Quantum-resistant Cryptosystems Research Articles

Method of encryption in the MST3 cryptosystem based on Automorphisms group of Suzuki's functional field

This article presents a new implementation of encryption based on MST, focused on generalized Suzuki 2-groups. The well-known MST cryptosystem, based on Suzuki groups, is constructed using a logarithmic signature at the center of the group, leading to a large array of logarithmic signatures. The proposed encryption is based on multi-parameter noncommutative groups, with a focus on the generalized multi-parameter Suzuki 2-group. This approach responds to the progress in the development of quantum computers, which may pose a threat to the security of many open cryptosystems, especially those based on factorization problems and discrete logarithms, such as RSA or ECC. The use of noncommutative groups to create quantum-resistant cryptosystems has been a known approach for the last two decades.
 The unsolvable word problem, proposed by Wagner and Magyarik, is used in the field of permutation groups and is key to the development of cryptosystems. Logarithmic signatures, introduced by Magliveras, represent a unique type of factorization suitable for finite groups. The latest version of such an implementation, known as MST3, is based on the Suzuki group. In 2008, Magliveras introduced the LS transitivity limit for the MST3 cryptosystem, and later Swaba proposed an improved version of the cryptosystem, eMST3. In 2018, T. van Trung suggested applying the MST3 approach using strong aperiodic logarithmic signatures for abelian p-groups. Kong and his colleagues conducted a deep analysis of MST3 and noted that due to the absence of publications on the quantum vulnerability of this algorithm, it can be considered a potential candidate for use in the post-quantum era.
 The main distinction of the new system is the use of homomorphic encryption to construct logarithmic signature coverings for all group parameters, which improves the secrecy of the cryptosystem, particularly against brute-force attacks.

Post-quantum distributed ledger technology: a systematic survey

Blockchain technology finds widespread application across various fields due to its key features such as immutability, reduced costs, decentralization, and transparency. The security of blockchain relies on elements like hashing, digital signatures, and cryptography. However, the emergence of quantum computers and supporting algorithms poses a threat to blockchain security. These quantum algorithms pose a significant threat to both public-key cryptography and hash functions, compelling the redesign of blockchain architectures. This paper investigates the status quo of the post-quantum, quantum-safe, or quantum-resistant cryptosystems within the framework of blockchain. This study starts with a fundamental overview of both blockchain and quantum computing, examining their reciprocal influence and evolution. Subsequently, a comprehensive literature review is conducted focusing on Post-Quantum Distributed Ledger Technology (PQDLT). This research emphasizes the practical implementation of these protocols and algorithms providing extensive comparisons of characteristics and performance. This work will help to foster further research at the intersection of post-quantum cryptography and blockchain systems and give prospective directions for future PQDLT researchers and developers.

Method of encryption in the MST3 cryptosystem based on Automorphisms group of Suzuki's functional field

This article presents a new encryption method based on the group of automorphisms of Suzuki's functional field, which enhances the security level of the existing MST3 cryptosystem. This approach is a response to the progress in developing powerful quantum computers, which can threaten the security of many public-key encryption systems, particularly those based on factorization and discrete logarithm problems, such as RSA or ECC. Using non-commutative groups to create quantum-resistant cryptosystems has been a known approach over the last two decades. The unsolvable word problem, proposed by Wagner and Magyarik, applied in the realm of permutation groups, is key to the development of cryptosystems. Logarithmic signatures, introduced by Magliveras, represent a unique type of factorization suitable for finite groups. The latest version of such implementation, known as MST3, is based on the Suzuki group. In 2008, Magliveras presented the LS limit of transitivity for the MST3 cryptosystem. Later, Svaba proposed an improved version of the cryptosystem eMST3 with enhanced protective features, including secret homomorphic covering. In 2018, T. van Trung suggested the application of the MST3 approach using strong aperiodic logarithmic signatures for Abelian p-groups. Kong and his colleagues conducted an in-depth analysis of MST3 and noted that due to the lack of publications on the quantum vulnerability of this algorithm, it can be considered as a potential candidate for use in the post-quantum era. One of the key ideas is to increase encryption efficiency by optimizing computational resources, particularly through reducing the size of the key space. This method is applied to the calculations of logarithmic signatures within the group. It was implemented over finite fields of small sizes.

A novel attack on McEliece's cryptosystem

This report proposes a new and novel attack on McEliece's cryptosystem that improves on the probability of attacks formerly proposed by Stern, and Lee and Brickell. Modern day encryption standards have been long since proven insecure to quantum attack, and quantum-resistant cryptosystems are now at the forefront of research. Since 2016, the National Institute of Standards and Technology (NIST) has presided over a public competition to establish new standards for public-key encryption that will secure our data in the post-quantum world. Now in its final round, one of the remaining candidates is McEliece's cryptosystem, a code-based cryptosystem proposed in 1978 by Robert J. McEliece. With a few minor alterations since its conception, McEliece's cryptosystem has, so far, proven resistant to quantum attacks, making it an ideal finalist candidate. The cryptosystem has not, however, escaped the attention of attack and, over the last four decades, a variety of algorithms have been proposed with the intention of exploiting it to recover the plaintext. This paper initially provides an overview of McEliece's cryptosystem and two existing attacks proposed by Stern, and Lee and Brickell in the 1980s. Observations are made on the shared probabilistic nature of Stern's algorithm, and Lee and Brickell's attack. It is noted that the first step of both algorithms involves the random selection of a subset of n indexes. In Stern's algorithm, n−k of n columns in a matrix H are chosen at random and, in Lee and Brickell's attack, k of n bits of the ciphertext are selected, also at random. This relationship is exploited to compound the two attacks and propose a new, novel attack. The complexity and probability of the new attack are discussed and an analysis is conducted to compare it against both Stern's algorithm and Lee and Brickell's attack. This analysis suggests that the probability of successful attack comes close to combining those of the two original attacks. Furthermore, the results suggest that the novel attack can successfully recover a message faster than Stern's algorithm. Improvements to the attack are suggested, concluding that further study should be conducted into fully analysing it and its implications on the security of McEliece's cryptosystem.

Construction of a three-parameter encryption scheme on Hermitian groups in the MST3 cryptosystem

The article proposes a method for constructing a three-parameter encryption scheme based on Hermitian groups, which improves the security parameters of the existing MST3 cryptosystem. The challenge of improving existing approaches to building cryptosystems is driven by successes in building a quantum computer with sufficient computing power to render many public-key cryptosystems insecure. In particular, we are talking about those cryptosystems based on the complexity of factorization or the discrete logarithm problem, such as RSA, ECC, etc. There are several proposals that have become classic over the past almost 20 years for using non-commutative groups to build quantum-resistant cryptosystems The unsolvable word problem is an interesting area of research for cryptosystem construction. It was formulated by Wagner and Magyarik and lies in the plane of application of permutation groups. Logarithmic signatures (LS) were proposed by Magliveras. In this context, the logarithmic signature is a special type of factorization, it is applied to finite groups. The latest version of this implementation is known as MST3 and is based on the Suzuki group.
 In 2008, Magliveras demonstrated a transitive limit of LS for the MST3 cryptosystem. Svaba later proposed the eMST3 cryptosystem with improved security options. A secret homomorphic cover was added for this improvement. Then, in 2018, T. van Trung proposed an MST3 approach using strong aperiodic LS for abelian p-groups. Kong and colleagues conducted an extensive analysis of MST3 and noted that since there are no publications yet on the quantum vulnerability of the algorithm, it can be considered a candidate for the post-quantum era.
 One valuable idea is to improve encryption efficiency by optimizing the computational overhead. This is done while reducing the large size of the key space. This approach can be applied to LS calculations outside the center of the group. And this was done over the final fields of the small dimensions using groups with high order.

Interpolation-based decoding of folded variants of linearized and skew Reed–Solomon codes

The sum-rank metric is a hybrid between the Hamming metric and the rank metric and suitable for error correction in multishot network coding and distributed storage as well as for the design of quantum-resistant cryptosystems. In this work, we consider the construction and decoding of folded linearized Reed–Solomon (FLRS) codes, which are shown to be maximum sum-rank distance (MSRD) for appropriate parameter choices. We derive an efficient interpolation-based decoding algorithm for FLRS codes that can be used as a list decoder or as a probabilistic unique decoder. The proposed decoding scheme can correct sum-rank errors beyond the unique decoding radius with a computational complexity that is quadratic in the length of the unfolded code. We show how the error-correction capability can be optimized for high-rate codes by an alternative choice of interpolation points. We derive a heuristic upper bound on the decoding failure probability of the probabilistic unique decoder and verify its tightness by Monte Carlo simulations. Further, we study the construction and decoding of folded skew Reed-Solomon codes in the skew metric. Up to our knowledge, FLRS codes are the first MSRD codes with different block sizes that come along with an efficient decoding algorithm.

Quantum Computing and Blockchain Technologies over IoT Applications: A R

In cybersecurity, blockchain is essential for achieving privacy of IoT data as it provides the immutability feature. Blockchain is currently one of the most cutting-edge technologies, and it has received a lot of attention and research. The majority of current cryptography techniques could be broken with the efforts made to realize large-scale quantum computers. In order to create blockchain frameworks, a quantum tool must be able to execute at the level of digital computers and fend off potential attacks from both digital and quantum computers. A thorough analysis of the current literature on implementing Blockchain and Quantum Computing over IoT network will be crucial. In order to comprehend the state-of-the-art of this developing technology, this work aims to highlight unresolved issues that the IoT community will need to deal with in the upcoming years. This article also presents a comprehensive review of Quantum Computing and Blockchain Technologies deployed over IoT applications. Keywords Quantum Computing, Internet-of-Things (IoT), Blockchain, Cryptography, Quantum resistant Cryptosystems

DEVELOPMENTOFMETHODSFORTESTINGTHELIBRARYOFCRYPTOGRAPHICTRANSFORMATIONSONTHEEXAMPLEOFTHEMST3CRYPTOSYSTEMBASEDONGENERALIZEDSUZUKI2-GROUPS

The article proposes a methodology for testing a library of cryptographic transformations with the implementation of an improved encryption scheme on generalized Suzuki 2-groups in the MST3 cryptosystem. The need to improve existing methods of cryptosystem creation is driven by progress in quantum computer development, which possess sufficient computational power to compromise many existing public key cryptosystems. This is especially true for systems based on factorization and discrete logarithm, such as RSA and ECC. Over the last nearly 20 years, there have been proposals for using non-commutative groups to develop quantum-resistant cryptosystems. The unsolved word problem, formulated by Wagner and Magyarik, uses permutation groups and is a promising direction in cryptosystem development. Magliveras proposed logarithmic signatures, a special type of factorization applied to finite groups, and the latest version of this technology is known as MST3, based on the Suzuki group. The first implementation of the cryptosystem on the generalized Suzuki 2-group had limitations in encryption and protection against brute force attacks. Over the past years, many proposals have been made to improve the basic design. The research conducted by the authors expanded the possibilities of using public cryptography by refining parameters based on non-Abelian groups. The article demonstrates the methodology for conducting tests of the practical implementation of the library of cryptographic transformations with the implementation of an improved encryption scheme on Suzuki 2-groups, confirming its functionality.

CONSTRUCTION OF AN IMPROVED ENCRYPTION SCHEME ON GENERALIZED SUZUKI 2-GROUPS IN THE MST3 CRYPTOSYSTEM

This paper proposes a method for constructing an improved encryption scheme on generalized Suzuki 2-groups for the MST3 cryptosystem, which improves the security parameters of the original approach. The challenge of improving existing cryptosystem design approaches is driven by advances in building quantum computers with sufficient computing power to render many public-key cryptosystems insecure. In particular, this includes cryptosystems based on the factorization problem or the discrete logarithm problem, such as RSA and ECC. There have been several proposals in the past two decades for using non-commutative groups to create quantum-resistant cryptosystems. The unsolvable word problem is a promising area of research for building cryptosystems. It was formulated by Wagner and Magyarik and lies in the realm of permutation groups. Magliveras proposed logarithmic signatures, which are a special type of factorization that applies to finite groups. The latest version of this implementation, known as MST3, is based on the Suzuki group. In 2008, Magliveras demonstrated a transitive LS limit for the MST3 cryptosystem. Later, Svaba proposed the eMST3 cryptosystem with improved security parameters, achieved by adding a secret homomorphic cover. In 2018, T. van Trung proposed an MST3 approach using strong aperiodic logarithmic signatures for abelian p-groups. Kong and his colleagues conducted an extensive analysis of MST3 and noted that, since there are currently no publications on the quantum vulnerability of the algorithm, it can be considered a candidate for use in the post-quantum era. The first implementation of the cryptosystem on the generalized Suzuki 2-group does not provide encryption of the entire Suzuki 2-group and does not protect against attacks with sequential key recovery by the brute-force method. Further work has developed the idea of public cryptography using non-Abelian refinements of parameters. This paper proposes a method for constructing an encryption scheme on Suzuki 2-groups that improves the security parameters of the existing MST3 cryptosystem and solves its security problems.

On matrix sequences represented by negative indices Pell and Pell–Lucas number with the decoding of lucas blocking error correcting codes

This paper focuses on the matrix sequences that represent negative indices Pell and Pell–Lucas number. It will specifically provide the important relationships between negative indices Pell and Pell–Lucas number and Pell and Pell–Lucas matrix sequences. The research of new error correcting codes has received a lot of attention in recent years, specifically due to its potential use in quantum-resistant cryptosystems. These matrices are then applied in cryptology and have a great opportunity to report some cases that might be a mistake between the sender and the receiver by using Lucas blocking algorithm.

Information security in the post quantum era for 5G and beyond networks: Threats to existing cryptography, and post-quantum cryptography

Quantum computing is an emerging field that uses the concepts of quantum mechanics to outperform classical computers. Quantum computing finds plethora of applications in the 5G and Beyond networks. It can process data at an exponential rate, which can address numerous business and scientific challenges. In this paper, we provide a detailed review of the field, starting from the most important applications of quantum computers and then diving into the future of cryptography. The major applications of quantum computing include unstructured search, quantum simulation, and optimization. It can also provide improvements in terms of speed and accuracy for some existing technologies, such as machine learning. These technologies have numerous applications in 5G and beyond networks. However, due to such abilities of quantum computing, it can also pose a serious risk to many existing security systems, especially the asymmetric key cryptography schemes. The risk of quantum computing has also influenced the mobile broadband standards to move from a symmetric key cryptography techniques to PKI-based trust model. We also discuss in detail various alternate cryptosystems based on mathematical problems that are believed to be hard even for a quantum computer to solve. In parallel, we discuss the developments in the field of quantum key distribution that makes use of quantum phenomenon to develop a quantum-resistant crypto systems. Such quantum-resistant systems have a great potential in provisioning secure 5G and beyond networks.

Towards Post-Quantum Blockchain: A Review on Blockchain Cryptography Resistant to Quantum Computing Attacks

Blockchain and other Distributed Ledger Technologies (DLTs) have evolved significantly in the last years and their use has been suggested for numerous applications due to their ability to provide transparency, redundancy and accountability. In the case of blockchain, such characteristics are provided through public-key cryptography and hash functions. However, the fast progress of quantum computing has opened the possibility of performing attacks based on Grover’s and Shor’s algorithms in the near future. Such algorithms threaten both public-key cryptography and hash functions, forcing to redesign blockchains to make use of cryptosystems that withstand quantum attacks, thus creating which are known as post-quantum, quantum-proof, quantum-safe or quantum-resistant cryptosystems. For such a purpose, this article first studies current state of the art on post-quantum cryptosystems and how they can be applied to blockchains and DLTs. Moreover, the most relevant post-quantum blockchain systems are studied, as well as their main challenges. Furthermore, extensive comparisons are provided on the characteristics and performance of the most promising post-quantum public-key encryption and digital signature schemes for blockchains. Thus, this article seeks to provide a broad view and useful guidelines on post-quantum blockchain security to future blockchain researchers and developers.

High-Level Synthesis of Number-Theoretic Transform: A Case Study for Future Cryptosystems

Compared to traditional hardware development methodologies, high-level synthesis (HLS) offers a faster time-to-market and lower design-cost at the expense of implementation efficiency. Although HLS tools are becoming popular in some applications, such as digital signal processing and neural network classification, their usability on cryptographic applications is largely unexplored. This feasibility is critical especially for cryptosystems that are under development, such as the next-generation public-key cryptosystems needed for quantum-resistance. This letter provides a thorough investigation of HLS on number theoretic transform (NTT)-the core arithmetic function of lattice-based quantum-resistant cryptosystems. We demonstrate a fast yet extensive design space exploration of NTT through the Vivado HLS tool, analyze the shortcomings/challenges of optimized configurations, and quantitatively compare the results to software-based and hand-coded hardware designs.

From Pre-Quantum to Post-Quantum IoT Security: A Survey on Quantum-Resistant Cryptosystems for the Internet of Things

Although quantum computing is still in its nascent age, its evolution threatens the most popular public-key encryption systems. Such systems are essential for today’s Internet security due to their ability for solving the key distribution problem and for providing high security in insecure communications channels that allow for accessing websites or for exchanging e-mails, financial transactions, digitally signed documents, military communications or medical data. Cryptosystems like Rivest–Shamir–Adleman (RSA), elliptic curve cryptography (ECC) or Diffie–Hellman have spread worldwide and are part of diverse key Internet standards like Transport Layer Security (TLS), which are used both by traditional computers and Internet of Things (IoT) devices. It is especially difficult to provide high security to IoT devices, mainly because many of them rely on batteries and are resource constrained in terms of computational power and memory, which implies that specific energy-efficient and lightweight algorithms need to be designed and implemented for them. These restrictions become relevant challenges when implementing cryptosystems that involve intensive mathematical operations and demand substantial computational resources, which are often required in applications where data privacy has to be preserved for the long term, like IoT applications for defense, mission-critical scenarios or smart healthcare. Quantum computing threatens such a long-term IoT device security and researchers are currently developing solutions to mitigate such a threat. This article provides a survey on what can be called post-quantum IoT systems (IoT systems protected from the currently known quantum computing attacks): the main post-quantum cryptosystems and initiatives are reviewed, the most relevant IoT architectures and challenges are analyzed, and the expected future trends are indicated. Thus, this article is aimed at providing a wide view of post-quantum IoT security and give useful guidelines to the future post-quantum IoT developers.

Improved Strongly Deniable Authenticated Key Exchanges for Secure Messaging

Abstract A deniable authenticated key exchange (DAKE) protocol establishes a secure channel without producing cryptographic evidence of communication. A DAKE offers strong deniability if transcripts provide no evidence even if long-term key material is compromised (offline deniability) and no outsider can obtain evidence even when interactively colluding with an insider (online deniability). Unfortunately, existing strongly deniable DAKEs have not been adopted by secure messaging tools due to security and deployability weaknesses. In this work, we propose three new strongly deniable key exchange protocols—DAKEZ, ZDH, and XZDH—that are designed to be used in modern secure messaging applications while eliminating the weaknesses of previous approaches. DAKEZ offers strong deniability in synchronous network environments, while ZDH and XZDH can be used to construct asynchronous secure messaging systems with offline and partial online deniability. DAKEZ and XZDH provide forward secrecy against active adversaries, and all three protocols can provide forward secrecy against future quantum adversaries while remaining classically secure if attacks against quantum-resistant cryptosystems are found. We seek to reduce barriers to adoption by describing our protocols from a practitioner’s perspective, including complete algebraic specifications, cryptographic primitive recommendations, and prototype implementations. We evaluate concrete instantiations of our DAKEs and show that they are the most efficient strongly deniable schemes; with all of our classical security guarantees, our exchanges require only 1 ms of CPU time on a typical desktop computer and at most 464 bytes of data transmission. Our constructions are nearly as efficient as key exchanges with weaker deniability, such as the ones used by the popular OTR and Signal protocols.

Securing the Internet of Things in a Quantum World

Currently, we rely on cryptographic algorithms such as elliptic curve cryptosystems (ECCs) as basic building blocks to secure the communication in the IoT. However, public key schemes like ECC can easily be broken by the upcoming quantum computers. Due to recent advances in quantum computing, we should act now to prepare the IoT for the quantum world. In this article, we focus on the current state of the art and recent developments in the area of quantum-resistant cryptosystems for securing the IoT. We first demonstrate the impacts of quantum computers on the security of the cryptographic schemes used today, and then give an overview of the recommendations for cryptographic schemes that can be secure under the attacks of both classical and quantum computers. After that, we present the existing implementations of quantum-resistant cryptographic schemes on constrained devices suitable for the IoT. Finally, we give an introduction to ongoing projects for quantum-resistant schemes that will help develop future security solutions for the IoT.

Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies

Abstract We present new candidates for quantum-resistant public-key cryptosystems based on the conjectured difficulty of finding isogenies between supersingular elliptic curves. The main technical idea in our scheme is that we transmit the images of torsion bases under the isogeny in order to allow the parties to construct a shared commutative square despite the non-commutativity of the endomorphism ring. We give a precise formulation of the necessary computational assumptions along with a discussion of their validity, and prove the security of our protocols under these assumptions. In addition, we present implementation results showing that our protocols are multiple orders of magnitude faster than previous isogeny-based cryptosystems over ordinary curves. This paper is an extended version of [Lecture Notes in Comput. Sci. 7071, Springer (2011), 19–34]. We add a new zero-knowledge identification scheme and detailed security proofs for the protocols. We also present a new, asymptotically faster, algorithm for key generation, a thorough study of its optimization, and new experimental data.