arXiv subjects

Prashant Nalini Vasudevan

Publications and source records attributed to Prashant Nalini Vasudevan.

13 recordsLinked to original sources

arXiv2026

Sample-Preserving Search-to-Decision Reduction for Noisy Linear Equations over Large Moduli

The Noisy Linear Equations problem involves finding the solution to a random linear system over a finite field given noisy evaluations. This is a generalisation of various learning problems widely used in cryptography, including Learning with Errors (LWE), Learning with Rounding (LWR), and Learning Parity with Noise (LPN). We present a new sample-preserving search-to-decision reduction for Noisy Linear Equations that has complexity $\mathrm{poly}(n,m,\log q, q/σ)$, where $n$ is the dimension of the secret, $m$ is the number of samples, $q$ is the modulus, and $σ$ is the magnitude of the noise (defined suitably for the kind of noise involved). In particular, this implies search-to-decision reductions for LWE and LWR that run in time $\mathrm{poly}(n)$ even if the modulus $q$ is exponentially large, as long as the noise-to-modulus ratio $σ/q$ is non-negligible. Our reduction works for prime moduli $q$. If $q$ is composite, it instead produces a list that contains a small multiple of the solution. Prior to this work, known search-to-decision reductions for LWE and LWR either did not preserve the parameters of the problem (i.e., $m$, $n$, $σ$, and $q$), or had complexity that scaled polynomially with either $σ$ or the largest prime factor of $q$.

cs.CR↗

arXiv2026

Provably Subexponential Algorithms for Bounded-Error LWE with Few Samples

We show that a variant of the Wagner/BKW-style algorithm for LWE of Kirchner and Fouque [KF15] solves decision and search LWE with secrets of size $n$ over prime moduli $q=n^{κ+o(1)}$, for any fixed $κ>0$, with errors bounded by $\exp((\ln n)^{o(1)})$, in time $2^{O(n/\ln\ln n)}$ given $n+O(n/\ln\ln n)$ original samples. Further, if each coordinate of the secret is also bounded by $\exp((\ln n)^{o(1)})$, the same time bound holds with errors bounded by $n^{o(1)}$ and only $O(n/\ln\ln q)$ samples. Earlier, it was known that Kirchner and Fouque's algorithm could solve LWE with these parameters in subexponential time with a much larger number of samples $m = Ω(n\log{n})$ [HKM18]. Ducas et al. [DEL25] showed that an adaptation of this algorithm could solve the corresponding dual, the $\mathrm{SIS}^{\infty}$ problem, with a suitable norm bound $q/\operatorname{polylog}(n)$ and $m=n+ω(n/\ln\ln n)$ variables in subexponential time. Whereas their analysis involves interpreting the algorithm in terms of chains of lattices, our approach is quite elementary and is based on bounding the lower moments of a Fourier statistic.

cs.CR↗

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

Weak Zero-Knowledge and One-Way Functions

We study the implications of the existence of weak Zero-Knowledge (ZK) protocols for worst-case hard languages. These are protocols that have completeness, soundness, and zero-knowledge errors (denoted $ε_c$, $ε_s$, and $ε_z$, respectively) that might not be negligible. Under the assumption that there are worst-case hard languages in NP, we show the following: 1. If all languages in NP have NIZK proofs or arguments satisfying $ ε_c+ε_s+ ε_z < 1 $, then One-Way Functions (OWFs) exist. This covers all possible non-trivial values for these error rates. It additionally implies that if all languages in NP have such NIZK proofs and $ε_c$ is negligible, then they also have NIZK proofs where all errors are negligible. Previously, these results were known under the more restrictive condition $ ε_c+\sqrt{ε_s}+ε_z < 1 $ [Chakraborty et al., CRYPTO 2025]. 2. If all languages in NP have $k$-round public-coin ZK proofs or arguments satisfying $ ε_c+ε_s+(2k-1).ε_z < 1 $, then OWFs exist. 3. If, for some constant $k$, all languages in NP have $k$-round public-coin ZK proofs or arguments satisfying $ ε_c+ε_s+k.ε_z < 1 $, then infinitely-often OWFs exist.

cs.CR↗

arXiv2026

Improved Search-to-Decision Reduction for Random Local Functions

A random local function defined by a $d$-ary predicate $P$ is one where each output bit is computed by applying $P$ to $d$ randomly chosen bits of its input. These represent natural distributions of instances for constraint satisfaction problems. They were put forward by Goldreich as candidates for low-complexity one-way functions, and have subsequently been widely studied also as potential pseudo-random generators. We present a new search-to-decision reduction for random local functions defined by any predicate of constant arity. Given any efficient algorithm that can distinguish, with advantage $ε$, the output of a random local function with $m$ outputs and $n$ inputs from random, our reduction produces an efficient algorithm that can invert such functions with $\tilde{O}(m(n/ε)^2)$ outputs, succeeding with probability $Ω(ε)$. This implies that if a family of local functions is one-way, then a related family with shorter output length is family of pseudo-random generators. Prior to our work, all such reductions that were known required the predicate to have additional sensitivity properties, whereas our reduction works for any predicate. Our results also generalise to some super-constant values of the arity $d$, and to noisy predicates.

cs.CR↗

arXiv2025

Decoding Balanced Linear Codes With Preprocessing

Prange's information set algorithm is a decoding algorithm for arbitrary linear codes. It decodes corrupted codewords of any $\mathbb{F}_2$-linear code $C$ of message length $n$ up to relative error rate $O(\log n / n)$ in $\mathsf{poly}(n)$ time. We show that the error rate can be improved to $O((\log n)^2 / n)$, provided: (1) the decoder has access to a polynomial-length advice string that depends on $C$ only, and (2) $C$ is $n^{-Ω(1)}$-balanced. As a consequence we improve the error tolerance in decoding random linear codes if inefficient preprocessing of the code is allowed. This reveals potential vulnerabilities in cryptographic applications of Learning Noisy Parities with low noise rate. Our main technical result is that the Hamming weight of $Hw$, where $H$ is a random sample of *short dual* codewords, measures the proximity of a word $w$ to the code in the regime of interest. Given such $H$ as advice, our algorithm corrects errors by locally minimizing this measure. We show that for most codes, the error rate tolerated by our decoder is asymptotically optimal among all algorithms whose decision is based on thresholding $Hw$ for an arbitrary polynomial-size advice matrix $H$.

cs.CC↗

arXiv2025

Public-Key Encryption from the MinRank Problem

We construct a public-key encryption scheme from the hardness of the (planted) MinRank problem over uniformly random instances. This corresponds to the hardness of decoding random linear rank-metric codes. Existing constructions of public-key encryption from such problems require hardness for structured instances arising from the masking of efficiently decodable codes. Central to our construction is the development of a new notion of duality for rank-metric codes.

cs.CR↗

arXiv2024

On Wagner's k-Tree Algorithm Over Integers

The k-Tree algorithm [Wagner 02] is a non-trivial algorithm for the average-case k-SUM problem that has found widespread use in cryptanalysis. Its input consists of k lists, each containing n integers from a range of size m. Wagner's original heuristic analysis suggested that this algorithm succeeds with constant probability if n = m^{1/(\log{k}+1)}, and that in this case it runs in time O(kn). Subsequent rigorous analysis of the algorithm [Lyubashevsky 05, Shallue 08, Joux-Kippen-Loss 24] has shown that it succeeds with high probability if the input list sizes are significantly larger than this. We present a broader rigorous analysis of the k-Tree algorithm, showing upper and lower bounds on its success probability and complexity for any size of the input lists. Our results confirm Wagner's heuristic conclusions, and also give meaningful bounds for a wide range of list sizes that are not covered by existing analyses. We present analytical bounds that are asymptotically tight, as well as an efficient algorithm that computes (provably correct) bounds for a wide range of concrete parameter settings. We also do the same for the k-Tree algorithm over Z_m. Finally, we present experimental evaluation of the tightness of our results.

cs.CR↗

arXiv2023

Control, Confidentiality, and the Right to be Forgotten

Recent digital rights frameworks give users the right to delete their data from systems that store and process their personal information (e.g., the "right to be forgotten" in the GDPR). How should deletion be formalized in complex systems that interact with many users and store derivative information? We argue that prior approaches fall short. Definitions of machine unlearning Cao and Yang [2015] are too narrowly scoped and do not apply to general interactive settings. The natural approach of deletion-as-confidentiality Garg et al. [2020] is too restrictive: by requiring secrecy of deleted data, it rules out social functionalities. We propose a new formalism: deletion-as-control. It allows users' data to be freely used before deletion, while also imposing a meaningful requirement after deletion--thereby giving users more control. Deletion-as-control provides new ways of achieving deletion in diverse settings. We apply it to social functionalities, and give a new unified view of various machine unlearning definitions from the literature. This is done by way of a new adaptive generalization of history independence. Deletion-as-control also provides a new approach to the goal of machine unlearning, that is, to maintaining a model while honoring users' deletion requests. We show that publishing a sequence of updated models that are differentially private under continual release satisfies deletion-as-control. The accuracy of such an algorithm does not depend on the number of deleted points, in contrast to the machine unlearning literature.

cs.CR↗

arXiv2023

$k$-SUM in the Sparse Regime

In the average-case $k$-SUM problem, given $r$ integers chosen uniformly at random from $\{0,\dots,M-1\}$, the objective is to find a ``solution'' set of $k$ numbers that sum to $0$ modulo $M$. In the dense regime of $M \leq r^k$, where solutions exist with high probability, the complexity of these problems is well understood. Much less is known in the sparse regime of $M\gg r^k$, where solutions are unlikely to exist. In this work, we initiate the study of the sparse regime for $k$-SUM and its variant $k$-XOR, especially their planted versions, where a random solution is planted in a randomly generated instance and has to be recovered. We provide evidence for the hardness of these problems and suggest new applications to cryptography. Complexity. First we study the complexity of these problems in the sparse regime and show: - Conditional Lower Bounds. Assuming established conjectures about the hardness of average-case (non-planted) $k$-SUM/$k$-XOR when $M = r^k$, we provide non-trivial lower bounds on the running time of algorithms for planted $k$-SUM when $r^k\leq M\leq r^{2k}$. - Hardness Amplification. We show that for any $M \geq r^k$, if an algorithm running in time $T$ solves planted $k$-SUM/$k$-XOR with success probability $Ω(1/\text{polylog}(r))$, then there is an algorithm running in time $\tilde{O}(T)$ that solves it with probability $(1-o(1))$. - New Reductions and Algorithms. We provide reductions for $k$-SUM/$k$-XOR from search to decision, as well as worst-case and average-case reductions to the Subset Sum problem from $k$-SUM, as well as a new algorithm for average-case $k$-XOR at low densities. Cryptography. We show that by additionally assuming mild hardness of $k$-XOR, we can construct Public Key Encryption (PKE) from a weaker variant of the Learning Parity with Noise (LPN) problem than was known before.

cs.CC↗

arXiv2022

Deletion Inference, Reconstruction, and Compliance in Machine (Un)Learning

Privacy attacks on machine learning models aim to identify the data that is used to train such models. Such attacks, traditionally, are studied on static models that are trained once and are accessible by the adversary. Motivated to meet new legal requirements, many machine learning methods are recently extended to support machine unlearning, i.e., updating models as if certain examples are removed from their training sets, and meet new legal requirements. However, privacy attacks could potentially become more devastating in this new setting, since an attacker could now access both the original model before deletion and the new model after the deletion. In fact, the very act of deletion might make the deleted record more vulnerable to privacy attacks. Inspired by cryptographic definitions and the differential privacy framework, we formally study privacy implications of machine unlearning. We formalize (various forms of) deletion inference and deletion reconstruction attacks, in which the adversary aims to either identify which record is deleted or to reconstruct (perhaps part of) the deleted records. We then present successful deletion inference and reconstruction attacks for a variety of machine learning models and tasks such as classification, regression, and language models. Finally, we show that our attacks would provably be precluded if the schemes satisfy (variants of) Deletion Compliance (Garg, Goldwasser, and Vasudevan, Eurocrypt' 20).

cs.LG↗

arXiv2020

Formalizing Data Deletion in the Context of the Right to be Forgotten

The right of an individual to request the deletion of their personal data by an entity that might be storing it -- referred to as the right to be forgotten -- has been explicitly recognized, legislated, and exercised in several jurisdictions across the world, including the European Union, Argentina, and California. However, much of the discussion surrounding this right offers only an intuitive notion of what it means for it to be fulfilled -- of what it means for such personal data to be deleted. In this work, we provide a formal definitional framework for the right to be forgotten using tools and paradigms from cryptography. In particular, we provide a precise definition of what could be (or should be) expected from an entity that collects individuals' data when a request is made of it to delete some of this data. Our framework captures several, though not all, relevant aspects of typical systems involved in data processing. While it cannot be viewed as expressing the statements of current laws (especially since these are rather vague in this respect), our work offers technically precise definitions that represent possibilities for what the law could reasonably expect, and alternatives for what future versions of the law could explicitly require. Finally, with the goal of demonstrating the applicability of our framework and definitions, we consider various natural and simple scenarios where the right to be forgotten comes up. For each of these scenarios, we highlight the pitfalls that arise even in genuine attempts at implementing systems offering deletion guarantees, and also describe technological solutions that provably satisfy our definitions. These solutions bring together techniques built by various communities.

cs.CR↗

arXiv2017

On the Power of Statistical Zero Knowledge

We examine the power of statistical zero knowledge proofs (captured by the complexity class SZK) and their variants. First, we give the strongest known relativized evidence that SZK contains hard problems, by exhibiting an oracle relative to which SZK (indeed, even NISZK) is not contained in the class UPP, containing those problems solvable by randomized algorithms with unbounded error. This answers an open question of Watrous from 2002 [Aar]. Second, we "lift" this oracle separation to the setting of communication complexity, thereby answering a question of Göös et al. (ICALP 2016). Third, we give relativized evidence that perfect zero knowledge proofs (captured by the class PZK) are weaker than general zero knowledge proofs. Specifically, we exhibit oracles relative to which SZK is not contained in PZK, NISZK is not contained in NIPZK, and PZK is not equal to coPZK. The first of these results answers a question raised in 1991 by Aiello and Håstad (Information and Computation), and the second answers a question of Lovett and Zhang (2016). We also describe additional applications of these results outside of structural complexity. The technical core of our results is a stronger hardness amplification theorem for approximate degree, which roughly says that composing the gapped-majority function with any function of high approximate degree yields a function with high threshold degree.

cs.CC↗