arXiv subjects

Hemanta K. Maji

Publications and source records attributed to Hemanta K. Maji.

6 recordsLinked to original sources

arXiv2026

Energy Estimation of the Hamming Slice and its Applications

Let $R=\mathbb{Z}/(2^n-1)\mathbb{Z}$, where $n\geq 3$, and let $S_w\subseteq R$ be the residues whose canonical $n$-digit binary expansion has Hamming weight $w$. We obtain, in particular, an asymptotic formula for the additive energy of $S_w$ \[ E(S_w)=\frac{\left|S_w\right|^4}{|R|}+ \mathcal{O}\left(|R|^3 n^{-3} \right), \] which holds uniformly in $w$. The error term is optimal in order, with a matching lower bound for $w=\lfloor n/2+\sqrt{n} \rfloor$. It follows that triple sums of arbitrary unit dilates have asymptotically uniform representation counts when $\prod_{j=1}^{3} \left|S_{w_j} \right| /\left(|R| n^{-3/5}\right)^3\to\infty$, and that double sums have asymptotically full support when $\left|S_{w_1}\right| \left|S_{w_2} \right|/\left(|R| n^{-3/4}\right)^2\to\infty$. In the proof, modular collisions are represented using a cyclic binary carry automaton; this appears to be a novel approach in this area of problems.

math.CO↗

arXiv2026

Solving polynomial inequalities over spaces of convex sets and applications

We develop a symbolic elimination theory for finite systems of recursive containment inequalities whose unknowns are convex subsets of a finite-dimensional real vector space. The right-hand sides are formal expressions generated from variables and parameters by convex linear combinations, finite union, and a positive geometric join encoding strict convex combinations. We prove that every parameter assignment has a unique smallest convex-set-valued solution and give a finite Gaussian-elimination-type procedure that eliminates the unknowns while preserving this solution and produces parameter-only expressions for its coordinate sets. More generally, let $\mathcal B$ be a family of subsets containing $\emptyset$ and closed under finite unions, nonnegative dilation, Minkowski sums, positive geometric joins, and convex hulls. If all parameter sets lie in $\mathcal B$, then every coordinate set of the smallest solution lies in $\mathcal B$; when these operations are effective, so is the resulting description. In particular, if the parameters are finite unions of hemihedra---where a hemihedron is a bounded convex semi-linear set, equivalently a convex finite union of relative interiors of polytopes---then each coordinate set is a hemihedron and admits a quantifier-free semi-linear description. We apply this theory to lamination hulls. For \[ V=U\oplus\bigoplus_{i=1}^{k}W_i,\qquad \dim W_i=1,\qquad \Lambda=\bigcup_{i=1}^{k}(U+W_i), \] we prove that the lamination hull $G_{\Lambda}^{(\infty)}(S)$ of every finite $S\subset V$ is semi-algebraic and effectively computable by a quantifier-free formula over the reals.

math.CO↗

arXiv2024

GreedyML: A Parallel Algorithm for Maximizing Constrained Submodular Functions

We describe a parallel approximation algorithm for maximizing monotone submodular functions subject to hereditary constraints on distributed memory multiprocessors. Our work is motivated by the need to solve submodular optimization problems on massive data sets, for practical contexts such as data summarization, machine learning, and graph sparsification. Our work builds on the randomized distributed RandGreedi algorithm, proposed by Barbosa, Ene, Nguyen, and Ward (2015). This algorithm computes a distributed solution by randomly partitioning the data among all the processors and then employing \emph{a single} accumulation step in which all processors send their partial solutions to one processor. However, for large problems, the accumulation step exceeds the memory available on a processor, and the processor that performs the accumulation becomes a computational bottleneck. Hence we propose a generalization of the RandGreedi algorithm that employs multiple accumulation steps to reduce the memory required. We analyze the approximation ratio and the time complexity of the algorithm (in the BSP model). We evaluate the new GreedyML algorithm on three classes of problems, and report results from large-scale data sets with millions of elements. The results show that the GreedyML algorithm can solve problems where the sequential Greedy and distributed RandGreedi algorithms fail due to memory constraints. For certain computationally intensive problems, the GreedyML algorithm is faster than the RandGreedi algorithm. The observed approximation quality of the solutions computed by the GreedyML algorithm closely matches those obtained by the RandGreedi algorithm on these problems.

cs.DC↗

arXiv2019

Estimating Gaps in Martingales and Applications to Coin-Tossing: Constructions and Hardness

Consider designing a distributed coin-tossing protocol for n processors such that the probability of heads is X0 in [0,1], and an adversary can reset one processor to change the distribution of the final outcome. For X0=1/2, in the non-cryptographic setting, Blum's majority protocol is $\frac1{\sqrt{2\pi n}}$ insecure. For computationally bounded adversaries and any X0 in [0,1], the protocol of Moran,Naor,Segev (2009) is only O(1/n) insecure. In this paper, we study discrete-time martingales (X0,X1,..,Xn) such that Xi in [0,1], for all i in {0,..,n}, and Xn in {0,1}. In particular, for any X0 in [0,1], we construct martingales that yield $\frac12\sqrt{\frac{X_0(1-X_0)}{n}}$ insecure coin-tossing protocols with n-bit communication; irrespective of the number of bits required to represent the output distribution. Note that for sufficiently small X0, we achieve higher security than Moran et al's protocol even against computationally unbounded adversaries. For X0=1/2, our protocol requires only 40% of the processors to obtain the same security as the majority protocol. We introduce a new inductive technique that uses geometric transformations to estimate the large gaps in these martingales. For any X0 in [0,1], we show that there exists a stopping time $T$ such that $\mathbb{E}[|{X_T-X_{T-1}}|]\geq\frac2{\sqrt{2n-1}}\cdot X_0(1-X_0)$. The inductive technique also constructs martingales that demonstrate the optimality of our bound - we construct optimal martingales such that any $T$ has$\mathbb{E}[|{X_T-X_{T-1}}|]\leq\frac1{\sqrt{n}}\cdot\sqrt{X_0(1-X_0)}$. Our lower-bound holds for all X0 in [0,1]; while the previous bound of Cleve,Impagliazzo (1993) exists only for positive constant X0. Our approach only employs elementary techniques and avoids the complex probabilistic tools inherent to the approaches of Cleve,Impagliazzo (1993) and Beimel,Haitner,Makriyannis,Omri (2018).

cs.DM↗

arXiv2012

Limits of Random Oracles in Secure Computation

The seminal result of Impagliazzo and Rudich (STOC 1989) gave a black-box separation between one-way functions and public-key encryption: informally, a public-key encryption scheme cannot be constructed using one-way functions as the sole source of computational hardness. In addition, this implied a black-box separation between one-way functions and protocols for certain Secure Function Evaluation (SFE) functionalities (in particular, Oblivious Transfer). Surprisingly, however, {\em since then there has been no further progress in separating one-way functions and SFE functionalities} (though several other black-box separation results were shown). In this work, we present the complete picture for deterministic 2-party SFE functionalities. We show that one-way functions are black-box separated from {\em all such SFE functionalities}, except the ones which have unconditionally secure protocols (and hence do not rely on any computational hardness), when secure computation against semi-honest adversaries is considered. In the case of security against active adversaries, a black-box one-way function is indeed useful for SFE, but we show that it is useful only as much as access to an ideal commitment functionality is useful. Technically, our main result establishes the limitations of random oracles for secure computation.

cs.CR↗

arXiv2004

Towards Efficiently Solving Quantum Traveling Salesman Problem

We present a framework for efficiently solving Approximate Traveling Salesman Problem (Approximate TSP) for Quantum Computing Models. Existing representations of TSP introduce extra states which do not correspond to any permutation. We present an efficient and intuitive encoding for TSP in quantum computing paradigm. Using this representation and assuming a Gaussian distribution on tour-lengths, we give an algorithm to solve Approximate TSP (Euclidean) within BQP resource bounds. Generalizing this strategy for any distribution, we present an oracle based Quantum Algorithm to solve Approximate TSP. We present a realization of the oracle in the quantum counterpart of PP.

quant-ph↗