McEliece Cryptosystem Research Articles

Effective implementation of modern McEliece cryptosystem on generalized (L,G)-codes

Subject of Research. The paper presents the study of methods and approaches to implementation of the modern McEliece cryptosystem based on separable generalized (L, G)-codes. Method. A method is proposed based on the analysis of the well-known public sources on implementation of the modern McEliece cryptosystem that uses the generalized (L, G)-codes with locators of degree greater than or equal to the second one without using an extended field. Mаin Results. Approaches to implementation of the modern McEliece cryptosystem based on the generalized (L, G)-codes are developed, namely: creation of a parity check matrix for the generalized (L, G)-code using a separable Goppa polynomial and locators of various degrees, description of an approach to the implementation of encryption and decryption of messages in the modern McEliece cryptosystem, presentation of the Chein’s procedure for numerators of degree greater than or equal to the second one without expanding the field. Practical Relevance. The proposed methods can be used in the development of cryptographic systems that can withstand attacks from quantum computers and ensure data confidentiality, as well as improve the security and performance of cryptosystems. Aerospace, automobile, railway, network multimedia, telecommunication and mobile information protection systems can also be the scope of the work results.

Estimating the Dimension of the Subfield Subcodes of Hermitian Codes

In this paper, we study the behavior of the true dimension of the subfield subcodes of Hermitian codes. Our motivation is to use these classes of linear codes to improve the parameters of the McEliece cryptosystem, suchas key size and security level. The McEliece scheme is one of the promising alternative cryptographic schemes to the current public key schemes since in the last four decades, they resisted all known quantum computing attacks. By computing and analyzing a data collection of true dimensions of subfield subcodes, we concluded that they can be estimated by the extreme value distribution function.

PKC‐PC: A variant of the McEliece public‐key cryptosystem based on polar codes

Polar codes are novel and efficient error-correcting codes with low encoding and decoding complexities. These codes have a channel-dependent generator matrix, which is determined by the code dimension, code length and transmission channel parameters. A variant of the McEliece public-key cryptosystem based on polar codes, called the PKC-PC, is studied. Since the structure of the polar codes’ generator matrix depends on the parameters of the channel, the authors have used an efficient approach to conceal their generator matrix. The proposed approach is based on a random selection of rows of the matrix by which a random generator matrix is constructed. Using the characteristics of polar codes and introducing an efficient approach, they could reduce the public and secret key sizes, and computational complexity compared to the McEliece cryptosystem. Moreover, they show that PKC-PC yields an increased security level against conventional attacks as well as possible vulnerabilities to the code-based public-key cryptosystems. Furthermore, they prove the security of the authors’ cryptosystem and show that its security is reduced to solve NP-complete problems, called polar parameterised syndrome decoding and polar parameterised codeword existence.

Randomized symmetric McEliece cryptosystem based on generalized Reed-Solomon codes

One of the actual problems of modern cryptography is the design of practical post-quantum cryptosystems whose security is based on the complexity of solving one computationally challenging problem, in the same way as the security of the RSA cryptosystem is based on the complexity of integer factorization. A promising class of such cryptosystems is formed by code-based cryptosystems the first asymmetric of which is the McEliece cryptosystem. The purpose of this work is to design and research a symmetric version of the McEliece cryptosystem based on generalized Reed-Solomon codes. These codes were chosen because they exist for all natural values of the parameters (the length and dimension of the code) and they are maximal distance separable allowing a wide range to change the characteristics of the relevant cryptosystems. In addition, very fast decoding algorithms are known for these codes. Asymmetric cryptosystems based on the generalized Reed-Solomon codes are not secure because for them there are efficient algorithms for recovering private keys from public keys.A symmetric code cryptosystem is proposed that is more efficient (in terms of the length of the secret key for given security requirements) compared to the LPN-C cryptosystem. An estimate of the security of the proposed cryptosystem relative to an attack with the chosen plaintext is obtained and an algorithm for selecting parameters for constructing this cryptosystem is proposed. The proposed cryptosystem is compared with the LPN-C cryptosystem along the key length for a given lower limit of security to the attack in question.

A centralized key management scheme for space network with resistance of nonlinear channel noise

The channel noise in space is nonlinear and pseudorandom so that the efficiency and security of existing group key management schemes are constrained seriously. To solve these problems, we proposed a centralized and identity-based key management scheme by using McEliece public key cryptosystem. In this scheme, the node identity is used as the parameter to generate the public key. Thus the authentication can be embedded into the verification of the public key. The group key is distributed with the protection of public key so that it can be implemented safely. Furthermore, the error correction capacity provided by McEliece public cryptosystem can eliminate the disturbance of noise. It transfers the negative influence caused by pseudorandom noise to an enhancement of security and increases the efficiency of the group key distribution over the noisy channel. The security of public key generation, forward secrecy and backward secrecy is analyzed. The performance is analyzed and compared with other schemes. The error correction capacity is simulated. The results show that our scheme can provide confidentiality, integrity, authentication, non-repudiation, failure tolerance and error correction with lower computation overhead and interaction rounds.

Cryptanalysis of a system based on twisted Reed–Solomon codes

Twisted Reed-Solomon (TRS) codes are a family of codes that contains a large number of maximum distance separable codes that are non-equivalent to Reed--Solomon codes. TRS codes were recently proposed as an alternative to Goppa codes for the McEliece code-based cryptosystem, resulting in a potential reduction of key sizes. The use of TRS codes in the McEliece cryptosystem has been motivated by the fact that a large subfamily of TRS codes is resilient to a direct use of known algebraic key-recovery methods. In this paper, an efficient key-recovery attack on the TRS variant that was used in the McEliece cryptosystem is presented. The algorithm exploits a new approach based on recovering the structure of a well-chosen subfield subcode of the public code. It is proved that the attack always succeeds and breaks the system for all practical parameters in $O(n^4)$ field operations. A software implementation of the algorithm retrieves a valid private key from the public key within a few minutes, for parameters claiming a security level of 128 bits. The success of the attack also indicates that, contrary to common beliefs, subfield subcodes of the public code need to be precisely analyzed when proposing a McEliece-type code-based cryptosystem. Finally, the paper discusses an attempt to repair the scheme and a modification of the attack aiming at Gabidulin-Paramonov-Tretjakov cryptosystems based on twisted Gabidulin codes.

Single Trace Analysis against HyMES by Exploitation of Joint Distributions of Leakages

Beginning with the proposal of the McEliece cryptosystem in 1978, code-based cryptography has positioned itself as one of main categories in post-quantum cryptography (PQC). To date, the algebraic security of certain variants of McEliece cryptosystems has been challenged many times, although some of the variants have remained secure. However, recent studies on code-based cryptography have focused on the side-channel resistance since previous studies have indicated that the existing algorithms were vulnerable to side-channel analysis. In this paper, we propose the first side-channel attack on the Hybrid McEliece Scheme (HyMES) using only a single power consumption trace. HyMES is a variant of the McEliece system that provides smaller keys, along with faster encryption and decryption speed. By exploiting joint distributions of nonlinear functions in the decryption process, we were able to recover the private key of HyMES. To the best of our knowledge, this is the first work proposing a side-channel analysis based on a joint distribution of the leakages on the public-key system.

Connecting the Dots: Privacy Leakage via Write-Access Patterns to the Main Memory

Data-dependent access patterns of an application to an untrusted storage system are notorious for leaking sensitive information about the user's data. Previous research has shown how an adversary capable of monitoring both read and write requests issued to the memory can correlate them with the application to learn its sensitive data. However, information leakage through only the write access patterns is less obvious and not well studied in the current literature. In this work, we demonstrate an actual attack on power-side-channel resistant Montgomery's ladder based modular exponentiation algorithm commonly used in public key cryptography. We infer the complete 512-bit secret exponent in ~ 3.5 minutes by virtue of just the write access patterns of the algorithm to the main memory. In order to learn the victim algorithm's write access patterns under realistic settings, we exploit a compromised DMA device to take frequent snapshots of the application's address space, and then run a simple differential analysis on these snapshots to find the write access sequence. The attack has been shown on an Intel Core(TM) i7-4790 3.60GHz processor based system. We further discuss a possible attack on McEliece public-key cryptosystem that also exploits the write-access patterns to learn the secret key.

Структура конечной групповой алгебры одного полупрямого произведения абелевых групп и её приложения

В 1978 году Р. Мак-Элисом построена первая асимметричная кодовая криптосистема, основанная на применении помехоустойчивых кодов Гоппы, при этом эффективные атаки на секретный ключ этой криптосистемы до сих пор не найдены. К настоящему времени известно много криптосистем, основанных на теории помехоустойчивого кодирования. Одним из способов построения таких криптосистем является модификация криптосистемы Мак-Элиса с помощью замены кодов Гоппы на другие классы кодов. Однако, известно что криптографическая стойкость многих таких модификаций уступает стойкости классической криптосистемы Мак-Элиса. В связи с развитием квантовых вычислений кодовые криптосистемы, наряду с криптосистемамми на решётках, рассматриваются как альтернатива теоретико-числовым. Поэтому актуальна задача поиска перспективных классов кодов, применимых в криптографии. Представляется, что для этого можно использовать некоммутативные групповые коды, т.е. левые идеалы в конечных некоммутативных групповых алгебрах.Для исследования некоммутативных групповых кодов полезной является теорема Веддерберна, доказывающая существование изоморфизма групповой алгебры на прямую сумму матричных алгебр. Однако конкретный вид слагаемых и конструкция изоморфизма этой теоремой не определены, и поэтому для каждой группы стоит задача конструктивного описания разложения Веддерберна. Это разложение позволяет легко получить все левые идеалы групповой алгебры, т.е. групповые коды. В работе рассматривается полупрямое произведение $$Q_{m,n} = (\mathbb{Z}_m \times \mathbb{Z}_n) \leftthreetimes (\mathbb{Z}_2 \times \mathbb{Z}_2)$$ абелевых групп и конечная групповая алгебра $$\mathbb{F}_q Q_{m,n}$$ этой группы. Для этой алгебры при условиях $$n \mid q -1$$ и $$\text{НОД}(2mn, q) = 1$$ построено разложение Веддербёрна. В случае поля чётной характеристики, когда эта групповая алгебра не является полупростой, также получена сходная структурная теорема. Описаны все неразложимые центральные идемпотенты этой групповой алгебры. Полученные результаты используются для алгебраического описания всех групповых кодов над $$Q_{m,n}.$$

Cryptosystem design based on Hermitian curves for IoT security

The ultimate goal of modern cryptography is to protect the information resource and make it absolutely unbreakable and beyond compromise. However, throughout the history of cryptography, thousands of cryptosystems emerged and believed to be invincible and yet attackers were able to break and compromise their security. The main objective of this paper is to design a robust cryptosystem that will be suitable to be implemented in Internet of Things. The proposed cryptosystem is based on algebraic geometric curves, more specifically on Hermitian curves. The new cryptosystem design is called Hermitian-based cryptosystem (HBC). During the development of the HBC design, Kerckhoffs’s desideratum was the main guidance principle, which has been satisfied by choosing the Hermitian curves as the core of the proposed design. The proposed HBC inherits all the advantageous characteristics of Hermitian curve which are large number of points that satisfy the curve and high genus curves. The aforementioned characteristics play a crucial role in generating a large size encryption key for HBC and determine the block size of plaintext. Due to the fact that HBC used algebraic geometric codes over Hermitian curve, it has the ability to perform error correction in addition to data encryption. The error correction is another advantage of HBC compared with many existing cryptosystems such as McEliece cryptosystem. The number of errors that can be corrected by HBC is larger (high data rate) than other algebraic geometric codes such as elliptic and hyperelliptic curves. It also uses non-binary representation which increases its attack resistance. In this paper, the proposed HBC has been mathematically compared with elliptic curve cryptosystem. The results show that HBC has many advantages over the elliptic curves in terms of number of points and genus of the curve.

Encryption scheme based on expanded Reed-Solomon codes

We present a code-based public-key cryptosystem, in which we use Reed-Solomon codes over an extension field as secret codes and disguise it by considering its shortened expanded code over the base field. Considering shortened expanded codes provides a safeguard against distinguisher attacks based on the Schur product. Moreover, without using a cyclic or a quasi-cyclic structure we obtain a key size reduction of nearly $45 \%$ compared to the classic McEliece cryptosystem proposed by Bernstein et al.

McEliece cryptosystem: simulation and security vulnerabilities

While symmetric key cryptosystems utilise single private key at both ends for encryption and decryption purposes and this mentioned scenario gives advantages of better speed-up as compare to public key cryptosystems. Still, security is a challenge while performing compatibly efficient key distribution and secure private data transfer among entities in an untrusted environment. McEliece cryptosystem, designed in 1978, is a public key based cryptosystem whose security is based on some unknown natured error-correcting codes. This paper gives state-of-the-art detailed overview on this cryptosystem, its component-wise algorithmic description and implementation. Various attacks on McEliece cryptosystem are discussed separately. The experimental results employing Goppa codes are also presented in the paper where the simulations are performed on various extension degrees. Based on the simulations performed, we concluded the results along with the various issues faced while implementation.

A Centralized Key Management Scheme Based on McEliece PKC for Space Network

Constrained by the limited resource, high-latency and high bit error rate, the existing group key management schemes for the space network are inefficient. To solve this problem, we proposed a centralized and identity-based key management scheme by using McEliece public-key cryptosystem (PKC). In this scheme, the node identity is used as the parameter to generate the public key. Therefore, the authentication can be embedded into the verification of the public key without needing the PKI. The group key is distributed with the protection of the public key so that the group key management scheme can be implemented safely. Furthermore, the McEliece public-key cryptosystem can resist the quantum attack and provide error correction capacity. It improves the efficiency of the group key distribution over the noisy channel. The proposed key management scheme is simulated on OPNET. The security of public-key generation, forward secrecy, backward secrecy and performance are analyzed. The results show that our scheme can provide confidentiality, integrity, authentication, non-repudiation, failure tolerance and error correction. In addition, the computation overhead and rounds of interaction are lower than former work.

Public-key cryptosystem based on quantum BCH codes and its quantum digital signature

There is a security threat to the traditional cryptography because of the emergence of quantum computation. Quantum computers can break through many cryptosystems based on mathematical puzzles. The post-cryptography has been established for the sake of the security. It has been proved that McEliece public-key cryptosystem based on error-correcting codes can resist quantum attacks. In this paper, several new generating algorithms of the quantum Bose-Chaudhuri-Hocquenghem (BCH) codes have been proposed, and the quantum BCH codes can correct the errors including bit-flip and phase-flip. At the same time, the quantum McEliece public-key cryptosystem and quantum Niederreiter public key cryptosystem have been designed by us on the basis of quantum BCH codes in this paper, and their encryption and decryption procedures are created in detail. Our cryptosystem not only retains the advantages of postquantum computation, but also can encrypt or decrypt in quantum state. From the perspective of computational complexity theory, both cryptosystems can resist the attacks of Shor algorithm and Grover algorithm very well. The Niederreiter classical digital signature is designed by us according to the structure characteristics of quantum BCH codes.

McEliece cryptosystem: simulation and security vulnerabilities

McEliece cryptosystem: simulation and security vulnerabilities

Advanced code-based electronic digital signature scheme

The article is devoted to the study and research of the properties of code-based cryptosystems. They provide a high level of security even in the conditions of quantum cryptographic analysis, i.e. belong to the new generation of cryptosystems for post-quantum application. The main disadvantage of the known code-based digital signature schemes is the long time to generate a signature. This is due to the large number of attempts to decode a randomly generated vector (which is interpreted as a syndrome vector). The high complexity of such a procedure requires the search for new mechanisms and algorithms that would accelerate the formation of code-base electronic signatures. The article presents the results of two research vectors. First, we propose a new code-based digital signature scheme on the use of a one-way function from the classical McEliece cryptosystem and not only provides a proper level of resistance to classical cryptanalysis and cryptanalysis using quantum computers, but also, compared to known alternatives, provides protection against special types of attacks, such as simultaneous counterfeit attacks. Quantitative estimates of the reliability and speed of the new cryptographic algorithm, which were obtained by experimental verification on the BCH codes, are also given. The second vector of research concerns the study of a new direction, which is associated with the modification of the decoder by artificially increasing the corrective code ability. Thanks to the improved decoder scheme, we can significantly reduce the generation time of signatures. The paper confirms the effectiveness of the proposed decoder modification in the implementation of a new digital signature scheme in comparison with the classic Peterson-Gorenstein-Zierler decoder in the context of comparing the speed of signature formation and the number of required decoding attempts.

CODE-BASED HYBRID CRYPTOSYSTEM: COMPARATIVE STUDIES AND ANALYSIS OF EFFICIENCY

In this paper the basic principles of construction and operation of McEliece and Niederreiter cryptosystems based on the use of error-correcting codes were considered. A new hybrid cryptosystem that combines the rules of encryption according to the above-mentioned schemes is proposed. Also, this paper presents the analysis and comparative studies from the standpoint of stability, the volume of public and private keys, length of ciphertext and relative speed of information transmission of the new proposed scheme and McEliece and Niederreiter cryptosystems. It is considered from an analytical point of view and with the help of graphic images. Comparative studies revealed that the hybrid cryptosystem retains the positive aspects of its predecessors, as well as allows us to increase the relative transmission rate with the preservation of the stability indicator to the classical and quantum cryptanalysis. One disadvantage is the increase in decoding time by adding information extracted as in Niederreiter scheme, but the increase in this indicator is not critical. Despite the demonstrated benefits, it remains open to all cryptosystems to reduce the amount of the used key data, which, in the case of quantum computers to maintain stability, still needs to be increased once.

Computer Algebra Tales on Goppa Codes and McEliece Cryptography

The 40-year old McEliece public-key crypto-system is revisited with the help of recently developed resources: an improved Peterson–Gorenstein–Zierler decoder for alternant error-correcting codes; PyECC, a purely Python CAS; a package of PyECC functional utilities for the computations involved in defining, coding and decoding error-correcting codes; and a Web page with free-access to companion materials. The last two tales trace the evolution of the McEliece systems and provide an account about their current security levels.

On the Support Splitting Algorithm for Induced Codes

As shown by N. Sendrier in 2000, if a $$[n{\text{,}}\,k{\text{,}}\,d]$$-linear code $$C( \subseteq \mathbb{F}_{q}^{n})$$ with length $$n$$, dimensionality $$k$$ and code distance $$d$$ has a trivial group of automorphisms $${\text{PAut}}(C)$$, it allows one to construct a determined support splitting algorithm in order to find a permutation $$\sigma $$ for a code $$D$$, being permutation-equivalent to the code $$C$$, such that $$\sigma (C) = D$$. This algorithm can be used for attacking the McEliece cryptosystem based on the code$$C$$. This work aims the construction and analysis of the support splitting algorithm for the code $$\mathbb{F}_{q}^{l} \otimes C$$, induced by the code $$C$$, $$l \in \mathbb{N}$$. Since the group of automorphisms PAut$$(\mathbb{F}_{q}^{l} \otimes C)$$ is nontrivial even in the case of that trivial for the base code $$C$$, it enables one to assume a potentially high resistance of the McEliece cryptosystem on the code $$\mathbb{F}_{q}^{l} \otimes C$$ to the attack based on a carrier split. The support splitting algorithm is being constructed for the code $$\mathbb{F}_{q}^{l} \otimes C$$ and its efficiency is compared with the attack to a McEliece cryptosystem based on the code $$\mathbb{F}_{q}^{l} \otimes C.$$

Codes in a Dihedral Group Algebra

In 1978, Robert McEliece constructed the first asymmetric code-based cryptosystem using noise-immune Goppa codes; no effective key attacks has been described for it yet. By now, quite a lot of code-based cryptosystems are known; however, their cryptographic security is inferior to that of the classical McEliece cryptosystem. In connection with the development of quantum computing, code-based cryptosystems are considered as an alternative to number theoretical ones; therefore, the problem of seeking promising classes of codes to construct new secure code-based cryptosystems is relevant. For this purpose, noncommutative codes can be used, that is, ideals in group algebras $${{\mathbb{F}}_{q}}G$$ over finite noncommutative groups $$G$$. The security of cryptosystems based on codes induced by subgroup codes has been studied earlier. The Artin–Wedderburn theorem, which proves the existence of an isomorphism of a group algebra to the direct sum of matrix algebras, is important for studying noncommutative codes. However, the particular form of terms and the construction of the isomorphism are not specified by this theorem; thus, for each group, there remains the problem of constructing the Wedderburn representation. The complete Wedderburn decomposition for the group algebra $${{\mathbb{F}}_{q}}{{D}_{{2n}}}$$ over the dihedral group $${{D}_{{2n}}}$$ has been obtained by F.E. Brochero Martinez in the case when the cardinality of the field and the order of the group are relatively prime numbers. Using these results, we study codes in the group algebra $${{\mathbb{F}}_{q}}{{D}_{{2n}}}$$ in this paper. The problem on the structure of all codes is solved, and the structure of codes induced by codes over cyclic subgroups of $${{D}_{{2n}}}$$ is described, which is of interest for cryptographic applications.