arXiv subjects

Rishav Gupta

Publications and source records attributed to Rishav Gupta.

4 recordsLinked to original sources

arXiv2026

Towards Worst-case Hardness for Low-Noise LPN

The hardness of the Learning Parity with Noise (LPN) problem is a foundational assumption in cryptography, forming the basis of constructions ranging from symmetric-key primitives to public-key encryption and beyond. A central open question is whether the average-case hardness of LPN can be based on worst-case complexity assumptions, as has been achieved for the analogous Learning With Errors (LWE) problem. Existing worst-case-to-average-case reductions for LPN [BLVW19, YZ21] rely on statistical smoothing of linear codes, which inherently limits the resulting average-case hardness to noise rates as large as $1/2 - 1/\mathrm{poly}(n)$, which is insufficient for public-key applications. We explore a new approach towards obtaining such reductions: rather than requiring that random sparse combinations of the rows of the generator matrix of a code be statistically close to uniform, we only require that they be computationally indistinguishable from uniform. This leads to a clean win-win structure: we show that any efficient LPN solver can be transformed into a pair of efficient algorithms $(S, D)$ such that for every matrix $A$ of appropriate dimensions over $\mathbb{F}_2$, either $S$ decodes the code generated by $A$ from random noise, or $D$ distinguishes random noisy codewords of the dual of this code from uniform. By instantiating this reduction with appropriate parameters, we obtain the average-case hardness of LPN with inverse-polynomial noise rate $n^{-α}$ for any constant $α< 1$, assuming the worst-case simultaneous hardness of decoding a code from random noise and distinguishing random noisy codewords of its dual from uniform. In particular, setting $α= 1/2$, our reduction yields LPN hardness in the parameter regime required for Alekhnovich's construction of public-key encryption [Ale03], a regime that was previously inaccessible via worst-case reductions.

cs.CR↗

arXiv2026

Hardness Amplification for (Sparse) LPN

We prove new hardness amplification results for Learning Parity with Noise ($\mathsf{LPN}$) and its sparse variants. In $\mathsf{LPN}_{η,n,m}$, the goal is to recover a secret $\vec s\in\mathbb{F}_2^n$ from $m$ noisy linear samples $(\vec a,b)$, where $\vec a\leftarrow \mathbb{F}_2^n$ is uniform and $b=\langle \vec a,\vec s\rangle + e$ with $e\leftarrow \mathrm{Ber}(η)$. Building on the direct-product framework introduced by Hirahara and Shimizu [HS23], we show an 'instance-fraction amplification' theorem: for any $\varepsilon,δ>0$, any algorithm that solves $\mathsf{LPN}_{η,n,m}$ with success probability $\varepsilon$ can be transformed into an algorithm that succeeds with probability $1-δ$ on a related $\mathsf{LPN}$ distribution with scaled parameters $\mathsf{LPN}_{η/k,\;n/k,\;m}$, where $ k=Θ\!\left(\frac{1}δ\log\frac{1}{\varepsilon}\right). $ Equivalently, an algorithm that solves $\mathsf{LPN}$ on a 'small fraction of instances' can be converted into an algorithm that solves $\mathsf{LPN}$ on 'almost all instances', yielding a self-amplification for a wide range of parameters. We extend the same amplification approach to $\mathsf{LPN}$ over $\mathbb{F}_q$ and to Sparse-$\mathsf{LPN}$, where each query vector $\vec a$ has exactly $σ$ nonzero entries. Together, these results establish hardness self-amplification for a broad family of $\mathsf{LPN}$-type problems, strengthening the foundations for assuming the average-case hardness of $\mathsf{LPN}$ and its sparse variants.

cs.CR↗

arXiv2026

Tight Lower Bound for Approximating Parametrized Maximum Likelihood Decoding under ETH

We present a simple deterministic reduction which, assuming the Exponential Time Hypothesis ($\mathsf{ETH}$), yields tight lower bounds for approximating the parameterized Maximum Likelihood Decoding problem ($\mathsf{MLD}$) and the parameterized Nearest Codeword Problem ($\mathsf{NCP}$) within some fixed constant factor. Our starting point is the ETH-based exponential-time hardness of $(c,s)$-Gap-$\mathsf{MAXLIN}$ established in [BHI+24]. We transform a $(c,s)$-Gap-$\mathsf{MAXLIN}$ instance into an instance of $γ$-Gap $k$-$\mathsf{MLD}$ via a novel combinatorial object that we call a cover family. We provide both a randomized construction of the required cover families and a subsequent derandomization. Prior to our work, $n^{Ω(k)}$ hardness for constant-factor approximation was only shown under the randomized Gap Exponential Time Hypothesis Gap-$\mathsf{ETH}$ [Man20], which is a much stronger assumption than $\mathsf{ETH}$. Under $\mathsf{ETH}$, the strongest known lower bound was $n^{Ω(k/\operatorname{poly} \log k)}$ due to [BKM25]. Unlike previous approaches that rely on reductions from the hardness of approximating $2$-$\mathsf{CSP}$, our reduction provides a more direct and conceptually simpler route to achieving the optimal lower bounds.

cs.CC↗

arXiv2026

Mind the Gap? Not for SVP Hardness under ETH!

We prove new hardness results for fundamental lattice problems under the Exponential Time Hypothesis (ETH). Building on a recent breakthrough by Bitansky et al.\ \cite{BHIRW24}, who gave a polynomial-time reduction from $\mathsf{3SAT}$ to the (gap) $\mathsf{MAXLIN}$ problem-a class of CSPs with linear equations over finite fields-we derive ETH hardness for several lattice problems. First, we show that for any $p \in [1, \infty)$, there exists an explicit constant $γ> 1$ such that $\mathsf{CVP}_{p,γ}$ (the $\ell_p$-norm approximate Closest Vector Problem) does not admit a $2^{o(n)}$-time algorithm unless ETH is false. Our reduction is deterministic and proceeds via a direct reduction from (gap) $\mathsf{MAXLIN}$ to $\mathsf{CVP}_{p,γ}$. Our main contribution is a randomized ETH hardness result for $\mathsf{SVP}_{p,γ}$ (the $\ell_p$-norm approximate Shortest Vector Problem) for all $p \in (2, \infty)$. This result relies on a novel geometric property of the integer lattice $\mathbb{Z}^n$ in the $\ell_p$ norm, which says that for any $p \in (2, \infty)$, the number of lattice vectors close to $\frac{1}{2}\vec{1}_n$ (in the $\ell_p$ norm) is exponentially larger than the number of short vectors (namely those close to the origin). We establish this property via a new inequality for the Theta function, which we use to get a randomized reduction from $\mathsf{CVP}_{p,γ}$ to $\mathsf{SVP}_{p,γ'}$. Finally, we also use our ideas to give some minor improvements over prior reductions from $\mathsf{3SAT}$ to $\mathsf{BDD}_{p,α}$ (the Bounded Distance Decoding Problem), yielding better ETH hardness results for $\mathsf{BDD}_{p,α}$ for any $p \in [1, \infty)$ and $α> α_p^{\ddagger}$, where $α_p^{\ddagger}$ is an explicit threshold depending on $p$.

cs.CC↗