A new non-commutative key exchange protocol on combinatorial semigroup

In this paper we present a cryptographic primitive based on non-commutative cryptography. This primitive is used for key exchange protocol (KEP) construction. We prove that the security of this primitive relies on a nondeterministic polynomial complete (NP-Complete) decisional problem. Recently there are no known quantum cryptanalysis algorithms effectively solving NP-Complete problems. So far, KEPs are widely used in secure communication channel creation, e.g., in hypertext transfer protocol secure (https://) and are based on traditional cryptographic primitives representing commutative cryptography. However, the security of these protocols does not rely on NP-Complete problems and hence, according to P. W. Shorr, they are vulnerable to quantum cryptanalysis. We use one of seven non-commuting groups of order 16 which is not isomorphic to any other group to define a platform group for a key exchange protocol based on previously considered matrix power function (MPF). By investigating basic properties on the group M16 and their implementation for our goals we fix the order of actions in MPF from left to right. Furthermore, we define a special form of the base matrix and separate templates for left and right power matrices. Using properties of the specified templates and Schaeffer criteria we prove that the security of the proposed key exchange relies on an NP-Complete decisional problem.

Key exchange protocols, first introduced by Diffie and Hellman in 1976, are one of the most widely-deployed cryptographic protocols. They allow two parties, that have never interacted before, to establish shared secrets. These shared cryptographic keys may subsequently be used to establish a secure communication channel. Use cases include the classic client-server setting that is for example at play when browsing the internet, but also chats via end-to-end-encrypted instant messaging applications. Security-wise, we generally demand of key exchange protocols to achieve key secrecy and authentication. While, informally, authentication ensures that the communicating parties have confidence in the identity of their peers, key secrecy ensures that any shared cryptographic key that is established via the key exchange protocol is only known to the participants in the protocol and can be used securely in cryptographic protocols, i.e., is sufficiently random. In 1993, Bellare and Rogaway gave a first formalization of key exchange protocol security that captures these properties with respect to powerful adversaries with full control over the network. Their model constitutes the basis of the many subsequent treatments of authenticated key exchange security, including the models presented in this thesis. The common methodological approach underlying all of these formalizations is the provable security paradigm, which has become a standard tool in assessing the security of cryptographic protocols and primitives. So-called security models specify the expected security guarantees of the scheme in question with regards to a well-defined class of adversaries. Proofs that validate these security claims do so by reducing the security of the overall scheme to the security of the underlying cryptographic primitives and hardness assumptions. However, advances in computational power and more sophisticated cryptanalytic capabilities often render exactly these components insecure. Especially the advent of quantum computers will have a devastating effect on much of today's public key cryptography. This is especially true for key exchange protocols since they rely crucially on public-key algorithms. In this thesis, our focus in future-proofing key exchange protocols is two-fold. First, we focus on extending security models for key exchange protocols to capture the (un)expected break of cryptographic primitives and hardness assumptions. The aim is to gain assurances with respect to future adversaries and to investigate the effects of primitive failures on key exchange protocols. More specifically, we explore how key exchange protocols can be safely transitioned to new, post-quantum secure algorithms with hybrid techniques. Hybrids combine classical and post-quantum algorithms such that the overall key agreement scheme remains secure as long as one of the two base schemes remains secure. For this, we introduce security notions for key encapsulation mechanisms that account for adversaries with varying levels of quantum capabilities and present three new constructions for hybrid key encapsulation mechanisms. Our hybrid designs are practice-inspired and for example capture draft proposals for hybrid modes in the Transport Layer Security (TLS) protocol, which is one of the most widely-deployed cryptographic protocols that enables key agreement. Furthermore, our notion of breakdown resilience for key exchange protocols allows to gauge the security of past session keys in the event of a failure of a cryptographic component in the key exchange. We exercise our model on variants of the post-quantum secure key exchange protocol NewHope by Alkim et al. Thereby, we confirm the intuition that, in order to guard against adversaries that only have access to quantum computing power in the (more distant) future, it is sufficient to use classically-secure authentication mechanisms alongside post-quantum key agreement to achieve authenticated key exchange. As with any mathematical statement, theorems in the provable security paradigm are only as valid as the underlying assumptions. A careful consideration of any newly made assumption is thus essential to ensure the meaningfulness of the statement itself and make the assumption a viable tool for future analyses. Thus, secondly, we systematically classify the PRF-ODH assumption, a complexity-theoretic hardness assumption that has been used in key exchange security analyses of such prominent protocols as TLS, Signal, and Wireguard. In particular, we give a unified, parametrized definition of the assumption encompassing different variants that are present in the literature. We relate the resulting parametrized notions in terms of their strength and show where these assumptions fit in the collection of well-understood related hardness assumptions. We finally sketch our result on the impossibility of instantiating this assumption in the standard model, thereby disposing of the uncertainty in the community whether PRF-ODH is in fact a standard model assumption, i.e., removes the usage of some idealized assumptions in key exchange protocol proofs.

A new enhanced matrix power function (MPF) is presented for the construction of cryptographic primitives. According to the definition in previously published papers, an MPF is an action of two matrices powering some base matrix on the left and right. The MPF inversion equations, corresponding to the MPF problem, are derived and have some structural similarity with classical multivariate quadratic (MQ) problem equations. Unlike the MQ problem, the MPF problem seems to be more complicated, since its equations are not defined over the field, but are represented as left–right action of two matrices defined over the infinite near-semiring on the matrix defined over the certain infinite, additive, noncommuting semigroup. The main results are the following: (1) the proposition of infinite, nonsymmetric, and noncommuting algebraic structures for the construction of the enhanced MPF, satisfying associativity conditions, which are necessary for cryptographic applications; (2) the proof that MPF inversion is polynomially equivalent to the solution of a certain kind of generalized multivariate quadratic (MQ) problem which can be reckoned as hard; (3) the estimation of the effectiveness of direct MPF value computation; and (4) the presentation of preliminary security analysis, the determination of the security parameter, and specification of its secure value. These results allow us to make a conjecture that enhanced MPF can be a candidate one-way function (OWF), since the effective (polynomial-time) inversion algorithm for it is not yet known. An example of the application of the proposed MPF for the Key Agreement Protocol (KAP) is presented. Since the direct MPF value is computed effectively, the proposed MPF is suitable for the realization of cryptographic protocols in devices with restricted computation resources.

In this paper we propose a key exchange protocol (KEP) based on the so-called matrix power function (MPF) defined over a non-commuting platform group. In general, it is not possible to construct KEP using a non-commuting platform group. Therefore we proposed special templates for our public parameters thus allowing us to construct KEP relying on the basic properties of our MPF. Security analysis is based on the decisional Diffie-Hellman (DDH) attack game. We proved that the distribution of the entries of the public session parameter matrices and the shared key matrix asymptotically approaches to uniform with exponential rate. Hence proposed KEP is secure under the DDH assumption. This implies that our protocol is not vulnerable to the computational Diffie-Hellman (CDH) attack. We presented the evidence of CDH security by numerical simulation of linearization attack and showed that it is infeasible.KeywordsNon-commutative cryptographyMatrix power functionKey exchange protocol

Shor’s quantum algorithm establishes a polynomial time attack on the discrete logarithm problem in any group. Recent announcements of progress in building quantum computers highlight the need for new concepts to create cryptosystems that are resistant to quantum attacks. In this paper, we present а new message encryption scheme. To enhance the security of the scheme, we suggest double key-exchange protocol (KEP). The first stage of the key exchange uses a matrix power function (MPF) in a tropical semiring. These functions are based on the action of a matrix semiring acting on some matrix set. MPFs can be considered as one-way functions because they are based on some generalized satisfiability problems that are potentially NP-complete. The obtained shared secret key at the first stage of the key exchange serves as an input for the second stage. The security of the second phase relies on the difficulty of the semiring action problem. In our protocol, we suggest using left or right action of the tropical semiring (which can be both min-plus and max-plus) on the group of commutative matrices (circulant matrices, in our case). The fact that the key-exchange protocol works in two phases contributes to its security, since an attacker needs to solve two difficult problems in order to break it. The main advantages of the presented protocol are: the increased efficiency and improved security.

A Virtual Private Network (VPN) ensures the confidentiality and integrity of data transferred between two endpoints, even if the means of transport are insecure. One popular protocol is Internet Protocol Security (IPSec) which operates at OSI layer 3 and protects all protocols at higher layers. Cryptographic keys in IPSec are negotiated using the Internet Key Exchange (IKE) protocol. IKE negotiates security parameters for IPSec sessions. In particular, the IKEv2 protocol uses the Elliptic Curve Diffie-Hellman (ECDH) algorithm to establish a secret key shared between two nodes on a network. Although solving such a problem remains difficult with current computing power, it is believed that generic quantum computers will be able to solve this problem, which implies that the security of IKEv2 is compromised. There are, however, several cryptographic systems that are trusted to be resistant to attacks by quantum computers. This family of cryptosystems is known as quantum-resistant cryptography (QRC). In this paper, after highlighting the requirements for a secure key exchange protocol, we briefly review the QRC solutions that have been proposed in the recent literature.

The development of quantum computing systems poses a great threat to the security of existing public key-based systems. As a result, the National Institute of Standards and Technology (NIST) started a Post-Quantum Cryptography (PQC) standardization project in 2015, and currently active research is being conducted to apply PQC to various cryptographic protocols. Unlike elliptic curve cryptography (ECC)-based schemes, PQC requires a large memory footprint and key/signature size. Therefore, when migrating PQC to a protocol, depending on the PQC and protocol specifications, it can be hard to migrate PQC. In the case of the WAVE protocol, it is difficult to satisfy the accuracy of a specific PQC algorithm because segmentation of the signature occurs during transmission due to the limitation of the maximum packet size. Therefore, in this paper, we present two methodologies that can apply PQC while complying with IEEE 1609.2 standards to the WAVE protocol in the V2V environment. Whereas previous migration studies have focused on designing a hybrid mode of protocols, this paper explores solutions more intuitively at the application layer of protocols. We analyzed two postquantum digital signature algorithms (Crystals-Dilithium and Falcon) and the structure of basic-safety messages (BSMs) of the V2V protocol on the size side. Through this, we propose methods that can perform an independent signature verification process without waiting for all divided signatures in the WAVE protocol. Our methodology overcomes the limitation that schemes with large signature sizes cannot be mounted into the WAVE protocol. We also note that the architecture used as an on-board unit (OBU) in an autonomous driving environment is mainly a microprocessor. We investigated an optimized PQC implementation in the OBU environment and simulated our methodology with the V2Verifier. Finally, we measured the accurate latency through simulation in Jetson Xavier, which is mainly used as an OBU in the V2V communication network.

Man in the middle Attack (MIMA) problem of Diffie-Hellman key exchange (D-H) protocol, has led to introduce the Hash Diffie-Hellman key exchange (H-D-H) protocol. Which was cracked by applying the brute force attack (BFA) results of hash function. For this paper, a system will be suggested that focusses on an improved key exchange (D-H) protocol, and distributed transform encoder (DTE). That system utilized for enhanced (D-H) protocol algorithm when (D-H) is applied for generating the keys used for encrypting data of long messages. Hash256, with two secret keys and one public key are used for D-H protocol improvements. Finally, DTE where applied, this cryptosystem led to increase the efficiency of data transfer security with strengthening the shared secret key code. Also, it has removed the important problems such as MITM and BFA, as compared to the previous work.

A tropical encryption scheme is analyzed in this paper, which uses double key exchange protocol (KEP). The key exchange protocol is divided into two stages: The first stage of the key exchange uses matrix power function in a tropical semiring; the obtained shared key at the first phase of the key exchange serves as an input for the second phase. This paper proves that the common secret key of the first key exchange phase can be obtained by solving linear equations, and when the order of the matrix is 50, the time to solve the shared key is less than 1 second. Finally, the common secret key of the second phase can be obtained through KU attack and common secret key of the first key exchange. So the protocol isn’t secure.

Although postquantum cryptography is of growing practical concern, not many works have been devoted to implementation security issues related to postquantum schemes. In this paper, we look in particular at fault attacks against implementations of lattice-based signatures and key exchange protocols. For signature schemes, we are interested both in Fiat–Shamir type constructions (particularly BLISS, but also GLP, PASSSign, and Ring-TESLA) and in hash-and-sign schemes (particularly the GPV-based scheme of Ducas–Prest–Lyubashevsky). For key exchange protocols, we study the implementations of NewHope, Frodo, and Kyber. These schemes form a representative sample of modern, practical lattice-based signatures and key exchange protocols, and achieve a high level of efficiency in both software and hardware. We present several fault attacks against those schemes that recover the entire key recovery with only a few faulty executions (sometimes only one), show that those attacks can be mounted in practice based on concrete experiments in hardware, and discuss possible countermeasures against them.

This review explores the current state of public key cryptography based on idempotent semirings, with an emphasis on tropical semirings. It examines key hard problems, such as the tropical discrete logarithm problem, semidirect tropical product problem, the factorization of tropical polynomials, and the matrix power function, that underpin the security of these protocols. Given the significant number of compromised protocols based on tropical semirings, most of which are variations of the Stickel protocol, we present three algorithms and classify schemes of this type. The analysis is further illustrated with a figure that maps the relationships between tropical Stickel’s-like protocols and the attacks targeting them. Additionally, the review provides an in-depth exploration of the vulnerabilities that have led to many tropical semiring-based cryptosystems being compromised. To address these challenges, the review highlights promising alternative approaches, including non-tropical idempotent platforms and non-idempotent options, such as supertropical semirings, which offer potential solutions to overcome known limitations. Furthermore, a discussion on the interplay between tropical cryptography and post-quantum cryptography is presented, raising the following question: what is the future of tropical cryptography?

This paper is a continuation of our previous publication of enhanced matrix power function (MPF) as a conjectured one-way function. We are considering a problem introduced in our previous paper and prove that tis problem is NP-Complete. The proof is based on the dual interpretation of well known multivariate quadratic (MQ) problem defined over the binary field as a system of MQ equations, and as a general satisfiability (GSAT) problem. Due to this interpretation the necessary constraints to MPF function for cryptographic protocols construction can be added to initial GSAT problem. Then it is proved that obtained GSAT problem is NP-Complete using Schaefer dichotomy theorem. Referencing to this result, GSAT problem by polynomial-time reduction is reduced to the sub-problem of enhanced MPF, hence the latter is NP-Complete as well.

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.

Encryption has become increasingly prevalent in many applications and for various purposes, but its use also brings big challenges to network security. In this paper, we take the first steps towards addressing some of these challenges by introducing a novel system to identify key exchange protocols. These protocols are usually required if encryption keys are not shared in advance. We observed that key exchange protocols yield certain patterns of high-entropy data blocks, such as those found in key material. We propose a multi-resolution approach to accurately detect high-entropy data blocks and a method of generating fingerprints for cryptographic protocols. We provide experimental evidence that our approach has the potential to identify cryptographic protocols by their unique key exchanges, leading to the ability to detect malware traffic that includes customized key exchange protocols.

New NP-complete problem, named as multivariate quadratic power (MQP) problem, is presented. It is based on solution of multivariate quadratic power system of equations over the semigroup Zn, denoted by MQP(Zn), where n is positive integer. Two sequential polynomial-time reductions from known NP-complete multivariate quadratic (MQ) problem over the field Z2, i.e. MQ(Z2) to MQP(Zn) are constructed. It is proved that certain restricted MQP(Zn) problem over some subgroup of Zn is equivalent to MQ(Z2) problem. This allow us to prove that MQP(Zn) is NP-complete also.MQP problem is linked to some author’s previously declared matrix power function (MPF) used for several cryptographic protocols construction. Obtained NP-complete problem will be used to create new candidate one-way function (OWF) based on MPF for new cryptographic primitives’ construction.DOI: http://dx.doi.org/10.5755/j01.itc.41.1.821