Postquantum Cryptography, Part 2

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.

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.

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.

Lattice-based cryptography is a novel approach to public key cryptography (PKC), of which the mathematical investigation (so far) resists attacks from quantum computers. By choosing a module learning with errors (MLWE) algorithm as the next standard, the National Institute of Standards and Technology (NIST) follows this approach. The multiplication of polynomials is the central bottleneck in the computation of lattice-based cryptography. Because PKC is mostly used to establish common secret keys, the focus is on compact area, power, and energy budget and, to a lesser extent, on throughput or latency. While most other work focuses on optimizing number theoretic transform (NTT)-based multiplications, in this article, we highly optimize a Toom–Cook-based multiplier. We demonstrate that a memory-efficient striding Toom–Cook with lazy interpolation results in a highly compact, low-power implementation, which, on top, enables a very regular memory access scheme. To demonstrate the efficiency, we integrate this multiplier into a Saber post-quantum accelerator, one of the four NIST finalists. Algorithmic innovation to reduce active memory, timely clock gating, and shift-add multiplier has helped to achieve 38% less power than state-of-the-art post-quantum cryptography (PQC) core, 4 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times $ </tex-math></inline-formula> less memory, 36.8% reduction in multiplier energy, and 118 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times $ </tex-math></inline-formula> reduction in active power with respect to state-of-the-art Saber accelerator (not silicon verified). This accelerator consumes 0.158- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text {mm}^{2}$ </tex-math></inline-formula> active area, which is the lowest reported to date despite the process disadvantages of the state-of-the-art designs.

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.

Post-quantum cryptography is an emerging solution to the expected security breach, introduced by quantum computers, to the currently used public key cryptosystems. This paper presents an experimental study on the effect of the overhead associated with the wireless transmission of some of the newly developed post quantum cryptography algorithms through the Internet of Things networks. Experimental results of this work include the energy and time measurements for wireless transmission of chosen sample messages of these new cryptographic algorithms. In addition, recommendations are given for appropriate modules configurations and choice of suitable security algorithm for the Internet of Things networks.

Emerging network applications are based on collaborative environments that rely on the secure group-based communication paradigm. These protocols use traditional public-key cryptography (PKC) for authenticating the group members and exchange secret keys. However, the foreseeable breakthrough of quantum computers represents a risk for the secure systems based on PKC. In order to prepare for such an event, quantum-secure group-based communications must be established. Post-quantumcryptography (PQC) and quantum-key distribution (QKD) are two quantum-resistant technologies that can be used for enabling the long term security in such collaborative environments. This work proposes four quantum-secure group-based communication solutions, based on contributory and decentralized protocols. PQC and hybrid technologies (PQC/QKD) are explored. The solutions use PQC Key-exchange algorithms or QKD protocols (decoy-BB84 and T12) for key generation and PQC signature algorithms for authentication. For the first time hybrid schemes for group-based technologies are designed and explored. Results show that the group-based decentralized protocol (PQC SLIMCAST) outperforms the contributory protocol PQC GAKE. The PQC-based approaches have a better performance than the QKD versions.

Abstract: This book presents a comprehensive exploration of quantum cryptography as an emergent paradigm in secure communications, enhanced through the integration of quantum computing and machine learning methodologies. Grounded in the principles of quantum mechanics—superposition, entanglement, and the no-cloning theorem—it develops a conceptual and mathematical framework for information-theoretic security resistant to both classical and quantum adversaries. The work identifies the limitations of classical cryptographic systems in the advent of large-scale quantum computation and proposes hybrid architectures that leverage machine learning for adaptive threat detection, protocol optimization, and anomaly classification in quantum key distribution (QKD) environments. Methodologically, the text synthesizes theoretical analysis, formal security proofs, and empirical evaluation using both quantum simulation platforms and experimental testbeds. Key results demonstrate that quantum–machine learning synergies can improve key generation rates, enhance noise tolerance, and accelerate real-time countermeasure deployment. The implications extend to the design of scalable, interoperable cryptographic systems compatible with emerging quantum internet infrastructures, while also addressing ethical and regulatory considerations in deployment. By combining rigorous academic analysis with practical implementation insights, this book establishes a scholarly and applied foundation for next-generation secure communication systems in the quantum era. Keywords Quantum cryptography, quantum key distribution, quantum computing, machine learning, quantum machine learning, cryptographic security, post-quantum cryptography, hybrid architectures, adaptive protocol optimization, information-theoretic security, secure communications, quantum internet, ethical implications in cryptography

This Research to Practice Full Paper discusses an approach to motivating graduate computer science and electrical engineering students in the study of quantum computing. The U.S. Congress and other governmental and industrial organizations have recognized that teaching Quantum Computing in university-level Computer Science and Electrical Engineering programs is strategically important for workforce development along with technology development. Success in this endeavor requires generating genuine interest in the topic and motivating potential students. Quantum Computing requires an understanding of relatively difficult related subjects including physics, math and computer science. Many computer science and electrical engineering students shy away from quantum computing because of its complexity. This paper discusses our success with generating interest in quantum computing among graduate students and motivating them through exploring, in related courses, some aspects of quantum computing that have the potential to generate discontinuities. The approach is also examined in terms of the Scientific Learning Cycle as discussed by Wankat & Oreovicz. Opportunities and challenges of adding Quantum Computing to Computer Science and Electrical Engineering programs were studied through a student small group project in a cybersecurity course. The work built on extensive research into the effectiveness of small group projects and the role of “play” in learning. Our small group project on quantum computing was introduced three years ago into a cybersecurity course which is required for all students in our university's Masters in Computer Science and Masters in Electrical Engineering programs. The basic goals of the project are to teach students some basics of quantum computing by allowing them to “play with it” and to alert students to the fact that quantum computing is a rapidly developing new field that they need to follow during at least the next decade. The small group project helped students to explore the potential of quantum computing to substantially undermine current Public Key Cryptography and, at the same time provide a new approach to encryption that is even more secure than Public Key Cryptography. A qualitative analysis of discussions and assigned tasks suggests that this strategy succeeded in generating significant interest in quantum computing among students. They became motivated and immersed in a difficult subject despite the requirement for investment of significant time and effort to understand a complex combination of physics, math and computer science.

Consortium blockchain has been widely used in many application scenarios, where there is the demand for a universal user authentication and key exchange mechanism for all the application users in the system like Know Your Customer. Since current solutions heavily rely on traditional public-key cryptosystems that are vulnerable to attacks from quantum computers, we design and implement the first post-quantum (PQ) user authentication and key exchange system for consortium blockchain, which is integrated with all the PQ public-key (i.e., signature and encryption/KEM) algorithms in the current round of NIST call for national standard. Furthermore, we also provide chaincodes, related APIs together with client codes for further development. Last but not least, we perform a systematic evaluation on the performance of the system including the consumed time of chaincodes execution and the needed on-chain storage space. Based on the experiment results, we discuss the implications of our findings, which are helpful for the PQ blockchain-based application developers, the undergoing NIST call and the developers of the PQ algorithms.

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

Post-quantum cryptography aims to find quantum-resistant public-key cryptographic schemes, that is, schemes which remain secure against both quantum and standard computers. In fact in 1994, Shor published an algorithm [85] breaking the nowadays used public-key cryptoschemes in polynomial time on a quantum computer. The security of public-key algorithms is based on computationally hard mathematical problems. Among the post-quantum proposals there is the Multivariate Problem: Solving a multivariate polynomial system over a finite field. The general and the most computationally efficient method for solving such systems is computing a Groöbner basis of the ideal they generate. In 2021, Beullens [22] used a strategy involving the MinRank Problem for breaching the most promising multivariate scheme at the time. All the details regarding public-key schemes and in particular the multivariate one can be found within the Introduction. The thesis is divided into three blocks. The goal of Chapter 1 is twofold. It is a collection of algebraic concepts and classical facts that are used through these pages, in particular, Section 1.1 serves this scope. In addition, the problem of solving polynomial systems via Gr¨bner basis is fully explained in Section 1.2. In Chapter 2 and Chapter 3, I present the two main works of my PhD, during which I mainly focused on the analysis of multivariate cryptoschemes twofolds. Firstly, together with my advisor Elisa Gorla, we propose a new definition of ‘random system’ using the concept of generality mutued from algebraic geometry. The details of this work are given in Chapter 2. On the other hand, with the help of Daniel Cabarcas (Universidad Nacional de Colombia), we investigated the SupportMinors Modeling, which is a method for modeling the MinRank Problem used by Beullens in his attack. The specifics of this work can be found in Chapter 3.

In recent years public key cryptography has been revolutionary developed in cryptosystems. Algorithms such as RSA, ECC and DSA are commonly used as public key cryptography in a variety of applications. The drawback of these protocols is vulnerability against quantum computing. These protocols rely on the computational hardness of finding factorization but Peter Shor has proved that quantum computing can break all the public key cryptosystems that are relying on this factorization. Hash-Based Cryptography has been subject of debates for past few decades since it is known to be resistant to quantum computer attacks and there have been some advancements over this field. But all the available methods are concentrated on the digital signature schemes. None of them have the ability to encrypt and decrypt a plain text. This paper will describe some methods that can be used as encryption and decryption of plain text using hash functions.KeywordsHash functionsCryptographyPublic key cryptographyPhysically unclonable functions

Ensuring communication security in real-time automotive networks is of paramount importance given the sensitivity of the exchanged information and highly safety critical nature of its operation. The first step towards ensuring security is to securely authenticate all the computational nodes through use of authentication protocols based on public-key cryptography. But, traditional public-key cryptographic primitives we use today are believed to be breakable by large scale quantum computers of the future. Thus, NIST is currently running a global global level standardization process for quantum-resistant public-key cryptography, better known as post-quantum cryptography. In this work, we perform a first of its kind practical implementation of a secure authentication protocol for automotive systems with post-quantum cryptographic algorithms and perform a detailed comparative evaluation of the speed and communication bandwidth performance against their pre-quantum counterparts.

The exponential development in Quantum Computing is becoming a threat to the existing asymmetric cryptographic algorithms like RSA (Rivest, Shamir, Adleman). A threat to these public key cryptographic algorithms is a threat to the entire Internet itself, as algorithms like RSA are the basic building blocks of existing secure Internet communication. Thus, a new area of quantum cryptography emerged, that lead to the development of Post-Quantum Cryptography (PQC) algorithms and Quantum-safe Internet protocol Quantum Key Distribution (QKD). These systems, which are generally known as Quantum Cryptographic Systems, have many practical limitations that make them complex and expensive to be implemented or even experimented with, in a real-life scenario. Therefore, to develop new protocols and analyze the feasibility of the existing protocols in a cost-effective manner, we need efficient simulation frameworks. So, in this paper, we are profiling some of the recent Quantum Cryptographic based networks and Quantum Key Distribution (QKD) simulation frameworks and conducting a systematic and thorough analysis of the implementation and functionalities of these experimental platforms, with an aim to build a universal quantum testbed in the future.