On improving security of GPT cryptosystems

The public key cryptosystem based on rank error correcting codes (the GPT cryptosystem) was proposed in 1991. Use of rank codes in cryptographic applications is advantageous since it is practically impossible to utilize combinatoric decoding. This enabled using public keys of a smaller size. Several attacks against this system were published, including Gibson's attacks and more recently Overbeck's attacks. A few modifications were proposed withstanding Gibson's attack but at least one of them was broken by the stronger attacks by Overbeck. A tool to prevent Overbeck's attack is presented in [12]. In this paper, we apply this approach to other variants of the GPT cryptosystem.Comment: 5 pages. submitted ISIT 2009.Processed on IEEE ISIT201

A Smart Approach for GPT Cryptosystem Based on Rank Codes

The concept of Public- key cryptosystem was innovated by McEliece's cryptosystem. The public key cryptosystem based on rank codes was presented in 1991 by Gabidulin -Paramonov-Trejtakov(GPT). The use of rank codes in cryptographic applications is advantageous since it is practically impossible to utilize combinatoric decoding. This has enabled using public keys of a smaller size. Respective structural attacks against this system were proposed by Gibson and recently by Overbeck. Overbeck's attacks break many versions of the GPT cryptosystem and are turned out to be either polynomial or exponential depending on parameters of the cryptosystem. In this paper, we introduce a new approach, called the Smart approach, which is based on a proper choice of the distortion matrix X. The Smart approach allows for withstanding all known attacks even if the column scrambler matrix P over the base field Fq.Comment: 5 pages. to appear in Proceedings of IEEE ISIT201

Характеристика генеалогической структуры стада маток абердин-ангусской породы по группам крови и ДНК-маркеру CASTUOGC282G

Analysis of the research results showed that the antigen’s occurrence frequency by loci was distributed with high variability from 0 to 75%. At locus C, the concentration of R1 antigen was the highest in the groups of cows of Bismarck sires - 60% and Design - 75%. In the group of cows of the Bismarck bull, antigens W’ were not detected in the C locus and antigen V in the F-V locus, and antigens B’, 0’ in the B locus and H” in the S’ locus were not detected in the group of cows of the Design bull. Comparative characteristics of the main breeding traits of cows of two groups of locus B of the blood group did not reveal a significant difference in their productivity. At the same time, cows of the 2nd group with a live weight of 629.3±17.6 kg were superior to their peers of the 1st group by 7.4% (Р≤0.05), in terms of height in the sacrum (133.80±0.88 cm) - by 3.2% (Р≤0.01). The revealed difference in the indicators of breeding traits between groups of cows is associated with the genetic dominance of the sire Design bull and its prepotency. According to the results of genotyping of animals of two eco-groups according to the allelic composition of the CASTUOGC282G gene, the superiority of homozygous genotypes CC and GG relative to heterozygous CG in the Bismarck group by 0.12 and 0.24 units, in the Design group - by 0.25 and 0.17 units. The uterus of the Bismarck bull with the GG allelic set was significantly heavier than the females of the CC genotype - by 3.8% (P≤0.05), CC relative to CG - by 5.1 (P≤0.01) and GG close to CG - by 2.3% (P≤0.001). Similarly, female representatives of Design with genotypes GG had a significant advantage by 8.3% (P≤0.01) and CG by 10.7% (P≤0.001) compared with peers of the CC genotype. The cows of the sire group Design 1015 of three genotypes - GG, CC and CG were heavier than the cows of the Bismarck 5682 group by 11.6 (P≤0.001); 4.6 (P≤0.01) and 12.4% (P≤0.001). At the same time, the lowest indicator of the coefficient of variability was the height in the sacrum in both groups for all genotypes (3.66–2.19%), which indicates the genetic predisposition of animals to inherit this selection trait in offspring.Анализ результатов исследований показал, что частота встречаемости антигена по локусам распределилась с высокой вариабельностью от 0 до 75 %. В локусе С концентрация антигена R1 была максимальной в группах коров быков-производителей Бисмарка – 60% и Дизайна – 75%. В группе коров быка Бисмарка не выявлены антигены W' в локусе С и в локусе F-V антиген V, а в группе коров быка-производителя Дизайна – антигены В', 0' в локусе В и H'' в локусе S'. Сравнительная характеристика основных селекционных признаков коров двух групп локуса B группы крови не выявила достоверной разницы в показателях их продуктивности. Вместе с тем коровы 2-й группы с живой массой 629,3±17,6 кг превосходили сверстниц 1-й группы на 7,4% (Р≤0,05), по высоте в крестце (133,80±0,88 см) – на 3,2% (Р≤0,01). Выявленная разница показателей селекционных признаков между группами коров связана с генетическим доминированием быка-производителя Дизайна и его препотентностью. По результатам генотипирования животных двух экогрупп по аллельному составу гена CASTUOGC282G установлено превосходство гомозиготных генотипов CC и GG относительно гетерозиготных CG по группе Бисмарка на 0,12 и 0,24 ед., по группе Дизайна – на 0,25 и 0,17 ед. Матки быка Бисмарка с аллельным набором GG были достоверно тяжелее соплеменниц генотипа CC – на 3,8% (Р≤0,05), CC относительно CG – на 5,1 (Р≤0,01) и GG относительно CG – на 2,3% (Р≤0,001). Аналогично представительницы производителя Дизайна с генотипам GG по сравнению со сверстницами генотипа СС достоверно имели преимущество на 8,3% (Р≤0,01) и CG – на 10,7% (Р≤0,001). Коровы группы производителя Дизайна 1015 трех генотипов – GG, CC и CG были тяжелее коров группы Бисмарка 5682 на 11,6 (Р≤0,001); 4,6 (Р≤0,01) и 12,4% (Р≤0,001). При этом самый низкий показатель коэффициента изменчивости был по высоте в крестце в обеих группах по всем генотипам (3,66–2,19%), что свидетельствует о генетической предрасположенности животных наследовать данный селекционный признак потомством

From Skew-Cyclic Codes to Asymmetric Quantum Codes

We introduce an additive but not $\mathbb{F}_4$-linear map $S$ from $\mathbb{F}_4^{n}$ to $\mathbb{F}_4^{2n}$ and exhibit some of its interesting structural properties. If $C$ is a linear $[n,k,d]_4$-code, then $S(C)$ is an additive $(2n,2^{2k},2d)_4$-code. If $C$ is an additive cyclic code then $S(C)$ is an additive quasi-cyclic code of index $2$. Moreover, if $C$ is a module $\theta$-cyclic code, a recently introduced type of code which will be explained below, then $S(C)$ is equivalent to an additive cyclic code if $n$ is odd and to an additive quasi-cyclic code of index $2$ if $n$ is even. Given any $(n,M,d)_4$-code $C$, the code $S(C)$ is self-orthogonal under the trace Hermitian inner product. Since the mapping $S$ preserves nestedness, it can be used as a tool in constructing additive asymmetric quantum codes.Comment: 16 pages, 3 tables, submitted to Advances in Mathematics of Communication

Two attacks on rank metric code-based schemes: RankSign and an Identity-Based-Encryption scheme

RankSign [GRSZ14a] is a code-based signature scheme proposed to the NIST competition for quantum-safe cryptography [AGHRZ17] and, moreover, is a fundamental building block of a new Identity-Based-Encryption (IBE) [GHPT17a]. This signature scheme is based on the rank metric and enjoys remarkably small key sizes, about 10KBytes for an intended level of security of 128 bits. Unfortunately we will show that all the parameters proposed for this scheme in [AGHRZ17] can be broken by an algebraic attack that exploits the fact that the augmented LRPC codes used in this scheme have very low weight codewords. Therefore, without RankSign the IBE cannot be instantiated at this time. As a second contribution we will show that the problem is deeper than finding a new signature in rank-based cryptography, we also found an attack on the generic problem upon which its security reduction relies. However, contrarily to the RankSign scheme, it seems that the parameters of the IBE scheme could be chosen in order to avoid our attack. Finally, we have also shown that if one replaces the rank metric in the [GHPT17a] IBE scheme by the Hamming metric, then a devastating attack can be found

An IND-CCA-Secure Code-Based EncryptionScheme Using Rank Metric

The use of rank instead of Hamming metric has been proposed to address the main drawback of code-based cryptography: large key sizes. There exist several Key Encapsulation Mechanisms (KEM) and Public Key Encryption (PKE) schemes using rank metric including some submissions to the NIST call for standardization of Post-Quantum Cryptography. In this work, we present an IND-CCA PKE scheme based on the McEliece adaptation to rank metric proposed by Loidreau at PQC 2017. This IND-CCA PKE scheme based on rank metric does not use a hybrid construction KEM + symmetric encryption. Instead, we take advantage of the bigger message space obtained by the different parameters chosen in rank metric, being able to exchange multiple keys in one ciphertext. Our proposal is designed considering some specific properties of the random error generated during the encryption. We prove our proposal IND-CCA-secure in the QROM by using a security notion called disjoint simulatability introduced by Saito et al. in Eurocrypt 2018. Moreover, we provide security bounds by using the semi-oracles introduced by Ambainis et al

Good Random Matrices over Finite Fields

The random matrix uniformly distributed over the set of all m-by-n matrices over a finite field plays an important role in many branches of information theory. In this paper a generalization of this random matrix, called k-good random matrices, is studied. It is shown that a k-good random m-by-n matrix with a distribution of minimum support size is uniformly distributed over a maximum-rank-distance (MRD) code of minimum rank distance min{m,n}-k+1, and vice versa. Further examples of k-good random matrices are derived from homogeneous weights on matrix modules. Several applications of k-good random matrices are given, establishing links with some well-known combinatorial problems. Finally, the related combinatorial concept of a k-dense set of m-by-n matrices is studied, identifying such sets as blocking sets with respect to (m-k)-dimensional flats in a certain m-by-n matrix geometry and determining their minimum size in special cases.Comment: 25 pages, publishe

Two attacks on rank metric code-based schemes: RankSign and an IBE scheme

International audienceRankSign [29] is a code-based signature scheme proposed to the NIST competition for quantum-safe cryptography [5] and, moreover , is a fundamental building block of a new Identity-Based-Encryption (IBE) [25]. This signature scheme is based on the rank metric and enjoys remarkably small key sizes, about 10KBytes for an intended level of security of 128 bits. Unfortunately we will show that all the parameters proposed for this scheme in [5] can be broken by an algebraic attack that exploits the fact that the augmented LRPC codes used in this scheme have very low weight codewords. Therefore, without RankSign the IBE cannot be instantiated at this time. As a second contribution we will show that the problem is deeper than finding a new signature in rank-based cryptography, we also found an attack on the generic problem upon which its security reduction relies. However, contrarily to the RankSign scheme, it seems that the parameters of the IBE scheme could be chosen in order to avoid our attack. Finally, we have also shown that if one replaces the rank metric in the [25] IBE scheme by the Hamming metric, then a devastating attack can be found

An Algebraic Approach for Decoding Spread Codes

In this paper we study spread codes: a family of constant-dimension codes for random linear network coding. In other words, the codewords are full-rank matrices of size (k x n) with entries in a finite field F_q. Spread codes are a family of optimal codes with maximal minimum distance. We give a minimum-distance decoding algorithm which requires O((n-k)k^3) operations over an extension field F_{q^k}. Our algorithm is more efficient than the previous ones in the literature, when the dimension k of the codewords is small with respect to n. The decoding algorithm takes advantage of the algebraic structure of the code, and it uses original results on minors of a matrix and on the factorization of polynomials over finite fields