arXiv subjects

Shuhong Gao

Publications and source records attributed to Shuhong Gao.

16 recordsLinked to original sources

arXiv2026

Counterexamples to the Jacobian conjecture in dimensions greater than two

The Jacobian conjecture, open since 1939, asks whether every polynomial map of $\C^n$ whose Jacobian determinant is a nonzero constant must have a polynomial inverse. It was refuted in dimension three by Alpöge on July 19, 2026, with an infinite family by Gallagher (July 20) and a geometric explanation by Speyer (July 23): the counterexample sweeps the tangent lines of a plane curve --- a map that classical duality forces to hit most points several times. We give a self-contained account of this tangent-sweep mechanism and generalize it from plane curves to direction fields on hypersurfaces. The resulting construction produces counterexamples in every dimension greater than two and, in each dimension, of arbitrarily large geometric degree (the number of preimages of a typical point). We work it out in five new explicit maps: one three-dimensional of degree four, two four-dimensional of degrees five and ten, and two five-dimensional of degrees six and twelve. The counterexamples provide explicit examples of étale coverings $\C^n\to\C^n$ that are not proper: they are everywhere unramified, and fail to be injective only through points escaping to infinity. All identities were verified in exact rational arithmetic; an appendix determines exact fiber structures through Gröbner bases.

math.AG↗

arXiv2025

Security Analysis of Integer Learning with Errors with Rejection Sampling

At ASIACRYPT 2018, a digital attack based on linear least squares was introduced for a variant of the learning with errors (LWE) problem which omits modular reduction known as the integer learning with errors problem (ILWE). In this paper, we present a theoretical and experimental study of the effectiveness of the attack when applied directly to small parameter ILWE instances found in popular digital signature schemes such as CRYSTALS-Dilithium which utilize rejection sampling. Unlike other studies which form ILWE instances based on additional information obtained from side-channel attacks, we take a more direct approach to the problem by constructing our ILWE instance from only the obtained signatures. We outline and introduce novel techniques in our simulation designs such as modular polynomial arithmetic via matrices in $\mathbb{R}$, as well as algorithms for handling large sample sizes efficiently. Our experimental results reinforce the proclaimed security of signature schemes based on ILWE. We additionally discuss the implications of our work and digital signatures as a whole in regards to real-world applications such as in Intelligent Transportation Systems (ITS).

cs.CR↗

arXiv2025

Experimental Evaluation of Post-Quantum Homomorphic Encryption for Privacy-Preserving I2I Communication in ITS

This study experimentally evaluates the feasibility of post-quantum secure Homomorphic Encryption (HE) for privacy-preserving Infrastructure-to-Infrastructure (I2I) communication in Intelligent Transportation Systems (ITS). Unlike prior simulation-based efforts, this work implements three lattice-based HE schemes: Brakerski-Fan-Vercauteren (BFV), Brakerski-Gentry-Vaikuntanathan (BGV), and Cheon-Kim-Kim-Song (CKKS), within a real experimental pipeline representing roadside unit (RSU)-Cloud data exchange over Wi-Fi and Ethernet networks. The experiments benchmark encrypted addition and addition-plus-multiplication operations representing key analytical tasks, such as vehicle queue assessment and regional speed computation. Results show that while BFV achieves sub-5-second latency suitable for intersection-level analytics, BGV supports regional aggregation with 10 to 30-second updates. CKKS, though exhibiting higher latency (21-32 seconds), remains practical for minute-scale applications like eco-driving. These findings demonstrate that post-quantum HE can enable privacy-preserving ITS backhaul analytics when latency requirements align with application needs. The study also presents optimization pathways, including algorithmic tuning, network adaptation, and hardware acceleration, to reduce end-to-end delay.

cs.CR↗

arXiv2025

Bounded Distance Decoding for Random Lattices

The current paper investigates the bounded distance decoding (BDD) problem for ensembles of lattices whose generator matrices have sub-Gaussian entries. We first prove that, for these ensembles the BDD problem is NP-hard in the worst case. Then, we introduce a polynomial-time algorithm based on singular value decomposition (SVD) and establish, both theoretically and through extensive experiments, that, for a random selected lattice from the same ensemble, the algorithm solves the BDD problem with high probability. To the best of our knowledge, this work provides the first example of a lattice problem that is NP-hard in the worst case yet admits a polynomial time algorithm on the average case.

cs.CC↗

arXiv2024

On binomial Weil sums and an application

Let $p$ be a prime, and $N$ be a positive integer not divisible by $p$. Denote by ${\rm ord}_N(p)$ the multiplicative order of $p$ modulo $N$. Let $\mathbb{F}_q$ represent the finite field of order $q=p^{{\rm ord}_N(p)}$. For $a, b\in\mathbb{F}_q$, we define a binomial exponential sum by $$S_N(a,b):=\sum_{x\in\mathbb{F}_q\setminus\{0\}}χ(ax^{\frac{q-1}{N}}+bx),$$ where $χ$ is the canonical additive character of $\mathbb{F}_q$. In this paper, we provide an explicit evaluation of $S_{N}(a,b)$ for any odd prime $p$ and any $N$ satisfying ${\rm ord}_{N}(p)=ϕ(N)$. Our elementary and direct approach allows for the construction of a class of ternary linear codes, with their exact weight distribution determined. Furthermore, we prove that the dual codes achieve optimality with respect to the sphere packing bound, thereby generalizing previous results from even to odd characteristic fields.

math.NT↗

arXiv2023

Privacy-Preserving Discretized Spiking Neural Networks

The rapid development of artificial intelligence has brought considerable convenience, yet also introduces significant security risks. One of the research hotspots is to balance data privacy and utility in the real world of artificial intelligence. The present second-generation artificial neural networks have made tremendous advances, but some big models could have really high computational costs. The third-generation neural network, SNN (Spiking Neural Network), mimics real neurons by using discrete spike signals, whose sequences exhibit strong sparsity, providing advantages such as low energy consumption and high efficiency. In this paper, we construct a framework to evaluate the homomorphic computation of SNN named FHE-DiSNN that enables SNN to achieve good prediction performance on encrypted data. First, benefitting from the discrete nature of spike signals, our proposed model avoids the errors introduced by discretizing activation functions. Second, by applying bootstrapping, we design new private preserving functions FHE-Fire and FHE-Reset, through which noise can be refreshed, allowing us to evaluate SNN for an arbitrary number of operations. Furthermore, We improve the computational efficiency of FHE-DiSNN while maintaining a high level of accuracy. Finally, we evaluate our model on the MNIST dataset. The experiments show that FHE-DiSNN with 30 neurons in the hidden layer achieves a minimum prediction accuracy of 94.4%. Under optimal parameters, it achieves a 95.1% accuracy, with only a 0.6% decrease compared to the original SNN (95.7%). These results demonstrate the superiority of SNN over second-generation neural networks for homomorphic evaluation.

cs.CR↗

arXiv2022

Linear Complexity of A Family of Binary $pq^2$-periodic Sequences From Euler Quotients

We first introduce a family of binary $pq^2$-periodic sequences based on the Euler quotients modulo $pq$, where $p$ and $q$ are two distinct odd primes and $p$ divides $q-1$. The minimal polynomials and linear complexities are determined for the proposed sequences provided that $2^{q-1} \not\equiv 1 \mod{q^2}.$ The results show that the proposed sequences have high linear complexities.

cs.IT↗

arXiv2018

Optimal Bounds for Johnson-Lindenstrauss Transformations

In 1984, Johnson and Lindenstrauss proved that any finite set of data in a high-dimensional space can be projected to a lower-dimensional space while preserving the pairwise Euclidean distance between points up to a bounded relative error. If the desired dimension of the image is too small, however, Kane, Meka, and Nelson (2011) and Jayram and Woodruff (2013) independently proved that such a projection does not exist. In this paper, we provide a precise asymptotic threshold for the dimension of the image, above which, there exists a projection preserving the Euclidean distance, but, below which, there does not exist such a projection.

cs.DM↗

arXiv2018

The Complexity of Subdivision for Diameter-Distance Tests

We present a general framework for analyzing the complexity of subdivision-based algorithms whose tests are based on the sizes of regions and their distance to certain sets (often varieties) intrinsic to the problem under study. We call such tests diameter-distance tests. We illustrate that diameter-distance tests are common in the literature by proving that many interval arithmetic-based tests are, in fact, diameter-distance tests. For this class of algorithms, we provide both non-adaptive bounds for the complexity, based on separation bounds, as well as adaptive bounds, by applying the framework of continuous amortization. Using this structure, we provide the first complexity analysis for the algorithm by Plantinga and Vegeter for approximating real implicit curves and surfaces. We present both adaptive and non-adaptive a priori worst-case bounds on the complexity of this algorithm both in terms of the number of subregions constructed and in terms of the bit complexity for the construction. Finally, we construct families of hypersurfaces to prove that our bounds are tight.

cs.SC↗

arXiv2017

Gröbner Bases of Generic Ideals

Let $I = ( f_1, \dots, f_n )$ be a homogeneous ideal in the polynomial ring $K[x_1, \dots,x_n]$ over a field $K$ generated by generic polynomials. Using an incremental approach based on a method by Gao, Guan and Volny, and properties of the standard monomials of generic ideals, we show how a Gröbner basis for the ideal $(f_1, \dots, f_i)$ can be obtained from that of $(f_1, \dots, f_{i-1})$. If $deg f_i = d_i$, we are able to give a complete description of the initial ideal of $I$ in the case where $d_i \geq \left(\sum_{j=1}^{i-1}d_j\right) - i -1$. It was conjectured by Moreno-Socías that the initial ideal of $I$ is almost reverse lexicographic, which implies a conjecture by Fröberg on Hilbert series of generic algebras. As a result, we obtain a partial answer to Moreno-Socías Conjecture: the initial ideal of $I$ is almost reverse lexicographic if the degrees of generators satisfy the condition above. This result improves a result by Cho and Park. We hope this approach can be strengthened to prove the conjecture in full.

math.AC↗

arXiv2017

Counting Roots of Polynomials Over Prime Power Rings

Suppose $p$ is a prime, $t$ is a positive integer, and $f\!\in\!\mathbb{Z}[x]$ is a univariate polynomial of degree $d$ with coefficients of absolute value $<\!p^t$. We show that for any fixed $t$, we can compute the number of roots in $\mathbb{Z}/(p^t)$ of $f$ in deterministic time $(d+\log p)^{O(1)}$. This fixed parameter tractability appears to be new for $t\!\geq\!3$. A consequence for arithmetic geometry is that we can efficiently compute Igusa zeta functions $Z$, for univariate polynomials, assuming the degree of $Z$ is fixed.

math.NT↗

arXiv2016

Codes for distributed storage from 3-regular graphs

This paper considers distributed storage systems (DSSs) from a graph theoretic perspective. A DSS is constructed by means of the path decomposition of a 3- regular graph into P4 paths. The paths represent the disks of the DSS and the edges of the graph act as the blocks of storage. We deduce the properties of the DSS from a related graph and show their optimality.

cs.IT↗

arXiv2016

Sparse Univariate Polynomials with Many Roots Over Finite Fields

Suppose $q$ is a prime power and $f\in\mathbb{F}_q[x]$ is a univariate polynomial with exactly $t$ monomial terms and degree $<q-1$. To establish a finite field analogue of Descartes' Rule, Bi, Cheng, and Rojas (2013) proved an upper bound of $2(q-1)^{\frac{t-2}{t-1}}$ on the number of cosets in $\mathbb{F}^*_q$ needed to cover the roots of $f$ in $\mathbb{F}^*_q$. Here, we give explicit $f$ with root structure approaching this bound: For $q$ a $(t-1)$-st power of a prime we give an explicit $t$-nomial vanishing on $q^{\frac{t-2}{t-1}}$ distinct cosets of $\mathbb{F}^*_q$. Over prime fields $\mathbb{F}_p$, computational data we provide suggests that it is harder to construct explicit sparse polynomials with many roots. Nevertheless, assuming the Generalized Riemann Hypothesis, we find explicit trinomials having $Ω\left(\frac{\log p}{\log \log p}\right)$ distinct roots in $\mathbb{F}_p$.

math.NT↗

arXiv2012

Finite field elements of high order arising from modular curves

In this paper, we recursively construct explicit elements of provably high order in finite fields. We do this using the recursive formulas developed by Elkies to describe explicit modular towers. In particular, we give two explicit constructions based on two examples of his formulas and demonstrate that the resulting elements have high order. Between the two constructions, we are able to generate high order elements in every characteristic. Despite the use of the modular recursions of Elkies, our methods are quite elementary and require no knowledge of modular curves. We compare our results to a recent result of Voloch. In order to do this, we state and prove a slightly more refined version of a special case of his result.

math.NT↗

arXiv2011

Solving the 100 Swiss Francs Problem

Sturmfels offered 100 Swiss Francs in 2005 to a conjecture, which deals with a special case of the maximum likelihood estimation for a latent class model. This paper confirms the conjecture positively.

stat.CO↗

arXiv2008

Computing Irreducible Decomposition of Monomial Ideals

The paper presents two algorithms for finding irreducible decomposition of monomial ideals. The first one is recursive, derived from staircase structures of monomial ideals. This algorithm has a good performance for highly non-generic monomial ideals. The second one is an incremental algorithm, which computes decompositions of ideals by adding one generator at a time. Our analysis shows that the second algorithm is more efficient than the first one for generic monomial ideals. Furthermore, the time complexity of the second algorithm is at most $O(n^2p\ell)$ where $n$ is the number of variables, $p$ is the number of minimal generators and $\ell$ is the number of irreducible components. Another novelty of the second algorithm is that, for generic monomial ideals, the intermediate storage is always bounded by the final output size which may be exponential in the input size.

math.AC↗