arXiv subjects

Aude Maignan

Publications and source records attributed to Aude Maignan.

8 recordsLinked to original sources

arXiv2026

Algorithms for the local and the global postage stamp problem

We consider stamps with different values (denominations) and same dimensions, and an envelope with a fixed maximum number of stamp positions. The local postage stamp problem is to find the smallest value that cannot be realized by the sum of the stamps on the envelope. The global postage stamp problem is to find the set of denominations that maximize that smallest value for a fixed number of distinct denominations. The local problem is NP-hard and we propose here a novel algorithm that improves on both the time complexity bound and the amount of required memory. We also propose a polynomial approximation algorithm for the global problem together with its complexity analysis. Finally we show that our algorithms allow to improve secure multi-party computations on sets via a more efficient homomorphic evaluation of polynomials on ciphered values.

cs.DS↗

arXiv2026

Fuzzy Private Set Union via Oblivious Key Homomorphic Encryption Retrieval

Private Set Multi-Party Computations are protocols that allow parties to jointly and securely compute functions: apart from what is deducible from the output of the function, the input sets are kept private. Then, a Private Set Union (PSU), resp. Intersection (PSI), is a protocol that allows parties to jointly compute the union, resp. the intersection, between their private sets. Now a structured PSI, is a PSI where some structure of the sets can allow for more efficient protocols. For instance in Fuzzy PSI, elements only need to be close enough, instead of equal, to be part of the intersection. We present in this paper, Fuzzy PSU protocols (FPSU), able to efficiently take into account approximations in the union. For this, we introduce a new efficient sub-protocol, called Oblivious Key Homomorphic Encryption Retrieval (OKHER), improving on Oblivious Key-Value Retrieval (OKVR) techniques in our setting. In the fuzzy context, the receiver set $X=\{x_i\}_{1..n}$ is replaced by ${\mathcal B}_δ(X)$, the union of $n$ balls of dimension $d$ with radius $δ$, centered at the $x_i$. The sender set is just its $m$ points of dimension $d$. Then the FPSU functionality corresponds to $X \sqcup \{y \in Y, y \notin {\mathcal B}_δ(X)\}$. Thus, we formally define the FPSU functionality and security properties, and propose several protocols tuned to the patterns of the balls using the $l_\infty$ distance. Using our OKHER routine and homomorphic encryption, we are for instance able to obtain a FPSU protocols with an asymptotic communication volume bound ranging from $O(dm\log(δ{n}))$ to $O(d^2m\log(δ^2n))$, depending on the receiver data set structure.

cs.CR↗

arXiv2025

Optimal Communication Unbalanced Private Set Union

We present new two-party protocols for the Unbalanced Private Set Union (UPSU) problem. Here, the Sender holds a set of data points, and the Receiver holds another (possibly much larger) set, and they would like for the Receiver to learn the union of the two sets and nothing else. Furthermore, the Sender's computational cost, along with the communication complexity, should be smaller when the Sender has a smaller set. While the UPSU problem has numerous applications and has seen considerable recent attention in the literature, our protocols are the first where the Sender's computational cost and communication volume are linear in the size of the Sender's set only, and do not depend on the size of the Receiver's set. Our constructions combine linearly homomorphic encryption (LHE) with fully homomorphic encryption (FHE). The first construction uses multi-point polynomial evaluation (MEv) on FHE, and achieves optimal linear cost for the Sender, but has higher quadratic computational cost for the Receiver. In the second construction we explore another trade-off: the Receiver computes fast polynomial Euclidean remainder in FHE while the Sender computes a fast MEv, in LHE only. This reduces the Receiver's cost to quasi-linear, with a modest increase in the computational cost for the Sender. Preliminary experimental results using HElib indicate that, for example, a Sender holding 1000 elements can complete our first protocol using less than 2s of computation time and less than 10MB of communication volume, independently of the Receiver's set size.

cs.CR↗

arXiv2023

VESPo: Verified Evaluation of Secret Polynomials

Proofs of Retrievability are protocols which allow a Client to store data remotely and to efficiently ensure, via audits, that the entirety of that data is still intact. Dynamic Proofs of Retrievability (DPoR) also support efficient retrieval and update of any small portion of the data.We propose a novel protocol for arbitrary outsourced data storage that achieves both low remote storage size and audit complexity.A key ingredient, that can be also of intrinsic interest, reduces to efficiently evaluating a secret polynomial at given public points, when the (encrypted) polynomial is stored on an untrusted Server.The Server performs the evaluations and also returns associated certificates. A Client can check that the evaluations are correct using the certificates and some pre-computed keys, more efficiently than re-evaluating the polynomial.Our protocols support two important features: the polynomial itself can be encrypted on the Server, and it can be dynamically updated by changing individual coefficients cheaply without redoing the entire setup.Our methods rely on linearly homomorphic encryption and pairings, and our implementation shows good performance for polynomial evaluations with millions of coefficients, and efficient DPoR with terabytes of data.For instance, for a 1TB database, compared to the state of art, we can reduce the Client storage by 5000x, communication size by 20x, and client-side audit time by 2x, at the cost of one order of magnitude increase in server-side audit time.

cs.CR↗

arXiv2022

Organic Structures Emerging From Bio-Inspired Graph-Rewriting Automata

Graph-Rewriting Automata (GRA) are an extension of Cellular Automata to a dynamic structure using local graph-rewriting rules. This work introduces linear algebra based tools that allow for a practical investigation of their behavior in deeply extended time scales. A natural subset of GRA is explored in different ways thereby demonstrating the benefits of this method. Some elements of the subset were discovered to create chaotic patterns of growth and others to generate organic-looking graph structures. These phenomena suggest a strong relevance of GRA in the modeling natural complex systems. The approach presented here can be easily adapted to a wide range of GRA beyond the chosen subset.

math.DS↗

arXiv2021

Dynamic proofs of retrievability with low server storage

Proofs of Retrievability (PoRs) are protocols which allow a client to store data remotely and to efficiently ensure, via audits, that the entirety of that data is still intact. A dynamic PoR system also supports efficient retrieval and update of any small portion of the data. We propose new, simple protocols for dynamic PoR that are designed for practical efficiency, trading decreased persistent storage for increased server computation, and show in fact that this tradeoff is inherent via a lower bound proof of time-space for any PoR scheme. Notably, ours is the first dynamic PoR which does not require any special encoding of the data stored on the server, meaning it can be trivially composed with any database service or with existing techniques for encryption or redundancy. Our implementation and deployment on Google Cloud Platform demonstrates our solution is scalable: for example, auditing a 1TB file takes just less than 5 minutes and costs less than $0.08 USD. We also present several further enhancements, reducing the amount of client storage, or the communication bandwidth, or allowing public verifiability, wherein any untrusted third party may conduct an audit.

cs.CR↗

arXiv2017

On the Planar Split Thickness of Graphs

Motivated by applications in graph drawing and information visualization, we examine the planar split thickness of a graph, that is, the smallest $k$ such that the graph is $k$-splittable into a planar graph. A $k$-split operation substitutes a vertex $v$ by at most $k$ new vertices such that each neighbor of $v$ is connected to at least one of the new vertices. We first examine the planar split thickness of complete graphs, complete bipartite graphs, multipartite graphs, bounded degree graphs, and genus-1 graphs. We then prove that it is NP-hard to recognize graphs that are $2$-splittable into a planar graph, and show that one can approximate the planar split thickness of a graph within a constant factor. If the treewidth is bounded, then we can even verify $k$-splittability in linear time, for a constant $k$.

math.CO↗

arXiv2017

Parallel Graph Rewriting with Overlapping Rules

We tackle the problem of simultaneous transformations of networks represented as graphs. Roughly speaking, one may distinguish two kinds of simultaneous or parallel rewrite relations over complex structures such as graphs: (i) those which transform disjoint subgraphs in parallel and hence can be simulated by successive mere sequential and local transformations and (ii) those which transform overlapping subgraphs simultaneously. In the latter situations, parallel transformations cannot be simulated in general by means of successive local rewrite steps. We investigate this last problem in the framework of overlapping graph transformation systems. As parallel transformation of a graph does not produce a graph in general, we propose first some sufficient conditions that ensure the closure of graphs by parallel rewrite relations. Then we mainly introduce and discuss two parallel rewrite relations over graphs. One relation is functional and thus deterministic, the other one is not functional for which we propose sufficient conditions which ensure its confluence.

cs.FL↗