient way to solve it. In this paper, we give several further improvements on BKZ algorithm, which can be used for different SVP subroutines base on both enumeration and sieving. These improvements in combination provide a speed up of 2^{3~4} in total. It is significant in concrete attacks. Using these new techniques, we solved the 656 dimensional ideal lattice challenge in only 380 thread hours (also with a enumeration based SVP subroutine), much less than the previous records (which costs 4600 thread hours in total). With these improvements enabled, we can still simulate the new BKZ algorithm easily. One can also use this simulator to find the blocksize strategy (and the corresponding cost) to make Pot of the basis (defined in section 5.2) decrease as fast as possible, which means the length of the first basis vector decrease the fastest if we accept the GSA assumption. It is useful for analyzing concrete attacks on lattice-based cryptography.

Often times, the ability to distinguish between random data and a public key can leads to an attack against the cryptosystem itself. In this paper, we will show experimentally a very efficient distinguisher based on the distribution of ranks of the symmetric matrices associated with the central map in the multivariate cryptosystem HFE when the degree D of the central map is very small.

Our purpose is to compare how much the F5 algorithm can gain in efficiency compared to the F4 algorithm. This can be achieve as the F5 algorithm uses the concept of signatures to foresee potential useless computation which the F4 algorithm might make represented by zero rows in the reduction of a large matrix. We experimentally show that this is a modest increase in efficiency for the parameters we tested.

In 2017, Ward Beullenset al.submitted Lifted Unbalanced Oil andVinegar, which is a modification to the Unbalanced Oil and Vinegar Schemeby Patarin. Previously, Dinget al.proposed the Subfield Differential Attack which prompted a change of parameters by the authors of LUOV for the sec-ond round of the NIST post quantum standardization competition. In this paper we propose a modification to the Subfield Differential Attack called the Nested Subset Differential Attack which fully breaks half of the pa-rameter sets put forward. We also show by experimentation that this attack ispractically possible to do in under 210 minutes for the level I security param-eters and not just a theoretical attack. The Nested Subset Differential attack isa large improvement of the Subfield differential attack which can be used inreal world circumstances. Moreover, we will only use what is called the "lifted"structure of LUOV, and our attack can be thought as a development of solving"lifted" quadratic systems.