arXiv subjects

Rajendra Kumar

Publications and source records attributed to Rajendra Kumar.

18 recordsLinked to original sources

arXiv2026

Spectral Method attacks Sparse LWE, Sparse LPN and Beyond

Given a set of $k$-sparse linear equations over a ring $R$, we give algorithms to determine whether the right-hand sides are random or have a secret assignment planted with noise. For a parameter $k/2\leq l\leq n$, we give a spectral method to solve this problem in $\widetilde{O}\left(\binom{n}{l}\lvert{R}\rvert^l\right)$ time except with probability at most $n^{-\Omega(l)}$, provided the number of samples is roughly at least $\left(\frac{\lvert{R}\rvert n}{l}\right)^{k/2}$. This attack generalizes the Kikuchi method described by Wein et. al. (Journal of the ACM 2019) for $\mathbb{Z}_2$ to (commutative) rings of any finite size. We also give a simpler algorithm with better runtime than the spectral method and better sample complexity when $\lvert{R}\rvert=\omega(n/l)$. As a consequence, we obtain new sample-time tradeoffs for the decision problem of sparse LWE, sparse LPN over higher modulus $q$, and in general the distinguishing random vs planted $\mathbb{Z}_q$-linear equations for a large class of noise distributions. Our results imply a tightness of the hardness claims of Jain, Lin, Saha (Annual International Cryptology Conference, 2024) for sparse LWE.

cs.CR↗

arXiv2025

Structural optimization of lattice-matched Sc0.14Al0.86N/GaN superlattices for photonic applications

ScxAl1-xN is an emerging III-nitride material known for its high piezoelectric coefficient and ferroelectric properties. Integration of wide-bandgap ScxAl1-xN with GaN is particularly attractive for quantum photonic devices. Achieving low defect complex multilayers incorporating ScxAl1-xN, though, requires precise lattice-matching and carefully optimized growth parameters. This study systematically investigates the molecular-beam epitaxy of short-period ScxAl1-xN/GaN superlattices with total thicknesses of up to 600 nm on GaN templates. X-ray diffraction reciprocal space mapping confirmed lattice-matching at x = 0.14 Sc composition regardless of the thickness of GaN interlayers, as evidenced by symmetric superlattice satellites aligned in-plane with the underlying substrate peak. Superlattices with Sc compositions deviating from this lattice-matching condition exhibited strain-induced defects ranging from crack formation to partial relaxation. Scanning transmission electron microscopy (STEM) investigation of the ScxAl1-xN/GaN interfaces identified temperature-dependent intermixing as a major factor in setting the nitride composition variation and implicitly band structure profile along the growth direction. Energy-dispersive X-ray spectroscopy also revealed that Sc incorporation exhibits delays relative to Al at both onset and termination. Optimal growth conditions were observed at approximately 600{\deg}C and 550{\deg}C for superlattices with thick GaN layers (6 nm), and ultra-thin GaN layers (< 2 nm), respectively.

cond-mat.mtrl-sci↗

arXiv2025

Fine-Grained Complexity via Quantum Natural Proofs

Buhrman, Patro, and Speelman presented a framework of conjectures that together form a quantum analogue of the strong exponential-time hypothesis and its variants. They called it the QSETH framework. In this paper, using a notion of quantum natural proofs (built from natural proofs introduced by Razborov and Rudich), we show how part of the QSETH conjecture that requires properties to be `compression oblivious' can in many cases be replaced by assuming the existence of quantum-secure pseudorandom functions, a standard hardness assumption. Combined with techniques from Fourier analysis of Boolean functions, we show that properties such as PARITY and MAJORITY are compression oblivious for certain circuit class $\Lambda$ if subexponentially secure quantum pseudorandom functions exist in $\Lambda$, answering an open question in [Buhrman-Patro-Speelman 2021].

quant-ph↗

arXiv2025

Lattice Based Crypto breaks in a Superposition of Spacetimes

We explore the computational implications of a superposition of spacetimes, a phenomenon hypothesized in quantum gravity theories. This was initiated by Shmueli (2024) where the author introduced the complexity class $\mathbf{BQP^{OI}}$ consisting of promise problems decidable by quantum polynomial time algorithms with access to an oracle for computing order interference. In this work, it was shown that the Graph Isomorphism problem and the Gap Closest Vector Problem (with approximation factor $\mathcal{O}(n^{3/2})$) are in $\mathbf{BQP^{OI}}$. We extend this result by showing that the entire complexity class $\mathbf{SZK}$ (Statistical Zero Knowledge) is contained within $\mathbf{BQP^{OI}}$. This immediately implies that the security of numerous lattice based cryptography schemes will be compromised in a computational model based on superposition of spacetimes, since these often rely on the hardness of the Learning with Errors problem, which is in $\mathbf{SZK}$.

cs.CC↗

arXiv2025

On Beating $2^n$ for the Closest Vector Problem

The Closest Vector Problem (CVP) is a computational problem in lattices that is central to modern cryptography. The study of its fine-grained complexity has gained momentum in the last few years, partly due to the upcoming deployment of lattice-based cryptosystems in practice. A main motivating question has been if there is a $(2-\varepsilon)^n$ time algorithm on lattices of rank $n$, or whether it can be ruled out by SETH. Previous work came tantalizingly close to a negative answer by showing a $2^{(1-o(1))n}$ lower bound under SETH if the underlying distance metric is changed from the standard $\ell_2$ norm to other $\ell_p$ norms. Moreover, barriers toward proving such results for $\ell_2$ (and any even $p$) were established. In this paper we show \emph{positive results} for a natural special case of the problem that has hitherto seemed just as hard, namely $(0,1)$-$\mathsf{CVP}$ where the lattice vectors are restricted to be sums of subsets of basis vectors (meaning that all coefficients are $0$ or $1$). All previous hardness results applied to this problem, and none of the previous algorithmic techniques could benefit from it. We prove the following results, which follow from new reductions from $(0,1)$-$\mathsf{CVP}$ to weighted Max-SAT and minimum-weight $k$-Clique. 1. An $O(1.7299^n)$ time algorithm for exact $(0,1)$-$\mathsf{CVP}_2$ in Euclidean norm, breaking the natural $2^n$ barrier, as long as the absolute value of all coordinates in the input vectors is $2^{o(n)}$. 2. A computational equivalence between $(0,1)$-$\mathsf{CVP}_p$ and Max-$p$-SAT for all even $p$. 3. The minimum-weight-$k$-Clique conjecture from fine-grained complexity and its numerous consequences (which include the APSP conjecture) can now be supported by the hardness of a lattice problem, namely $(0,1)$-$\mathsf{CVP}_2$.

cs.DS↗

arXiv2024

Pinpointing Lattice-Matched Conditions for Wurtzite ScxAl1-xN/GaN Heterostructures with X-Ray Reciprocal Space Analysis

Using comprehensive x-ray reciprocal space mapping, we establish the precise lattice-matching composition for wurtzite $Sc_xAl_{1-x}N$ layers on (0001) GaN to be x = 0.14+/-0.01. 100-nm thick $Sc_xAl_{1-x}N$ films (x = 0.09 - 0.19) were grown in small composition increments on c-plane GaN templates by plasma-assisted molecular beam epitaxy. The alloy composition was estimated from the fit of the (0002) x-ray peak positions assuming the c-lattice parameter of ScAlN films coherently-strained on GaN increases linearly with Sc-content determined independently by Rutherford Backscattering Spectrometry. Reciprocal space maps obtained from high-resolution x-ray diffraction measurements of the (10-15) reflection reveal that $Sc_xAl_{1-x}N$ films with x = 0.14+/-0.01 are coherently strained with the GaN substrate while the other compositions show evidence of relaxation. The in-plane lattice-matching with GaN is further confirmed for a 300-nm thick $Sc_{0.14}Al_{0.86}N$ layer. The full-width-at-half-maximum of the (0002) reflection rocking curve for this $Sc_{0.14}Al_{0.86}N$ film is 106 arcseconds and corresponds to the lowest value reported in the literature for wurtzite ScAlN films.

cond-mat.mtrl-sci↗

arXiv2023

QSETH strikes again: finer quantum lower bounds for lattice problem, strong simulation, hitting set problem, and more

While seemingly undesirable, it is not a surprising fact that there are certain problems for which quantum computers offer no computational advantage over their respective classical counterparts. Moreover, there are problems for which there is no `useful' computational advantage possible with the current quantum hardware. This situation however can be beneficial if we don't want quantum computers to solve certain problems fast - say problems relevant to post-quantum cryptography. In such a situation, we would like to have evidence that it is difficult to solve those problems on quantum computers; but what is their exact complexity? To do so one has to prove lower bounds, but proving unconditional time lower bounds has never been easy. As a result, resorting to conditional lower bounds has been quite popular in the classical community and is gaining momentum in the quantum community. In this paper, by the use of the QSETH framework [Buhrman-Patro-Speelman 2021], we are able to understand the quantum complexity of a few natural variants of CNFSAT, such as parity-CNFSAT or counting-CNFSAT, and also are able to comment on the non-trivial complexity of approximate-#CNFSAT; both of these have interesting implications about the complexity of (variations of) lattice problems, strong simulation and hitting set problem, and more. In the process, we explore the QSETH framework in greater detail than was (required and) discussed in the original paper, thus also serving as a useful guide on how to effectively use the QSETH framework.

quant-ph↗

arXiv2022

Lattice Problems Beyond Polynomial Time

We study the complexity of lattice problems in a world where algorithms, reductions, and protocols can run in superpolynomial time, revisiting four foundational results: two worst-case to average-case reductions and two protocols. We also show a novel protocol. 1. We prove that secret-key cryptography exists if $\widetilde{O}(\sqrt{n})$-approximate SVP is hard for $2^{\varepsilon n}$-time algorithms. I.e., we extend to our setting (Micciancio and Regev's improved version of) Ajtai's celebrated polynomial-time worst-case to average-case reduction from $\widetilde{O}(n)$-approximate SVP to SIS. 2. We prove that public-key cryptography exists if $\widetilde{O}(n)$-approximate SVP is hard for $2^{\varepsilon n}$-time algorithms. This extends to our setting Regev's celebrated polynomial-time worst-case to average-case reduction from $\widetilde{O}(n^{1.5})$-approximate SVP to LWE. In fact, Regev's reduction is quantum, but ours is classical, generalizing Peikert's polynomial-time classical reduction from $\widetilde{O}(n^2)$-approximate SVP. 3. We show a $2^{\varepsilon n}$-time coAM protocol for $O(1)$-approximate CVP, generalizing the celebrated polynomial-time protocol for $O(\sqrt{n/\log n})$-CVP due to Goldreich and Goldwasser. These results show complexity-theoretic barriers to extending the recent line of fine-grained hardness results for CVP and SVP to larger approximation factors. (This result also extends to arbitrary norms.) 4. We show a $2^{\varepsilon n}$-time co-non-deterministic protocol for $O(\sqrt{\log n})$-approximate SVP, generalizing the (also celebrated!) polynomial-time protocol for $O(\sqrt{n})$-CVP due to Aharonov and Regev. 5. We give a novel coMA protocol for $O(1)$-approximate CVP with a $2^{\varepsilon n}$-time verifier. All of the results described above are special cases of more general theorems that achieve time-approximation factor tradeoffs.

cs.CC↗

arXiv2022

Why we couldn't prove SETH hardness of the Closest Vector Problem for even norms!

Recent work [BGS17,ABGS19] has shown SETH hardness of CVP in the $\ell_p$ norm for any $p$ that is not an even integer. This result was shown by giving a Karp reduction from $k$-SAT on $n$ variables to CVP on a lattice of rank $n$. In this work, we show a barrier towards proving a similar result for CVP in the $\ell_p$ norm where $p$ is an even integer. We show that for any $c>0$, if for every $k > 0$, there exists an efficient reduction that maps a $k$-SAT instance on $n$ variables to a CVP instance for a lattice of rank at most $n^{c}$ in the Euclidean norm, then $\mathsf{coNP} \subset \mathsf{NP/Poly}$. We prove a similar result for CVP for all even norms under a mild additional promise that the ratio of the distance of the target from the lattice and the shortest non-zero vector in the lattice is bounded by $exp(n^{O(1)})$. Furthermore, we show that for any $c> 0$, and any even integer $p$, if for every $k > 0$, there exists an efficient reduction that maps a $k$-SAT instance on $n$ variables to a $SVP_p$ instance for a lattice of rank at most $n^{c}$, then $\mathsf{coNP} \subset \mathsf{NP/Poly}$. The result for SVP does not require any additional promise. While prior results have indicated that lattice problems in the $\ell_2$ norm (Euclidean norm) are easier than lattice problems in other norms, this is the first result that shows a separation between these problems. We achieve this by using a result by Dell and van Melkebeek [JACM, 2014] on the impossibility of the existence of a reduction that compresses an arbitrary $k$-SAT instance into a string of length $\mathcal{O}(n^{k-\epsilon})$ for any $\epsilon>0$. In addition to CVP, we also show that the same result holds for the Subset-Sum problem using similar techniques.

cs.CC↗

arXiv2022

3D pride without 2D prejudice: Bias-controlled multi-level generative models for structure-based ligand design

Generative models for structure-based molecular design hold significant promise for drug discovery, with the potential to speed up the hit-to-lead development cycle, while improving the quality of drug candidates and reducing costs. Data sparsity and bias are, however, two main roadblocks to the development of 3D-aware models. Here we propose a first-in-kind training protocol based on multi-level contrastive learning for improved bias control and data efficiency. The framework leverages the large data resources available for 2D generative modelling with datasets of ligand-protein complexes. The result are hierarchical generative models that are topologically unbiased, explainable and customizable. We show how, by deconvolving the generative posterior into chemical, topological and structural context factors, we not only avoid common pitfalls in the design and evaluation of generative models, but furthermore gain detailed insight into the generative process itself. This improved transparency significantly aids method development, besides allowing fine-grained control over novelty vs familiarity.

stat.ML↗

arXiv2021

Dimension-Preserving Reductions Between SVP and CVP in Different $p$-Norms

$ \newcommand{\SVP}{\textsf{SVP}} \newcommand{\CVP}{\textsf{CVP}} \newcommand{\eps}{\varepsilon} $We show a number of reductions between the Shortest Vector Problem and the Closest Vector Problem over lattices in different $\ell_p$ norms ($\SVP_p$ and $\CVP_p$ respectively). Specifically, we present the following $2^{\eps m}$-time reductions for $1 \leq p \leq q \leq \infty$, which all increase the rank $n$ and dimension $m$ of the input lattice by at most one: $\bullet$ a reduction from $\widetilde{O}(1/\eps^{1/p})\gamma$-approximate $\SVP_q$ to $\gamma$-approximate $\SVP_p$; $\bullet$ a reduction from $\widetilde{O}(1/\eps^{1/p}) \gamma$-approximate $\CVP_p$ to $\gamma$-approximate $\CVP_q$; and $\bullet$ a reduction from $\widetilde{O}(1/\eps^{1+1/p})$-$\CVP_q$ to $(1+\eps)$-unique $\SVP_p$ (which in turn trivially reduces to $(1+\eps)$-approximate $\SVP_p$). The last reduction is interesting even in the case $p = q$. In particular, this special case subsumes much prior work adapting $2^{O(m)}$-time $\SVP_p$ algorithms to solve $O(1)$-approximate $\CVP_p$. In the (important) special case when $p = q$, $1 \leq p \leq 2$, and the $\SVP_p$ oracle is exact, we show a stronger reduction, from $O(1/\eps^{1/p})\text{-}\CVP_p$ to (exact) $\SVP_p$ in $2^{\eps m}$ time. For example, taking $\eps = \log m/m$ and $p = 2$ gives a slight improvement over Kannan's celebrated polynomial-time reduction from $\sqrt{m}\text{-}\CVP_2$ to $\SVP_2$. We also note that the last two reductions can be combined to give a reduction from approximate-$\CVP_p$ to $\SVP_q$ for any $p$ and $q$, regardless of whether $p \leq q$ or $p > q$. Our techniques combine those from the recent breakthrough work of Eisenbrand and Venzin (which showed how to adapt the current fastest known algorithm for these problems in the $\ell_2$ norm to all $\ell_p$ norms) together with sparsification-based techniques.

cs.DS↗

arXiv2020

Hardness of Approximation of (Multi-)LCS over Small Alphabet

The problem of finding longest common subsequence (LCS) is one of the fundamental problems in computer science, which finds application in fields such as computational biology, text processing, information retrieval, data compression etc. It is well known that (decision version of) the problem of finding the length of a LCS of an arbitrary number of input sequences (which we refer to as Multi-LCS problem) is NP-complete. Jiang and Li [SICOMP'95] showed that if Max-Clique is hard to approximate within a factor of $s$ then Multi-LCS is also hard to approximate within a factor of $\Theta(s)$. By the NP-hardness of the problem of approximating Max-Clique by Zuckerman [ToC'07], for any constant $\delta>0$, the length of a LCS of arbitrary number of input sequences of length $n$ each, cannot be approximated within an $n^{1-\delta}$-factor in polynomial time unless {\tt{P}}$=${\NP}. However, the reduction of Jiang and Li assumes the alphabet size to be $\Omega(n)$. So far no hardness result is known for the problem of approximating Multi-LCS over sub-linear sized alphabet. On the other hand, it is easy to get $1/|\Sigma|$-factor approximation for strings of alphabet $\Sigma$. In this paper, we make a significant progress towards proving hardness of approximation over small alphabet by showing a polynomial-time reduction from the well-studied \emph{densest $k$-subgraph} problem with {\em perfect completeness} to approximating Multi-LCS over alphabet of size $poly(n/k)$. As a consequence, from the known hardness result of densest $k$-subgraph problem (e.g. [Manurangsi, STOC'17]) we get that no polynomial-time algorithm can give an $n^{-o(1)}$-factor approximation of Multi-LCS over an alphabet of size $n^{o(1)}$, unless the Exponential Time Hypothesis is false.

cs.CC↗

arXiv2020

Improved Classical and Quantum Algorithms for the Shortest Vector Problem via Bounded Distance Decoding

The most important computational problem on lattices is the Shortest Vector Problem (SVP). In this paper, we present new algorithms that improve the state-of-the-art for provable classical/quantum algorithms for SVP. We present the following results. $\bullet$ A new algorithm for SVP that provides a smooth tradeoff between time complexity and memory requirement. For any positive integer $4\leq q\leq \sqrt{n}$, our algorithm takes $q^{13n+o(n)}$ time and requires $poly(n)\cdot q^{16n/q^2}$ memory. This tradeoff which ranges from enumeration ($q=\sqrt{n}$) to sieving ($q$ constant), is a consequence of a new time-memory tradeoff for Discrete Gaussian sampling above the smoothing parameter. $\bullet$ A quantum algorithm for SVP that runs in time $2^{0.950n+o(n)}$ and requires $2^{0.5n+o(n)}$ classical memory and poly(n) qubits. In Quantum Random Access Memory (QRAM) model this algorithm takes only $2^{0.835n+o(n)}$ time and requires a QRAM of size $2^{0.293n+o(n)}$, poly(n) qubits and $2^{0.5n}$ classical space. This improves over the previously fastest classical (which is also the fastest quantum) algorithm due to [ADRS15] that has a time and space complexity $2^{n+o(n)}$. $\bullet$ A classical algorithm for SVP that runs in time $2^{1.669n+o(n)}$ time and $2^{0.5n+o(n)}$ space. This improves over an algorithm of [CCL18] that has the same space complexity. The time complexity of our classical and quantum algorithms are obtained using a known upper bound on a quantity related to the lattice kissing number which is $2^{0.402n}$. We conjecture that for most lattices this quantity is a $2^{o(n)}$. Assuming that this is the case, our classical algorithm runs in time $2^{1.292n+o(n)}$, our quantum algorithm runs in time $2^{0.750n+o(n)}$ and our quantum algorithm in QRAM model runs in time $2^{0.667n+o(n)}$.

cs.DS↗

arXiv2018

On the Maximum Distance Sublattice Problem and Closest Vector Problem

In this paper, we introduce the Maximum Distance Sublattice Problem (MDSP). We observed that the problem of solving an instance of the Closest Vector Problem (CVP) in a lattice $\mathcal{L}$ is the same as solving an instance of MDSP in the dual lattice of $\mathcal{L}$. We give an alternate reduction between the CVP and MDSP. This alternate reduction does not use the concept of dual lattice.

cs.CC↗

arXiv2018

Spectroscopic signatures of native charge compensation in Mg doped GaN Nanorods

We study the native charge compensation effect in Mg doped GaN nanorods (NRs), grown by Plasma Assisted Molecular Beam Epitaxy (PAMBE), using Raman, photoluminescence (PL) and X-ray photoelectron spectroscopies (XPS). The XPS valence band analysis shows that upon Mg incorporation the E$_F$-E$_{VBM}$ reduces, suggesting the compensation of the native n-type character of GaN NRs. Raman spectroscopic studies on these samples reveal that the line shape of longitudinal phonon plasmon (LPP) coupled mode is sensitive to Mg concentration and hence to background n-type carrier density. We estimate a two order of native charge compensation in GaN NRs upon Mg-doping with a concentration of 10$^{19}$-10$^{20}$ atoms cm$^{-3}$. Room temperature (RT) PL measurements and our previous electronic structure calculations are used to identify the atomistic origin of this compensation effect.

cond-mat.mes-hall↗

arXiv2017

Enhanced radial growth of Mg doped GaN nanorods: A combined experiment and first-principles based analysis

We discuss the microstructural origin of enhanced radial growth in magnesium (Mg) doped gallium nitride (GaN) nanorods (NRs) using electron microscopy and \textit{first-principles} Density Functional Theory calculations. Experimentally, we find the Mg incorporation increases surface coverage of the grown samples and the height of NRs decreases as a consequence of an increase radial growth rate. We also observed the coalescence of NRs becomes prominent and the critical height of coalescence decreases with the increase in Mg concentration. From \textit{first-principles} calculations, we find the surface free energy of Mg doped surface reduces with increasing Mg concentration in the samples. The calculations further suggests a reduction in the diffusion barrier of Ga adatoms along [11$\overline{2}$0] on the side wall surface of the NRs, possibly the primary reason for the observed enhancement in the radial growth.

cond-mat.mes-hall↗

arXiv2017

Edge enhanced growth induced shape transition in the formation of GaN Nanowall Network

We address the mechanism of early stages of growth and shape transition of the unique nanowall network (NwN) nanostructure of GaN by experimentally monitoring its controlled growth using PA-MBE and complementing it by \textit{first-principles} calculations. Using electron microscopy, we observe the formation of tetrahedron shaped (3 faced pyramid) islands at early stages of growth, which later grows anisotropically along their edges of the (20$\overline{2}$1) facets, to form the wall like structure. The mechanism of this crystal growth is discussed in light of surface free energies of the different surfaces, adsorption energy and diffusion barrier of Ga ad-atoms on the (20$\overline{2}$1) facets. By \textit{first-principles} calculations, we find that the diffusion barrier of ad-atoms decreases with decreasing width of facets, and is responsible for the anisotropic growth and formation of the nanowall network. This study suggest that formation of NwN is a archetype example of structure dependent attachment kinetic (SDAK) instability induced shape transition in thin film growth.

cond-mat.mes-hall↗

arXiv2016

Centralized group key management scheme for secure multicast communication without re-keying

In the secure group communication, data is transmitted in such a way that only the group members are able to receive the messages. The main problem in the solution using symmetric key is heavy re-keying cost. To reduce re-keying cost tree based architecture is used. But it requires extra overhead to balance the key- tree in order to achieve logarithmic re-keying cost. The main challenging issue in dynamic and secure multimedia multicast communication is to design a centralized group key management scheme with minimal computational, communicational and storages complexities without breaching security issues. Several authors have proposed different centralized group key management schemes, wherein one of them proposes reducing communicational complexity but increases computational and storage costs however another proposes decreasing the computational and storage costs which eventually breaches forward and backward secrecy. In this paper we propose a comparatively more efficient centralized group key management scheme that not only minimize the computational, communicational and storages complexities but also maintaining the security at the optimal level. The message encryptions and decryptions costs are also minimized. Further, we also provide an extended multicast scheme, in which the several requests towards leaving or joining the group can be done by large number of members simultaneously. In order to obtain better performance of multicast encryption, the symmetric-key and asymmetric-key cryptosystems may be combined.

cs.CR↗