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Gilles Zémor

Publications and source records attributed to Gilles Zémor.

At least 37 records · Page 2Linked to original sources
arXiv2020

HQC-RMRS, an instantiation of the HQC encryption framework with a more efficient auxiliary error-correcting code

The HQC encryption framework is a general code-based encryption scheme for which decryption returns a noisy version of the plaintext. Any instantiation of the scheme will therefore use an error-correcting procedure relying on a fixed auxiliary code. Unlike the McEliece encryption framework whose security is directly related to how well one can hide the structure of an error-correcting code, the security reduction of the HQC encryption framework is independent of the nature of the auxiliary decoding procedure which is publicly available. What is expected from it is that the decoding algorithm is both efficient and has a decoding failure rate which can be easily modelized and analyzed. The original error-correction procedure proposed for the HQC framework was to use tensor products of BCH codes and repetition codes. In this paper we consider another code family for removing the error vector deriving from the general framework: the concatenation of Reed-Muller and Reed-Solomon codes. We denote this instantiation of the HQC framework by HQC-RMRS. These codes yield better decoding results than the BCH and repetition codes: overall we gain roughly 17\% in the size of the key and the ciphertext, while keeping a simple modelization of the decoding error rate. The paper also presents a simplified and more precise analysis of the distribution of the error vector output by the HQC protocol.

cs.CR↗
arXiv2020

Decodable quantum LDPC codes beyond the $\sqrt{n}$ distance barrier using high dimensional expanders

Constructing quantum LDPC codes with a minimum distance that grows faster than a square root of the length has been a major challenge of the field. With this challenge in mind, we investigate constructions that come from high-dimensional expanders, in particular Ramanujan complexes. These naturally give rise to very unbalanced quantum error correcting codes that have a large $X$-distance but a much smaller $Z$-distance. However, together with a classical expander LDPC code and a tensoring method that generalises a construction of Hastings and also the Tillich-Zemor construction of quantum codes, we obtain quantum LDPC codes whose minimum distance exceeds the square root of the code length and whose dimension comes close to a square root of the code length. When the ingredient is a 3-dimensional Ramanujan complex, we show that its 2-systole behaves like a square of the log of the complex size, which results in an overall quantum code of minimum distance $n^{1/2}\log n$, and sets a new record for quantum LDPC codes. When we use a 2-dimensional Ramanujan complex, or the 2-skeleton of a 3-dimensional Ramanujan complex, we obtain a quantum LDPC code of minimum distance $n^{1/2}\log^{1/2}n$. We then exploit the expansion properties of the complex to devise the first polynomial time algorithm that decodes above the square root barrier for quantum LDPC codes.

quant-ph↗
arXiv2020

Linear-time Erasure List-decoding of Expander Codes

We give a linear-time erasure list-decoding algorithm for expander codes. More precisely, let $r > 0$ be any integer. Given an inner code $C_0$ of length $d$, and a $d$-regular bipartite expander graph $G$ with $n$ vertices on each side, we give an algorithm to list-decode the expander code $C = C(G, C_0)$ of length $nd$ from approximately $δδ_r nd$ erasures in time $n \cdot \mathrm{poly}(d2^r / δ)$, where $δ$ and $δ_r$ are the relative distance and the $r$'th generalized relative distance of $C_0$, respectively. To the best of our knowledge, this is the first linear-time algorithm that can list-decode expander codes from erasures beyond their (designed) distance of approximately $δ^2 nd$. To obtain our results, we show that an approach similar to that of (Hemenway and Wootters, Information and Computation, 2018) can be used to obtain such an erasure-list-decoding algorithm with an exponentially worse dependence of the running time on $r$ and $δ$; then we show how to improve the dependence of the running time on these parameters.

cs.IT↗
arXiv2019

Low Rank Parity Check Codes: New Decoding Algorithms and Applications to Cryptography

We introduce a new family of rank metric codes: Low Rank Parity Check codes (LRPC), for which we propose an efficient probabilistic decoding algorithm. This family of codes can be seen as the equivalent of classical LDPC codes for the rank metric. We then use these codes to design cryptosystems à la McEliece: more precisely we propose two schemes for key encapsulation mechanism (KEM) and public key encryption (PKE). Unlike rank metric codes used in previous encryption algorithms -notably Gabidulin codes - LRPC codes have a very weak algebraic structure. Our cryptosystems can be seen as an equivalent of the NTRU cryptosystem (and also to the more recent MDPC \cite{MTSB12} cryptosystem) in a rank metric context. The present paper is an extended version of the article introducing LRPC codes, with important new contributions. We have improved the decoder thanks to a new approach which allows for decoding of errors of higher rank weight, namely up to $\frac{2}{3}(n-k)$ when the previous decoding algorithm only decodes up to $\frac{n-k}{2}$ errors. Our codes therefore outperform the classical Gabidulin code decoder which deals with weights up to $\frac{n-k}{2}$. This comes at the expense of probabilistic decoding, but the decoding error probability can be made arbitrarily small. The new approach can also be used to decrease the decoding error probability of previous schemes, which is especially useful for cryptography. Finally, we introduce ideal rank codes, which generalize double-circulant rank codes and allow us to avoid known structural attacks based on folding. To conclude, we propose different parameter sizes for our schemes and we obtain a public key of 3337 bits for key exchange and 5893 bits for public key encryption, both for 128 bits of security.

cs.IT↗
arXiv2018

Towards a function field version of Freiman's Theorem

We discuss a multiplicative counterpart of Freiman's $3k-4$ theorem in the context of a function field $F$ over an algebraically closed field $K$. Such a theorem would give a precise description of subspaces $S$, such that the space $S^2$ spanned by products of elements of $S$ satisfies $\dim S^2 \leq 3 \dim S-4$. We make a step in this direction by giving a complete characterisation of spaces $S$ such that $\dim S^2 = 2 \dim S$. We show that, up to multiplication by a constant field element, such a space $S$ is included in a function field of genus $0$ or $1$. In particular if the genus is $1$ then this space is a Riemann-Roch space.

math.NT↗
arXiv2017

RankSign: an efficient signature algorithm based on the rank metric

In this paper we propose a new approach to code-based signatures that makes use in particular of rank metric codes. When the classical approach consists in finding the unique preimage of a syndrome through a decoding algorithm, we propose to introduce the notion of mixed decoding of erasures and errors for building signature schemes. In that case the difficult problem becomes, as is the case in lattice-based cryptography, finding a preimage of weight above the Gilbert-Varshamov bound (case where many solutions occur) rather than finding a unique preimage of weight below the Gilbert-Varshamov bound. The paper describes RankSign: a new signature algorithm for the rank metric based on a new mixed algorithm for decoding erasures and errors for the recently introduced Low Rank Parity Check (LRPC) codes. We explain how it is possible (depending on choices of parameters) to obtain a full decoding algorithm which is able to find a preimage of reasonable rank weight for any random syndrome with a very strong probability. We study the semantic security of our signature algorithm and show how it is possible to reduce the unforgeability to direct attacks on the public matrix, so that no information leaks through signatures. Finally, we give several examples of parameters for our scheme, some of which with public key of size $11,520$ bits and signature of size $1728$ bits. Moreover the scheme can be very fast for small base fields.

cs.CR↗
arXiv2017

Linear-Time Maximum Likelihood Decoding of Surface Codes over the Quantum Erasure Channel

Surface codes are among the best candidates to ensure the fault-tolerance of a quantum computer. In order to avoid the accumulation of errors during a computation, it is crucial to have at our disposal a fast decoding algorithm to quickly identify and correct errors as soon as they occur. We propose a linear-time maximum likelihood decoder for surface codes over the quantum erasure channel. This decoding algorithm for dealing with qubit loss is optimal both in terms of performance and speed.

quant-ph↗
arXiv2017

Revisiting Kneser's Theorem for Field Extensions

A Theorem of Hou, Leung and Xiang generalised Kneser's addition Theorem to field extensions. This theorem was known to be valid only in separable extensions, and it was a conjecture of Hou that it should be valid for all extensions. We give an alternative proof of the theorem that also holds in the non-separable case, thus solving Hou's conjecture. This result is a consequence of a strengthening of Hou et al.'s theorem that is a transposition to extension fields of an addition theorem of Balandraud.

math.NT↗
arXiv2016

Efficient Encryption from Random Quasi-Cyclic Codes

We propose a framework for constructing efficient code-based encryption schemes from codes that do not hide any structure in their public matrix. The framework is in the spirit of the schemes first proposed by Alekhnovich in 2003 and based on the difficulty of decoding random linear codes from random errors of low weight. We depart somewhat from Aleknovich's approach and propose an encryption scheme based on the difficulty of decoding random quasi-cyclic codes. We propose two new cryptosystems instantiated within our framework: the Hamming Quasi-Cyclic cryptosystem (HQC), based on the Hamming metric, and the Rank Quasi-Cyclic cryptosystem (RQC), based on the rank metric. We give a security proof, which reduces the IND-CPA security of our systems to a decisional version of the well known problem of decoding random families of quasi-cyclic codes for the Hamming and rank metrics (the respective QCSD and RQCSD problems). We also provide an analysis of the decryption failure probability of our scheme in the Hamming metric case: for the rank metric there is no decryption failure. Our schemes benefit from a very fast decryption algorithm together with small key sizes of only a few thousand bits. The cryptosystems are very efficient for low encryption rates and are very well suited to key exchange and authentication. Asymptotically, for λthe security parameter, the public key sizes are respectively in $O(λ^{2})$ for HQC and in $O(λ^{4/3})$ for RQC. Practical parameter compares well to systems based on ring-LPN or the recent MDPC system.

cs.CR↗
arXiv2016

Perfectly Secure Message Transmission in Two Rounds

In the model that has become known as "Perfectly Secure Message Transmission"(PSMT), a sender Alice is connected to a receiver Bob through n parallel two-way channels. A computationally unbounded adversary Eve controls t of these channels, meaning she can acquire and alter any data that is transmitted over these channels. The sender Alice wishes to communicate a secret message to Bob privately and reliably, i.e. in such a way that Eve will not get any information about the message while Bob will be able to recover it completely. In this paper, we focus on protocols that work in two transmission rounds for n= 2t+1. We break from previous work by following a conceptually simpler blueprint for achieving a PSMT protocol. We reduce the previously best-known communication complexity, i.e. the number of transmitted bits necessary to communicate a 1-bit secret, from O(n^3 log n) to O(n^2 log n). Our protocol also answers a question raised by Kurosawa and Suzuki and hitherto left open: their protocol reaches optimal transmission rate for a secret of size O(n^2 log n) bits, and the authors raised the problem of lowering this threshold. The present solution does this for a secret of O(n log n) bits. Additionally, we show how our protocol can be adapted to a Network Coding context.

cs.CR↗
arXiv2016

LDA Lattices Without Dithering Achieve Capacity on the Gaussian Channel

This paper deals with Low-Density Construction-A (LDA) lattices, which are obtained via Construction A from non-binary Low-Density Parity-Check codes. More precisely, a proof is provided that Voronoi constellations of LDA lattices achieve the capacity of the AWGN channel under lattice encoding and decoding. This is obtained after showing the same result for more general Construction-A lattice constellations. The theoretical analysis is carried out in a way that allows to describe how the prime number underlying Construction A behaves as a function of the lattice dimension. Moreover, no dithering is required in the transmission scheme, simplifying some previous solutions of the problem. Remarkably, capacity is achievable with LDA lattice codes whose parity-check matrices have constant row and column Hamming weights. Some expansion properties of random bipartite graphs constitute an extremely important tool for dealing with sparse matrices and allow to find a lower bound of the minimum Euclidean distance of LDA lattices in our ensemble.

cs.IT↗
arXiv2015

Critical pairs for the Product Singleton Bound

We characterize Product-MDS pairs of linear codes, i.e.\ pairs of codes $C,D$ whose product under coordinatewise multiplication has maximum possible minimum distance as a function of the code length and the dimensions $\dim C, \dim D$. We prove in particular, for $C=D$, that if the square of the code $C$ has minimum distance at least $2$, and $(C,C)$ is a Product-MDS pair, then either $C$ is a generalized Reed-Solomon code, or $C$ is a direct sum of self-dual codes. In passing we establish coding-theory analogues of classical theorems of additive combinatorics.

cs.IT↗
arXiv2015

Quantum Expander Codes

We present an efficient decoding algorithm for constant rate quantum hypergraph-product LDPC codes which provably corrects adversarial errors of weight $Ω(\sqrt{n})$ for codes of length $n$. The algorithm runs in time linear in the number of qubits, which makes its performance the strongest to date for linear-time decoding of quantum codes. The algorithm relies on expanding properties, not of the quantum code's factor graph directly, but of the factor graph of the original classical code it is constructed from.

quant-ph↗
arXiv2015

Squares of Random Linear Codes

Given a linear code $C$, one can define the $d$-th power of $C$ as the span of all componentwise products of $d$ elements of $C$. A power of $C$ may quickly fill the whole space. Our purpose is to answer the following question: does the square of a code "typically" fill the whole space? We give a positive answer, for codes of dimension $k$ and length roughly $\frac{1}{2}k^2$ or smaller. Moreover, the convergence speed is exponential if the difference $k(k+1)/2-n$ is at least linear in $k$. The proof uses random coding and combinatorial arguments, together with algebraic tools involving the precise computation of the number of quadratic forms of a given rank, and the number of their zeros.

cs.IT↗
arXiv2013

A Construction of Quantum LDPC Codes from Cayley Graphs

We study a construction of Quantum LDPC codes proposed by MacKay, Mitchison and Shokrollahi. It is based on the Cayley graph of Fn together with a set of generators regarded as the columns of the parity-check matrix of a classical code. We give a general lower bound on the minimum distance of the Quantum code in $\mathcal{O}(dn^2)$ where d is the minimum distance of the classical code. When the classical code is the $[n, 1, n]$ repetition code, we are able to compute the exact parameters of the associated Quantum code which are $[[2^n, 2^{\frac{n+1}{2}}, 2^{\frac{n-1}{2}}]]$.

cs.IT↗
arXiv2013

Upper Bounds on the Size of Grain-Correcting Codes

In this paper, we re-visit the combinatorial error model of Mazumdar et al. that models errors in high-density magnetic recording caused by lack of knowledge of grain boundaries in the recording medium. We present new upper bounds on the cardinality/rate of binary block codes that correct errors within this model.

cs.DM↗
arXiv2012

Upper Bounds on the Rate of Low Density Stabilizer Codes for the Quantum Erasure Channel

Using combinatorial arguments, we determine an upper bound on achievable rates of stabilizer codes used over the quantum erasure channel. This allows us to recover the no-cloning bound on the capacity of the quantum erasure channel, R is below 1-2p, for stabilizer codes: we also derive an improved upper bound of the form : R is below 1-2p-D(p) with a function D(p) that stays positive for 0 < p < 1/2 and for any family of stabilizer codes whose generators have weights bounded from above by a constant - low density stabilizer codes. We obtain an application to percolation theory for a family of self-dual tilings of the hyperbolic plane. We associate a family of low density stabilizer codes with appropriate finite quotients of these tilings. We then relate the probability of percolation to the probability of a decoding error for these codes on the quantum erasure channel. The application of our upper bound on achievable rates of low density stabilizer codes gives rise to an upper bound on the critical probability for these tilings.

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