Commutative algebra techniques in post-quantum cryptography

Information security is a fundamental and urgent issue in the digital transformation era. Cryptographic techniques and digital signatures have been applied to protect and authenticate relevant information. However, with the advent of quantum computers and quantum algorithms, classical cryptographic techniques have been in danger of collapsing because quantum computers can solve complex problems in polynomial time. Stemming from that risk, researchers worldwide have stepped up research on post-quantum algorithms to resist attack by quantum computers. In this review paper, we survey studies in recent years on post-quantum cryptography (PQC) and provide statistics on the number and content of publications, including a literature overview, detailed explanations of the most common methods so far, current implementation status, implementation comparisons, and discussion on future work. These studies focused on essential public cryptography techniques and digital signature schemes, and the US National Institute of Standards and Technology (NIST) launched a competition to select the best candidate for the expected standard. Recent studies have practically implemented the public key encryption/key encapsulation mechanism (PKE/KEM) and digital signature schemes on different hardware platforms and applied various optimization measures based on other criteria. Along with the increasing number of scientific publications, the recent trend of PQC research is increasingly evident and is the general trend in the cryptography industry. The movement opens up a promising avenue for researchers in public key cryptography and digital signatures, especially on algorithms selected by NIST.

Modern cryptography pushed forward the need of having provable security. Whereas ancient cryptography was only relying on heuristic assumptions and the secrecy of the designs, nowadays researchers try to make the security of schemes to rely on mathematical problems which are believed hard to solve. When doing these proofs, the capabilities of potential adversaries are modeled formally. For instance, the black-box model assumes that an adversary does not learn anything from the inner-state of a construction. While this assumption makes sense in some practical scenarios, it was shown that one can sometimes learn some information by other means, e.g., by timing how long the computation take. In this thesis, we focus on two different areas of cryptography. In both parts, we take first a theoretical point of view to obtain a result. We try then to adapt our results so that they are easily usable for implementers and for researchers working in practical cryptography. In the first part of this thesis, we take a look at post-quantum cryptography, i.e., at cryptographic primitives that are believed secure even in the case (reasonably big) quantum computers are built. We introduce HELEN, a new public-key cryptosystem based on the hardness of the learning from parity with noise problem (LPN). To make our results more concrete, we suggest some practical instances which make the system easily implementable. As stated above, the design of cryptographic primitives usually relies on some well-studied hard problems. However, to suggest concrete parameters for these primitives, one needs to know the precise complexity of algorithms solving the underlying hard problem. In this thesis, we focus on two recent hard-problems that became very popular in post-quantum cryptography: the learning with error (LWE) and the learning with rounding problem (LWR). We introduce a new algorithm that solves both problems and provide a careful complexity analysis so that these problems can be used to construct practical cryptographic primitives. In the second part, we look at leakage-resilient cryptography which studies adversaries able to get some side-channel information from a cryptographic primitive. In the past, two main disjoint models were considered. The first one, the threshold probing model, assumes that the adversary can put a limited number of probes in a circuit. He then learns all the values going through these probes. This model was used mostly by theoreticians as it allows very elegant and convenient proofs. The second model, the noisy-leakage model, assumes that every component of the circuit leaks but that the observed signal is noisy. Typically, some Gaussian noise is added to it. According to experiments, this model depicts closely the real behaviour of circuits. Hence, this model is cherished by the practical cryptographic community. In this thesis, we show that making a proof in the first model implies a proof in the second model which unifies the two models and reconciles both communities. We then look at this result with a more practical point-of-view. We show how it can help in the process of evaluating the security of a chip based solely on the more standard mutual information metric.

A roadmap from classical cryptography to post-quantum resistant cryptography for 5G-enabled IoT: Challenges, opportunities and solutions

Post-quantum cryptography is essential for securing digital communications against threats posed by quantum computers. Researchers have focused on developing algorithms that can withstand attacks from both classical and quantum computers, thereby ensuring the security of data transmissions over public networks. A critical component of this security is the key agreement protocol, which allows two parties to establish a shared secret key over an insecure channel. This paper introduces two novel post-quantum key agreement protocols that can be easily implemented on standard computers using rectangular or rank-deficient matrices, exploiting the generalizations of the matrix power function, which is a generator of NP-hard problems. We provide basic concepts and proofs, pseudocodes, and examples, along with a discussion of complexity.

Introduction. Widespread use of unmanned aerial vehicles in the civilian and military spheres requires the development of new algorithms for identification friend or foe of targets, as used in the Armed Forces of Ukraine (AFU) devices of the "Parol" system are designed to service approximately 110 objects military equipment. AFU automation systems allow the use of additional sources of information about various objects from civil or special data transmission networks, which can be the basis for building a networked multi-level system of state recognition. Predictions of the development of quantum computers foresee the possibility of breaking modern algorithms for information security in polynomial time in the next 5-10 years, which requires the development and implementation of new encryption algorithms and revision of modern parameters. The purpose of the article is to develop a new algorithm for state recognition of objects, which can be scaled to process the required number of manned and unmanned aerial vehicles. Potential threats to classical cryptographic protection algorithms for data networks, which will result in the execution of algorithms such as Grover and Shore on quantum computers, were also discussed. Results. The article proposes a new multilevel algorithm of state recognition based on modern cryptographic methods of information protection, which allows to perform reliable automated identification of objects, scale systems using data on potential targets from other sources through secure special networks. Grover's search algorithm does not give a strong increase in key search performance for symmetric encryption algorithms, so there is no need to increase the key lengths for this type of information security algorithms. Post-quantum asymmetric encryption algorithms require additional study and comprehensive testing of information security or increasing the key lengths of cryptographic algorithms, which corresponds to the number of qubits, i.e. more than twice. The most promising is the family of asymmetric post-quantum cryptographic algorithms based on supersingular isogenic elliptic curves. Conclusions. The developed algorithm of identification friend or foe of objects is more secure compared to existing algorithms and is focused on the use of modern on-board computers and programmable radio modems. Shore's algorithm and the like will be a significant threat to modern asymmetric cryptography algorithms when the number of qubits of quantum computers exceeds the number of bits in public keys more than twice. Keywords: identification friend or foe, symmetric encryption, asymmetric cryptography, quantum computer, post-quantum cryptography.

As federated learning (FL) keeps dominating the way in privacy-preserving machine learning (ML), notably in finance, its long-term sustainability is confronted with the approaching challenge of quantum computing. The introduction of quantum computing poses a challenge to traditional cryptographic technique on which FL relies RSA, elliptic curve cryptography (ECC), and some homomorphic encryption (HE) methods applied for encrypting model updates during training rendering this chapter to explore the interplay between the threat of quantum computing and the FL architectures, with emphasis on the implications this interaction introduces to cybersecurity and privacy in finance applications. Quantum computers are theoretically capable of solving hard math problems that have been regarded as computationally intractable for quantum computers. The well-known algorithms like Shor’s and Grover’s provides serious risk to the cryptographic procedures FL pipelines depends on. The finance sector, by its nature involving sensitive data and dependence on secure, distributed ML systems, is particularly vulnerable. The chapter begins by summarizing the fundamentals of FL and current encryption methods adapted to protect FL workflows. It then shifts towards the essence of quantum computing, emphasizing on the notable algorithms that threaten FL. We also explore potential attack points through which quantum computing could attack FL, e.g., model inversion attacks, poisoning attacks facilitated by quantum capabilities, and threats to secure aggregation. Post-quantum cryptography (PQC) is also presented as a potential counterattack, followed by a discussion of its use case in FL platforms. This chapter attempts to close the gap between FL and quantum computing threats to the financial sector. It is an introduction to practitioners and researchers to revisit examining security assumptions in FL and move towards the quantum-augmented cyber threat model.

Quantum computing furnishes exponential speed up over classical computing in specific areas. For example, Shor’s algorithm can factor two numbers in a polynomial time complexity. Thus, many encryption algorithms that rely on large number factorization are potentially vulnerable to quantum computers. In order to address this, the National Institute of Standard and Test (NIST) has organized a competition to evaluate several post quantum cryptography (PQC) algorithms, that are secure from the attacks from quantum computers. Several of these lattice-based PQC encryption algorithms are based on Learning With Errors (LWE) computation. Conversely, LWE is the heaviest computation in a classical computer, which incurs significant portion of the latency overhead for the entire encryption algorithm. In this paper, we design an optimized quantum circuit for LWE computation. The proposed quantum circuit does not need any ancillary qubits and scales efficiently and easily if there are more qubits available on a higher qubit quantum computer.

In the modern world, the financial sector faces a wide range of extraordinary events, from market shocks and geopolitical tensions to pandemics and cyberattacks. These diverse challenges significantly impact banks. Recently, particular attention has been drawn to the influence of artificial intelligence (AI) and quantum computing (QC) technologies on the stability of the financial system. This article analyzes the potential areas of impact of quantum technologies on the financial sphere in general and the resilience of financial institutions in particular.It's determined that quantum technologies can simultaneously pose both threats and new opportunities. Specifically, the article analyzes the potential of quantum technologies to break existing cryptographic algorithms that protect the confidential data of financial institutions' clients. This threat is exacerbated by the possibility of "harvest now, decrypt later" attacks, where malicious actors intercept data today with the intention of decrypting it in the future. This presents challenges and threats for online/mobile banking, payment operations, and corporate communications. Among the positive impacts of quantum technologies on banking, the article highlights improved financial risk assessment, particularly enhanced accuracy in stress testing, portfolio management, and derivatives pricing, by solving problems inaccessible to classical computing. The potential of quantum machine learning for fraud detection is also noted. Attention is given to the financial sector's response to these challenges. Initiatives for developing quantum-resistant (post-quantum) cryptography are considered as the primary method for mitigating the risks associated with the proliferation of quantum computing, along with new cryptographic protocols and quantum key distribution as a new opportunity to enhance security. The article emphasizes the responsibility of central banks in forming quantum-readiness strategies, including investments in post-quantum cryptography, expertise development, and the adaptation of regulatory frameworks. The conclusion drawn is that although the immediate benefits of quantum technologies may be modest, their long-term transformative potential for the financial sector is immense.

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

Recent developments in quantum computing have shed light on the shortcomings of the conventional public cryptosystem. Even while Shor’s algorithm cannot yet be implemented on quantum computers, it indicates that asymmetric key encryption will not be practicable or secure in the near future. The NIST has started looking for a post-quantum encryption algorithm that is resistant to the development of future quantum computers as a response to this security concern. The current focus is on standardizing asymmetric cryptography that should be impenetrable by a quantum computer. This has become increasingly important in recent years. Currently, the process of standardizing asymmetric cryptography is coming very close to being finished. This study evaluated the performance of two PQC algorithms, both of which were selected as NIST fourth-round finalists. The research assessed the key generation, encapsulation, and decapsulation operations, providing insights into their efficiency and suitability for real-world applications. Further research and standardization efforts are required to enable secure and efficient post-quantum encryption. When selecting appropriate post-quantum encryption algorithms for specific applications, factors such as security levels, performance requirements, key sizes, and platform compatibility should be taken into account. This paper provides helpful insight for post-quantum cryptography researchers and practitioners, assisting in the decision-making process for selecting appropriate algorithms to protect confidential data in the age of quantum computing.

Post-Quantum Cryptography (PQC) has emerged as a response of the cryptographic community to the danger of attacks performed using quantum computers. All PQC schemes can be implemented in software and hardware using conventional (non-quantum) computing systems. PQC is the biggest revolution in cryptography since the invention of public-key schemes in the mid-1970 s. Lattice-based key exchange schemes have emerged as leading candidates in the NIST PQC standardization process due to their relatively short public keys and ciphertexts. This paper presents novel high-speed hardware architectures for four lattice-based Key Encapsulation Mechanisms (KEMs) representing three NIST PQC finalists: NTRU (with two distinct variants, NTRU-HPS and NTRU-HRSS), CRYSTALS-Kyber, and Saber. We benchmark these candidates in terms of their performance and resource utilization in today's FPGAs. Our best architectures outperform the best designs from other groups reported to date in terms of the area-time product by factors ranging from 1.01 to 2.88, depending on the algorithm and security level. Additionally, our study demonstrates that CRYSTALS-Kyber and Saber have very similar hardware performance. Both outperform NTRU in terms of execution time by a factor 36-62 for key generation and 3-7 for decapsulation, assuming the same security level.

The security of public-key cryptography relies on well-studied hard problems, problems for which we do not have efficient algorithms. Factorization and discrete logarithm are the two most known and used hard problems. Unfortunately, they can be easily solved on a quantum computer by Shor's algorithm. Also, the research area of cryptography demands for crypto-diversity which says that we should offer a range of hard problems for public-key cryptography. If one hard problem proves to be easy, we should be able to provide alternative solutions. Some of the candidates for post-quantum hard problems, i.e. problems which are believed to be hard even on a quantum computer, are the Learning Parity with Noise (LPN), the Learning with Errors (LWE) and the Shortest Vector Problem (SVP). A thorough study of these problems is needed in order to assess their hardness. In this thesis we focus on the algorithmic study of LPN. LPN is a hard problem that is attractive, as it is believed to be post-quantum resistant and suitable for lightweight devices. In practice, it has been employed in several encryption schemes and authentication protocols. At the beginning of this thesis, we take a look at the existing LPN solving algorithms. We provide the theoretical analysis that assesses their complexity. We compare the theoretical results with practice by implementing these algorithms. We study the efficiency of all LPN solving algorithms which allow us to provide secure parameters that can be used in practice. We push further the state of the art by improving the existing algorithms with the help of two new frameworks. In the first framework, we split an LPN solving algorithm into atomic steps. We study their complexity, how they impact the other steps and we construct an algorithm that optimises their use. Given an LPN instance that is characterized by the noise level and the secret size, our algorithm provides the steps to follow in order to solve the instance with optimal complexity. In this way, we can assess if an LPN instance provides the security we require and we show what are the secure instances for the applications that rely on LPN. The second framework handles problems that can be decomposed into steps of equal complexity. Here, we assume that we have an adversary that has access to a finite or infinite number of instances of the same problem. The goal of the adversary is to succeed in just one instance as soon as possible. Our framework provides the strategy that achieves this. We characterize an LPN solving algorithm in this framework and show that we can improve its complexity in the scenario where the adversary is restricted. We show that other problems, like password guessing, can be modeled in the same framework.

The advances in quantum technologies and the fast move toward quantum computing are threatening classical cryptography and urge the deployment of post-quantum (PQ) schemes. The only isogeny-based candidate forming part of the third round of the standardization, the Supersingular Isogeny Key Encapsulation (SIKE) mechanism, is a subject of constant latency optimizations given its attractive compact key size lengths and, thus, its limited bandwidth and memory requirements. In this work, we present a new speed record of the SIKE protocol by implementing novel low-level finite field arithmetics targeting ARMv7-M architecture. We develop a handcrafted assembly code for the modular multiplication and squaring functions where we obtain 8.71% and 5.38% of speedup, respectively, compared to the last best-reported assembly implementations for p434. After deploying the finite field optimized architecture to the SIKE protocol, we observe 5.63%, 3.93%, 3.48%, and 1.61% of latency reduction for SIKE p434, p503, p610, and p751, respectively, targeting the NIST recommended STM32F407VG discovery board for our experiments.KeywordsSupersingular Isogeny Key Encapsulation (SIKE)Post-Quantum cryptography (PQC)ARM Cortex-M4

In today Internet era, confidential information transmitted over an insecure channel. With the significant development in the area of quantum computing, there is a need for unconditional security in confidential information. Quantum key distribution protocols are proven secure if all devices are perfect (in terms of technologies and proper protocol operations). The major challenges in quantum communication are secret key rate, distance, cost and size of QKD devices. The purpose of this survey article is to carry out a systematic review in the area of quantum cryptography by covering various aspects of non-deterministic quantum key distribution protocols, quantum secure direct communication, semi-quantum key distribution, secure multiparty communication protocol, post-quantum cryptography and device-independent cryptography techniques. In addition, we also discussed various experimental work carried out in the area of quantum cryptography, various attacks and challenges relative to the paradigm shift from classical cryptography to quantum cryptography. Quantum cryptography will become a future replacement of classical cryptography techniques after the development of the first physical quantum computer.

Because of public-key cryptography’s relevance and quantum computers’ increasingly realistic threat to this technology, it’s necessary to come up with practical and secure postquantum cryptography. This special issue aims to present the state of the art and the grand challenges in postquantum cryptography and to discuss the transition of real-world systems to the new technology.