arXiv subjects

Chiara Marcolla

Publications and source records attributed to Chiara Marcolla.

10 recordsLinked to original sources

arXiv2025

Accurate BGV Parameters Selection: Accounting for Secret and Public Key Dependencies in Average-Case Analysis

The Brakerski-Gentry-Vaikuntanathan (BGV) scheme is one of the most significant fully homomorphic encryption (FHE) schemes. It belongs to a class of FHE schemes whose security is based on the presumed intractability of the Learning with Errors (LWE) problem and its ring variant (RLWE). Such schemes deal with a quantity, called noise, which increases each time a homomorphic operation is performed. Specifically, in order for the scheme to work properly, it is essential that the noise remains below a certain threshold throughout the process. For BGV, this threshold strictly depends on the ciphertext modulus, which is one of the initial parameters whose selection heavily affects both the efficiency and security of the scheme. For an optimal parameter choice, it is crucial to accurately estimate the noise growth, particularly that arising from multiplication, which is the most complex operation. In this work, we propose a novel average-case approach that precisely models noise evolution and guides the selection of initial parameters, improving efficiency while ensuring security. The key innovation of our method lies in accounting for the dependencies among ciphertext errors generated with the same key, and in providing general guidelines for accurate parameter selection that are library-independent.

cs.CR↗

arXiv2020

On the decoding of 1-Fibonacci error correcting codes

The study of new error correcting codes has raised attention in the last years, especially because of their use in cryptosystems that are resistant to attacks running on quantum computers. In 2006, while leaving a more in-depth analysis for future research, Stakhov gave some interesting ideas on how to exploit Fibonacci numbers to derive an original error correcting code with a compact representation. In this work we provide an explicit formula to compute the redundancy of Stakhov codes, we identify some flows in the initial decoding procedure described by Stakhov, whose crucial point is to solve some non-trivial Diophantine equations, and provide a detailed discussion on how to avoid solving such equations in some cases and on how to detect and correct errors more efficiently.

math.CO↗

arXiv2018

Minimum-weight codewords of the Hermitian codes are supported on complete intersections

Let $\mathcal{H}$ be the Hermitian curve defined over a finite field $\mathbb{F}_{q^2}$. In this paper we complete the geometrical characterization of the supports of the minimum-weight codewords of the algebraic-geometry codes over $\mathcal{H}$, started in [1]: if $d$ is the distance of the code, the supports are all the sets of $d$ distinct $\mathbb{F}_{q^2}$-points on $\mathcal{H}$ complete intersection of two curves defined by polynomials with prescribed initial monomials w.r.t. \texttt{DegRevLex}. For most Hermitian codes, and especially for all those with distance $d\geq q^2-q$ studied in [1], one of the two curves is always the Hermitian curve $\mathcal{H}$ itself, while if $d<q$ the supports are complete intersection of two curves none of which can be $\mathcal{H}$. Finally, for some special codes among those with intermediate distance between $q$ and $q^2-q$, both possibilities occur. We provide simple and explicit numerical criteria that allow to decide for each code what kind of supports its minimum-weight codewords have and to obtain a parametric description of the family (or the two families) of the supports. [1] C. Marcolla and M. Roggero, Hermitian codes and complete intersections, arXiv preprint arXiv:1510.03670 (2015).

math.AC↗

arXiv2018

Hermitian codes and complete intersections

In this paper we present a geometrical characterization for the minimum-weight codewords of the Hermitian codes over the fields $\mathbb{F}_{q^2}$ in the third and fourth phase, namely with distance $d \geq q^2-q$. We consider the unique writing $μq + λ(q+1)$ of the distance $d$ with $μ, λ$ non negative integers, and $μ\leq q$, and prove that the minimum-weight codewords correspond to complete intersection divisors cut on the Hermitian curve $\mathcal{H}$ by curves $\mathcal X$ of degree $μ+λ$ having $x^μy^λ$ as leading term w.r.t. the $\texttt{DegRevLex}$ term ordering (with $y>x$). Moreover, we show that any such curve $\mathcal X$ corresponds to minimum-weight codewords provided that the complete intersection divisor $\mathcal{H}\cap \mathcal X$ is made of simple $\mathbb{F}_{q^2}$-points. Finally, using this geometric characterization, we propose an algorithm to compute the number of minimum weight codewords and we present comparison tables between our algorithm and MAGMA command $\mathtt{MinimumWords}$.

math.AC↗

arXiv2016

Key-Policy Multi-Authority Attribute-Based Encryption

Bilinear groups are often used to create Attribute-Based Encryption (ABE) algorithms. In particular, they have been used to create an ABE system with multi authorities, but limited to the ciphertext-policy instance. Here, for the first time, we propose a multi-authority key-policy ABE system. In our proposal, the authorities may be set up in any moment and without any coordination. A party can simply act as an ABE authority by creating its own public parameters and issuing private keys to the users. A user can thus encrypt data choosing both a set of attributes and a set of trusted authorities, maintaining full control unless all his chosen authorities collude against him. We prove our system secure under the bilinear Diffie-Hellman assumption.

cs.CR↗

arXiv2015

Higher Hamming weights for locally recoverable codes on algebraic curves

We study the locally recoverable codes on algebraic curves. In the first part of this article, we provide a bound of generalized Hamming weight of these codes. Whereas in the second part, we propose a new family of algebraic geometric LRC codes, that are LRC codes from Norm-Trace curve. Finally, using some properties of Hermitian codes, we improve the bounds of distance proposed in [1] for some Hermitian LRC codes. [1] A. Barg, I. Tamo, and S. Vlladut. Locally recoverable codes on algebraic curves. arXiv preprint arXiv:1501.04904, 2015.

math.AC↗

arXiv2014

On the Hermitian curve and its intersections with some conics

We classify completely the intersections of the Hermitian curve with parabolas in the affine plane. To obtain our results we employ well-known algebraic methods for finite fields and geometric properties of the curve automorphisms. In particular, we provide explicit counting formulas that have also applications to some Hermitian codes.

math.AC↗

arXiv2012

On the Hermitian curve, its intersections with some conics and their applications to affine-variety codes and Hermitian codes

For any affine-variety code we show how to construct an ideal whose solutions correspond to codewords with any assigned weight. We classify completely the intersections of the Hermitian curve with lines and parabolas (in the $\mathbb{F}_{q^2}$ affine plane). Starting from both results, we are able to obtain geometric characterizations for small-weight codewords for some families of Hermitian codes over any $\mathbb{F}_{q^2}$. From the geometric characterization, we obtain explicit formulae. In particular, we determine the number of minimum-weight codewords for all Hermitian codes with $d\leq q$ and all second-weight codewords for distance-$3,4$ codes.

math.AC↗

arXiv2011

Improved decoding of affine-variety codes

General error locator polynomials are polynomials able to decode any correctable syndrome for a given linear code. Such polynomials are known to exist for all cyclic codes and for a large class of linear codes. We provide some decoding techniques for affine-variety codes using some multidimensional extensions of general error locator polynomials. We prove the existence of such polynomials for any correctable affine-variety code and hence for any linear code. We propose two main different approaches, that depend on the underlying geometry. We compute some interesting cases, including Hermitian codes. To prove our coding theory results, we develop a theory for special classes of zero-dimensional ideals, that can be considered generalizations of stratified ideals. Our improvement with respect to stratified ideals is twofold: we generalize from one variable to many variables and we introduce points with multiplicities.

math.AC↗

arXiv2011

On the geometry of small weight codewords of dual algebraic geometric codes

We investigate the geometry of the support of small weight codewords of dual algebraic geometric codes on smooth complete intersections by applying the powerful tools recently developed by Alain Couvreur. In particular, by restricting ourselves to the case of Hermitian codes, we recover and extend previous results obtained by the second named author joint with Marco Pellegrini and Massimiliano Sala.

math.AG↗