Analysis of Existing Approaches and Algorithms of Post-Quantum Cryptography

The methodological approach of this study involves an analysis of the functioning features of various cryptosystems, comparing their efficiency and stability. The analysis considers not only the models that participated in the NIST competition for post-quantum cryptography algorithms but also systems that have been used for a long time. Including non-NIST algorithms allows for a broader exploration of the cryptographic landscape beyond the competition. These alternative algorithms may offer unique features, novel approaches, or different trade-offs in terms of efficiency, security, or practical implementation. Evaluating their performance and comparing them to NIST competition algorithms provides a more comprehensive understanding of the available options for post-quantum cryptography. Including non-NIST algorithms in the study enables a more holistic assessment of the cryptographic systems landscape, ensuring that the evaluation considers established systems alongside the latest competition participants. This approach helps identify the most effective technologies and provides insights into potential directions for future research and development in the field of post-quantum cryptography.

The study is conducted in six stages, each focusing on a specific aspect of post-quantum cryptography. These stages include exploring approaches and Merkle signatures, examining NTRU encryption and the advantages of multidimensional quadratic systems, studying cryptographic encryption with supersingular isogeny and the SIKE model, analyzing public-key encryption and key establishment algorithms with CRYSTALS-KYBER and SABER, investigating digital signature algorithms such as Dilithium, Falcon, and alternative approaches, and concluding with a comparative analysis, identifying strengths and weaknesses, and outlining future directions in post-quantum cryptography. The study employs detailed descriptions, explanations, and comparisons to assess the effectiveness and security of different algorithms, providing valuable insights for the field of post-quantum cryptography.

Exploring approaches and Merkle signatures in post-quantum cryptography. At the first stage, it was indicated which approaches of post-quantum cryptography exclude the attacks possibility. A detailed description of the Merkle signature was provided. The disadvantages of algorithms based on hash functions were described. An explanation of the essence of cryptography based on error correction codes was provided on the example of the McEliece system, the history of its occurrence was presented.

NTRU encryption and the advantages of multidimensional quadratic systems. At the second stage, the Nth-degree TRUncated polynomial ring (NTRU) encryption system was presented, its main differences from the McEliece system were indicated. It was noted what actions should be taken to obtain an NTRU security guarantee. The cryptography advantages on multidimensional quadratic systems and the mathematical model of the braid group were revealed. The principle of the cryptosystem functioning of the hidden field’s equation was indicated. The process of developing a braid group, the conditions under which this model works effectively, as well as the advantages and disadvantages of the algorithm were described.

Cryptographic encryption with supersingular isogeny and the SIKE model. At the third stage, cryptographic encryption algorithms using supersingular isogeny were considered. The definition of the “isogeny” notion was provided. The features of the SIKE model, which reached the final of the third round of the NIST competition, were revealed. A protocol for generating a common key was described, which is based on the isogenies operation of supersingular elliptic curves. Emphasis was made on what tasks should be solved in this area.

Public-key encryption and key establishment algorithms with CRYSTALS-KYBER and SABER. The fourth stage considers public-key encryption and key establishment algorithms, which also made it to the third round of NIST. The features of an algebraic cryptographic suite based on the CRYSTALS-KYBER lattice and modular lattices were described. What operations are required for KyBER and Dilithium cryptographic primitives were indicated. An explanation was provided for the difference in algebraic structure between the lattices used in KyBER and Dilithium from Ring-LWE. The features of the KyBER mechanism and its design were studied, the existing modifications of this model were listed. The features of SABER, which is a secure mechanism for encapsulating IND-CCA2 keys, were considered. Three security levels of the SABER-suite were compared to standard counterparts.

Study of digital signature algorithms, including dilithium, falcon, and alternatives. At the fifth stage, a study of digital signature algorithms was conducted. A description of the operation principle of Dilithium was provided. The advantages and disadvantages of Falcon and Rainbow were briefly described. In addition, alternative algorithms that did not reach the NIST finals were provided at this stage. These include: SPHINCS+, Picnic, GeMSS, Haraka, SHAKE-256, Winternitz, etc. Similarities and differences between GeMSS and Rainbow were indicated. The process of keys generation in the Winternitz algorithm was described, one-time signature scheme was provided. A comparative table of characteristics of the SPHINCS+, Dilithium, Rainbox and Falcon algorithms was provided. A description of the SPHINCS+ digital signature modification was provided.

Comparative analysis, strengths, and future directions in post-quantum cryptography. At the final stage, the obtained results were compared with existing publications regarding this topic. It was revealed what information was missed or insufficiently disclosed. The strengths and shortcomings of the article were identified, conclusions about the most effective technologies were made, and ways for future studies were outlined.

Following the emergence of powerful quantum computers, many researchers in the cryptography area spoke about the need to revise information security algorithms. Post-quantum cryptographic algorithms were analysed by Kravchenko and Cherkesova [14]. In their work, the authors explained the “quantum superiority” concept and, as an example, described the condition under which a quantum computer is able to hack the RSA algorithm, which is a modern standard.

The current research provides additional details and insights compared to previous studies in the field of post-quantum cryptography. Specifically, the current study offers a more detailed analysis of the properties and cryptographic strength of various systems. For example, the study highlights that with an adequate key length, the symmetric AES encryption algorithm becomes more resistant to attacks, emphasizing the importance of its architecture. The advantages and disadvantages of different systems are explored, such as the limited number of signatures in hash functions, the practical difficulties with the McEliece code addressed by the McEliece-Niederreiter system and the subsequent NTRU encryption system, and the trade-offs in computational speed and key length in multidimensional quadratic systems and braid groups. The study also analyzes the encryption algorithms that made it to the finals of the NIST competition, comparing their resistance to quantum attacks and highlighting the strengths and weaknesses of each. The most promising algorithms identified in the study include WOTS+, Dilithium, and SABER. The research concludes by emphasizing the need for further development, testing, and thorough stability assessment of new algorithms in the field of post-quantum cryptography.

Key management complexity and suitability of elliptic curve systems: exploring vulnerabilities, benefits, and performance. Key management is quite complex; the algorithm is vulnerable to attacks through third-party channels [22, 23]. With regard to elliptic curves, the researchers [24] believe that these systems are suitable for public key encryption and key exchange. Benefits were identified for blind signature and undeniable signature schemes. The algorithm also has a small key size. However, the system is slow and unsuitable for devices with limited resources. This system arose relatively recently and is therefore little studied.

Examining cryptographic systems: exploring advantages, disadvantages, and development trends. The advantages of error correction codes include high speed of encryption and decryption, the information protection degree increases significantly with increasing key length [25]. The main disadvantage is the fact that in order to increase the algorithm security, it is necessary to increase the public key size many times over. Cryptography on lattices, as noted by the authors, has an acceptable performance. The features of the hash functions application are described and an overview of the development trend of post-quantum algorithms is provided. Unlike the work of the researchers, this article covers existing cryptographic systems in more detail.

In the work of Fedorov, more attention is paid to the classical system of McEliece, SABER, KyBER and NTRU, which reached the final of the second round of NIST [26]. The mechanisms of the algorithms action are described in more detail and the degree of their reliability is evaluated. Improved algorithms based on the McEliece system, for example, a double public key encryption system, are considered. Such system is more efficient and safer. The proof of the algorithm stability to ROM and QROM attacks is provided. The KyBER structure description is provided, however, not as detailed as in this article. The author points out that the SABER system was designed to be easy and flexible to use. To do this, the same key components are used in the module structure at different security levels [27]. SABER was also compared with NTRU and LWE schemes.

Exploring multidimensional public key cryptography: assessing security, performance, and advantages over discrete logarithms and integer factorization. The features analysis of multidimensional public key cryptography is conducted by Zakharova and Makarova [28]. The study noted that the security of these algorithms is based on the hypothesis that solving systems of quadratic polynomials is an NP-complete problem. The disadvantages of public-key cryptosystems based on discrete logarithms and integer factorisation are considered by the authors to be low calculation speed and unreliability [29]. As an alternative, multidimensional quadratic cryptosystems are suggested. Despite the requirement for a big length of public keys, the speed of these systems is much higher. As an example, researchers provide a cryptosystem based on hidden field equations (HFE) and substantiate their security.

The concept of cryptographic strength and the essence of algorithms based on learning with errors are extensively discussed in the study by Zakharova and Makhotin [30]. The authors provide a definition of stable algorithms and highlight that establishing absolute resistance to attacks is often impractical, but it is possible to calculate the probability of hacking. These algorithms are categorized into three types based on their level of resistance: computational, information-theoretic, and provable stability. The study also presents methods for solving cryptosystems based on learning with errors.

In the research conducted by Razumov et al. [31], an in-depth study of the NTRUEncrypt cryptosystem and its modification is presented. Unlike the current article, this study includes an analysis of the algorithmic complexity of NTRUEncrypt and its optimized version. The researchers developed software to implement both systems and demonstrated that the modified version performs faster than the standard version while optimizing the use of internal memory. The cryptographic strength of the system was confirmed, and performance comparisons with RSA showed significant speed improvements. The standard NTRUEncrypt outperformed RSA by 80% in message encryption speed, and the modified version performed even better with an 84% improvement. In decryption, both NTRUEncrypt systems achieved a 99% higher information processing speed than RSA, with the modified version being 17% faster than the standard one.

Bhattacharyya and Chakrabarti [32] emphasize the importance of consistency and simplicity in the encryption and decryption algorithms of an effective cryptosystem. They review the historical development of cryptography, discuss classical systems and their drawbacks, and focus on signature schemes based on hashing. The authors [33] also delve into the differences between quantum and post-quantum cryptography. Ott and Peikert explore the challenge of replacing widely used systems like RSA, ECDSA, ECDH, and DSA with alternative post-quantum cryptography. They point out that NIST's efforts alone are insufficient to ensure a smooth transition to post-quantum cryptography, given the diverse range of computer systems in existence. The work presents an action plan for disseminating new NIST standards and highlights the greater memory and computational requirements of post-quantum cryptography systems due to their larger key sizes and increased algorithmic complexity. The authors [34-36] analyze various factors influencing the transition process and propose solutions to expedite it, including conducting studies to evaluate algorithm replacements in different conditions, addressing political and social aspects of the transition, and exploring the application of these processes to new branches of cryptography, such as confidential computing protocols, blockchain, and homomorphic encryption.

Another review conducted by Di Chiano et al. [37] focuses on the NIST third-round finalists and alternative candidates for digital signatures. The study compares algorithms such as CRYSTALS-DILITHIUM, SPHINCS+, Picnic, Falcon, Rainbow, and GeMSS. Similarly, the work by W. Beullens provides a simplified description of the unbalanced UOV scheme and its Rainbow modification [38]. The authors experimentally investigate two types of attacks: rectangular MinRank attack and intersection attack, which are shown to be more powerful than existing attacks [39]. The findings indicate that UOV and Rainbow are not NIST compliant, as the new attacks significantly reduce the cost of key recovery. It is worth noting that this work lacks experimental evaluations of post-quantum cryptography algorithms and does not cover the functioning of cryptosystems such as NTRUEncrypt and AES. Nevertheless, the article offers a comprehensive overview of the various technologies in the emerging field of post-quantum cryptography. The analysis of various cryptographic systems in the current study provides valuable insights that can inform the development of new post-quantum cryptographic algorithms or the refinement of existing ones. The study offers a detailed understanding of the advantages and disadvantages of different systems, allowing researchers to identify areas for improvement and optimization. For example, the study highlights the limitations of traditional public-key cryptosystems based on discrete logarithms and integer factorization, such as low calculation speed and unreliability.

In contrast, multidimensional quadratic cryptosystems are suggested as an alternative with higher speed and security, despite the requirement for larger key sizes [40, 41]. This insight can guide researchers in the development of new algorithms that leverage the strengths of multidimensional quadratic systems while addressing their limitations, such as exploring methods to reduce key size without compromising security. Furthermore, the study provides a comprehensive analysis of the encryption algorithms that made it to the finals of the NIST competition, comparing their resistance to quantum attacks and highlighting their strengths and weaknesses. This analysis can inform researchers in the refinement and optimization of these algorithms to enhance their security and efficiency.

To further advance the field of post-quantum cryptography, several potential areas for future research and challenges need to be addressed. Continued cryptanalysis and security evaluation of post-quantum cryptographic algorithms are essential to ensure their resilience against both classical and quantum attacks. Researchers should explore novel attack techniques and evaluate the resistance of algorithms in various scenarios. Additionally, further research is needed to develop efficient and reliable methods for assessing the security of post-quantum cryptographic schemes, including formal security proofs and analysis tools.

Also, many post-quantum cryptographic algorithms have larger key sizes and higher computational requirements compared to traditional schemes. Future research should focus on improving the scalability and efficiency of these algorithms to make them more practical for real-world applications. This includes exploring optimization techniques, hardware acceleration, and parallelization strategies to enhance the performance of post-quantum cryptographic operations, enabling their seamless integration into existing systems and networks.