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Lifted Projective Reed-Solomon Codes [article]

Julien Lavauzelle
2018 arXiv   pre-print
Lifted Reed-Solomon codes, introduced by Guo, Kopparty and Sudan in 2013, are known as one of the few families of high-rate locally correctable codes. They are built through the evaluation over the affine space of multivariate polynomials whose restriction along any affine line can be interpolated as a low degree univariate polynomial. In this work, we give a formal definition of their analogues over projective spaces, and we study some of their parameters and features. Local correcting
more » ... ms are first derived from the very nature of these codes, generalizing the well-known local correcting algorithms for Reed-Muller codes. We also prove that the lifting of both Reed-Solomon and projective Reed-Solomon codes are deeply linked through shortening and puncturing operations. It leads to recursive formulae on their dimension and their monomial bases. We finally emphasize the practicality of lifted projective Reed-Solomon codes by computing their information sets and by providing an implementation of the codes and their local correcting algorithms.
arXiv:1809.00931v1 fatcat:brcnahpejjf2josfm3vngkcpn4

RAMESSES, a Rank Metric Encryption Scheme with Short Keys [article]

Julien Lavauzelle and Pierre Loidreau and Ba-Duc Pham
2019 arXiv   pre-print
We present a rank metric code-based encryption scheme with key and ciphertext sizes comparable to that of isogeny-based cryptography for an equivalent security level. The system also benefits from efficient encryption and decryption algorithms, which rely on linear algebra operations over finite fields of moderate sizes. The security only relies on rank metric decoding problems, and does not require to hide the structure of a code. Based on the current knowledge, those problems cannot be
more » ... ntly solved by a quantum computer. Finally, the proposed scheme admits a failure probability that can be precisely controlled and made as low as possible.
arXiv:1911.13119v1 fatcat:63cp4l7kyfcxtawkyxgwbawxi4

Lifted projective Reed–Solomon codes

Julien Lavauzelle
2018 Designs, Codes and Cryptography  
Lifted Reed-Solomon codes, introduced by Guo, Kopparty and Sudan in 2013, are known as one of the few families of high-rate locally correctable codes. They are built through the evaluation over the affine space of multivariate polynomials whose restriction along any affine line can be interpolated as a low degree univariate polynomial. In this work, we give a formal definition of their analogues over projective spaces, and we study some of their parameters and features. Local correcting
more » ... ms are first derived from the very nature of these codes, generalizing the well-known local correcting algorithms for Reed-Muller codes. We also prove that the lifting of both Reed-Solomon and projective Reed-Solomon codes are deeply linked through shortening and puncturing operations. It leads to recursive formulae on their dimension and their monomial bases. We finally emphasize the practicality of lifted projective Reed-Solomon codes by computing their information sets and by providing an implementation of the codes and their local correcting algorithms. Introduction Motivation and previous works. Locally decodable codes (LDC) and locally correctable codes (LCC) are codes equipped with a probabilistic algorithm which can efficiently decode or correct a single symbol of a noisy codeword, by querying only a few of its symbols. Low degree Reed-Muller codes define a well-known family of LDCs/LCCs with reasonable rate. Indeed, when restricted to an affine line, a sufficiently low-degree multivariate polynomial can be interpolated by a low-degree univariate polynomial. However, the rate R of such Reed-Muller codes stays stuck below 1/2. Multiplicity codes [KSY14] were the first family of codes breaking the R = 1/2 barrier for correcting a constant fraction of errors. The construction was based on a generalization of Reed-Muller codes which introduce multiplicities in the evaluation map. Shortly after the multiplicity codes breakthrough, Guo, Kopparty and Sudan [GKS13] proposed another generalization of Reed-Muller codes and considered all the multivariate polynomials (i.e. not only the low-degree ones) which can be interpolated as lowdegree univariate polynomials when restricted to a line. Surprisingly, it sometimes appears that much more polynomials satisfy this property than the low-degree ones lying in Reed-Muller codes. Resulting codes are named lifted Reed-Solomon codes, and in this work, more shortly referred to as affine lifted codes. Organisation. In this work, we show how to build analogues of these codes in projective spaces, that we call projective lifted codes. Our construction relies on the notion of degree sets which also appears in [GKS13] and helps us to exhibit relations between affine and projective We denote by F q the finite field with q elements, and by F × q its non-zero elements. For m ≥ 1, the affine space of dimension m is the set of m-tuples with coordinates in F q , and is denoted A m . We also define the projective space of dimension m as where for a, b ∈ A m+1 \ {0}, the relation ∼ is given by A projective point will be denoted a = (a 0 : · · · : a m ) ∈ P m . It has (q − 1) different representatives, and we call standard representative the only one such that ∀j < i, a j = 0 and a i = 1. The projective space P m contains θ m,q := q m+1 −1 q−1 distinct points. The hyperplane at infinity Π ∞ := {a ∈ P m , a 0 = 0} is isomorphic to P m−1 , and the bijective map (a 1 , . . . , a m ) → (1 : a 1 : · · · : a m ) embeds A m into P m . A projective line is a (q + 1)-subset of P m of the form L a,b := {xa + yb, (x : y) ∈ P 1 } for some distinct points a, b ∈ P m . The line L a,b is the only one containing both a and b, and there are exactly θ m−1,q = |P m−1 | = q m −1 q−1 projective lines on which a given point a ∈ P m lies. Polynomials and degrees. We denote by F q [X] := F q [X 1 , . . . , X m ] the ring of m-variate polynomials over F q . Following the terminology given in is called the set of degrees of f and is denoted Deg( f ). For a subset D ⊆ N m , we denote by Poly(D) the vector space of polynomials generated by monomials X d for d ∈ D: Some subsets D are of particular interest. For instance, for v ∈ N, • the 1-norm ball B m 1 (v) := {d ∈ N m , ∑ m i=1 d i ≤ v} generates the space F q [X] v of multivariate polynomials of total degree bounded by v, 1 see https://bitbucket.org/jlavauzelle/lifted_codes
doi:10.1007/s10623-018-0552-8 fatcat:swq7fs34ebcctaqhroxq7zvlt4

On the privacy of a code-based single-server computational PIR scheme [article]

Sarah Bordage, Julien Lavauzelle
2020 arXiv   pre-print
We show that the single-server computational PIR protocol proposed by Holzbaur, Hollanti and Wachter-Zeh in 2020 is not private, in the sense that the server can recover in polynomial time the index of the desired file with very high probability. The attack relies on the following observation. Removing rows of the query matrix corresponding to the desired file yields a large decrease of the dimension over F_q of the vector space spanned by the rows of this punctured matrix. Such a dimension
more » ... only shows up with negligible probability when rows unrelated to the requested file are deleted.
arXiv:2004.00509v1 fatcat:u7dvxbhmfraubf3roqjoxdtig4

Weighted Lifted Codes: Local Correctabilities and Application to Robust Private Information Retrieval [article]

Julien Lavauzelle, Jade Nardi
2019 arXiv   pre-print
Low degree Reed-Muller codes are known to satisfy local decoding properties which find applications in private information retrieval (PIR) protocols, for instance. However, their practical instantiation encounters a first barrier due to their poor information rate in the low degree regime. This lead the community to design codes with similar local properties but larger dimension, namely the lifted Reed-Solomon codes. However, a second practical barrier appears when one requires that the PIR
more » ... ocol resists collusions of servers. In this paper, we propose a solution to this problem by considering weighted Reed-Muller codes. We prove that such codes allow us to build PIR protocols with optimal computation complexity and resisting to a small number of colluding servers. In order to improve the dimension of the codes, we then introduce an analogue of the lifting process for weigthed degrees. With a careful analysis of their degree sets, we notably show that the weighted lifting of Reed-Solomon codes produces families of codes with remarkable asymptotic parameters.
arXiv:1904.08696v1 fatcat:rv33e32ijfadhl266nn2t6xrf4

Private Information Retrieval Schemes with Regenerating Codes [article]

Julien Lavauzelle, Razane Tajeddine, Ragnar Freij-Hollanti, Camilla Hollanti
2018 arXiv   pre-print
Lavauzelle is partially funded by French ANR-15-CE39-0013-01 "Manta". Indeed, under the constraint β = 1, a PM-MBR code is an [nd, B] linear code over Fq, where B = kd − k(k−1) 2 .  ... 
arXiv:1811.02898v2 fatcat:o3vfmluppnfhplt7x4ldoblg74

Generic constructions of PoRs from codes and instantiations

Julien Lavauzelle, Françoise Levy-dit-Vehel
2019 Journal of Mathematical Cryptology  
Lavauzelle and F. Levy-dit-Vehel, Generic constructions of PoRs from codes and instantiations  ... 
doi:10.1515/jmc-2018-0018 fatcat:wl6szkaxf5fyrfneop24p3osjq

Cryptanalysis of a System Based on Twisted Reed-Solomon Codes [article]

Julien Lavauzelle, Julian Renner
2019 arXiv   pre-print
It was recently proved that twisted Reed--Solomon codes represent a family of codes which contain a large amount of MDS codes, non-equivalent to Reed--Solomon codes. As a consequence, they were proposed as an alternative to Goppa codes for the McEliece cryptosystem, resulting to a potential reduction of key sizes. In this paper, an efficient key-recovery attack is given on this variant of the McEliece cryptosystem. The algorithm is based on the recovery of the structure of subfield subcodes of
more » ... wisted Reed--Solomon codes, and it always succeeds. Its correctness is proved, and it is shown that the attack breaks the system for all practical parameters in O(n^4) field operations. A practical implementation is also provided and retrieves a valid private key from the public key within just a few minutes, for parameters claiming a security level of 128 bits. We also discuss a potential repair of the scheme and an application of the attack to GPT cryptosystems using twisted Gabidulin codes.
arXiv:1904.11785v1 fatcat:p7svrf6jvbeklofshvqnr4hjyy

Rank-metric codes over arbitrary Galois extensions and rank analogues of Reed-Muller codes [article]

Daniel Augot, Alain Couvreur, Julien Lavauzelle, Alessandro Neri
2020 arXiv   pre-print
Lavauzelle is funded by French Direction Générale l'Armement, through the Pôle d'excellence cyber. A. Neri is funded by Swiss National Science Foundation, through grant no. 187711.  ... 
arXiv:2006.14489v1 fatcat:y3m5ati3dvhmdbdbydtbcpaqdm

New proofs of retrievability using locally decodable codes

Julien Lavauzelle, Francoise Levy-dit-Vehel
2016 2016 IEEE International Symposium on Information Theory (ISIT)  
Proofs of retrievability (PoR) are probabilistic protocols which ensure that a client can recover a file he previously stored on a server. Good PoRs aim at reaching an efficient tradeoff between communication complexity and storage overhead, and should be usable an unlimited number of times. We present a new unbounded-use PoR construction based on a class of locally decodable codes, namely the lifted codes of Guo et. al.. Our protocols feature sublinear communication complexity and very low
more » ... age overhead. Moreover, the various parameters can be tuned so as to minimize the communication complexity (resp. the storage overhead) according to the setting of concern.
doi:10.1109/isit.2016.7541611 dblp:conf/isit/LavauzelleL16 fatcat:h5uhi6tpp5bu5jtduvcprtlb6e

Voxelisation in the 3-D Fly Algorithm for PET

Zainab Ali Abbood, Julien Lavauzelle, Évelyne Lutton, Jean-Marie Rocchisani, Jean Louchet, Franck P. Vidal
2017 Swarm and Evolutionary Computation  
Peer reviewed version Cyswllt i'r cyhoeddiad / Link to publication Dyfyniad o'r fersiwn a gyhoeddwyd / Citation for published version (APA): Abstract The Fly Algorithm was initially developed for 3-D robot vision applications. It consists in solving the inverse problem of shape reconstruction from projections by evolving a population of 3-D points in space (the 'flies'), using an evolutionary optimisation strategy. Here, in its version dedicated to tomographic reconstruction in medical imaging,
more » ... the flies are mimicking radioactive photon sources. Evolution is controlled using a fitness function based on the discrepancy of the projections simulated by the flies with the actual pattern received by the sensors. The reconstructed radioactive concentration is derived from the population of flies, i.e. a collection of points in the 3-D Euclidean space, after convergence. 'Good' flies were previously binned into voxels. In this paper, we study which flies to include in the final solution and how this information can be sampled to provide more accurate datasets in a reduced computation time. We investigate the use of density fields, based on Metaballs and on Gaussian functions respectively, to obtain a realistic output. The spread of each Gaussian kernel is modulated in function of the corresponding fly fitness. The resulting volumes are compared with previous work in terms of normalised-cross correlation. In our test-cases, data fidelity increases by more than 10% when density fields are used instead of binning. Our method also provides reconstructions comparable to those obtained using well-established techniques used in medicine (filtered back-projection and ordered subset expectation-maximisation)
doi:10.1016/j.swevo.2017.04.001 fatcat:4pfugiyoujb4xfch7tre5y4oeu

New proofs of retrievability using locally decodable codes New Proofs of Retrievability using Locally Decodable Codes

Julien Lavauzelle, Françoise Levy-Dit-Vehel, Julien Lix, Inria Bâtiment, Alan Turing
2016 International Symposium on Information Theory ISIT 2016   unpublished
To cite this version: Julien Lavauzelle, Françoise Levy-Dit-Vehel. New proofs of retrievability using locally decodable codes.  ... 
fatcat:g742a7tj3zbglg7bo75ukgr264

Page 410 of The English Historical Review Vol. 11, Issue 42 [page]

1896 The English Historical Review  
Havet (Julien), G@uvres de. 2 vol. Pp. 456, 526. Paris: Leroux. 25 f. JasionskI (L.) Histoire de l'art militaire, accompagnée de morceaux choisis des grands écrivains militaires. Limoges: Lavauzelle.  ... 

Page 378 of Nuova Antologia, Revista Di Lettere, Scienze Ed Arti Vol. 193, Issue [page]

1904 Nuova Antologia, Revista Di Lettere, Scienze Ed Arti  
RI Hector Berlioz et la Société de son temps, par JULIEN TIERSOT. HacHETTE.  ...  CHARETON. — Henri Charles- Lavauzelle. Fr. 3.50. L’Evangile da Pauvre, par Mor. BauNARD. — V* Ch. Poussielgue. Fr 3.50. Le Japon d'aujourd'hui Etudes sociales par G. WeuLE&sse. — Colin. Fr. 4.  ... 

Page 50 of National Union Catalog Vol. 9, Issue [page]

1948 National Union Catalog  
HJ1091.G9 336.44 51-16918 Guyon, Julien, 1905- Troublants plaisirs see ,Guyon, Julien, 1905- joyeuse veillée, comé- die humoristique ..._ _D’aprés le roman Troublants plaisirs, St. ty — -Reyseouze 1947  ...  ou d ; des cou: darithmétique rédigés conformément & Vinstruction - érale du 27 novembre 1933 pour I’admission dans les écoles de sous-officiers éléves officiers de l’armée active. 7. éd. zi Charles-Lavauzelle  ... 
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