The development of quantum computers is a threat to current cryptographic algorithms, such as the RSA algorithm. Due to this, the development of post quantum cryptography is a topic of research. One digital signature algorithm that is proposed as a quantum proof solution is the Dilithium digital signature system. Dilithium is based on a learning with errors problem (LWE) in which the value of generated signatures includes an error term. The security of Dilithium is based on previous re- search on topics such as the Module LWE problem, the Fiat Shamir transformation with aborts and the Module Short integer solution problem (MSIS ). The reference implementation includes optimization in the signature generation process. By using the number theoretic transform the multiplication of the ring elements is point-wise or in other words in linear time. This thesis presents the mathematical foundations on which the systems security is based on. In addition, the security of digital signatures, identification schemes and the LWE problem are discussed. In the final chapter these topics are brought together to present the Dilithium scheme.
The development of quantum computers is a threat to current cryptographic algorithms, such as the RSA algorithm. Due to this, the development of post quantum cryptography is a topic of research. One digital signature algorithm that is proposed as a quantum proof solution is the Dilithium digital signature system. Dilithium is based on a learning with errors problem (LWE) in which the value of generated signatures includes an error term. The security of Dilithium is based on previous re- search on topics such as the Module LWE problem, the Fiat Shamir transformation with aborts and the Module Short integer solution problem (MSIS ). The reference implementation includes optimization in the signature generation process. By using the number theoretic transform the multiplication of the ring elements is point-wise or in other words in linear time. This thesis presents the mathematical foundations on which the systems security is based on. In addition, the security of digital signatures, identification schemes and the LWE problem are discussed. In the final chapter these topics are brought together to present the Dilithium scheme.
The CRYSTALS-Dilithium digital signature algorithm, which was selected as the prototype of the new «Vershyna» digital signature algorithm, is analyzed in this paper. The characteristics of the National Digital Signature Standard Project and the construction of the «Vershyna» algorithm are also presented. During the analysis of the project, the predicted number of iterations that the algorithm must perform to create the correct signature was calculated. In addition, basic theoretical information about the structure of Fiat-Shamir with aborts and its security in quantum and classical models oracle models is also provided. We obtain our own results on the resistance of the «Vershyna» algorithm to the attack without the use of a message in classical and quantum oracle models. The resistance of the «Vershyna» algorithm to a key recovery attack is based on the assumption of the hardness of the MLWE~problem, and the resistance to existential signature forgery is based on the assumption of the hardness of the MSIS~problem. In this work, the expected level of hardness of SIS~and LWE~problems is calculated, to which there are reductions from MSIS~and MLWE~problems.
Since its birth by Chaum, blind signatures have become one of the fundamental components in the so-called e-cash or e-voting. The scheme ensures a message to be blinded before being signed. Among all lattice-based signatures submitted to NIST post-quantum cryptographic standardization, Dilithium is a very promising candidate. The scheme has some advantages such as simple to implement securely, conservative with parameters and minimal in total size of public key and signature. In this paper, we propose a blind signature scheme based on the framework of Dilithium in order to take advantages of Dilithium. The proposed scheme is blind and one-more unforgeable secure in the random oracle model under the hardness of the module Learning with Errors problem MLWE and the module short integer solution problem MSIS.
To overcome the issue of traditional blind signature schemes being vulnerable to quantum computing attacks during interactions over the consumer electronic Internet of Things (CIoT), a new blind signature scheme called CDBS has been developed. The post quantum digital signature technique CRYSTALS-Dilithium, which was standardized by NIST, serves as the foundation for the quantum-resistant blind signature scheme, CDBS, presented in this study. This system is appropriate for a variety of internet application scenarios, including e-ticketing, anonymous payments, and privacy protection, because it integrates the idea of blind signatures, which is not included in typical digital signature schemes. This scheme is designed to resist quantum computing attacks and is based on the post-quantum digital signature algorithm CRYSTALS-Dilithium, which NIST has selected. The whole technique utilizes the Fiat-Shamir signature structure, which consists of five stages: key creation, blinding, signing, unblinding, and verification. The security of the proposed scheme is dependent on the resolution of the Module Learning with Errors (MLWE) problem and the Short Integer Solution (SIS) problem. Additionally, the scheme must satisfy the criteria of correctness, blindness, and unforgeability. According to the test results, the CRYSTALS-Dilithium scheme requires an average of 401.89 μ s for a single round of verification and signing. The CDBS technique takes an average of 657.65 ls for one round of signing and verification since the blinding and unblinding modules are added on top of the original algorithm, increasing the algorithm’s complexity. The software testing results indicate that the average time required to complete a single full blind signature and verification operation is 657.65 microseconds. The suggested system offers a framework for expanding the CRYSTALS-Dilithium digital signature algorithm.
Digital signatures are indispensable for security on the Internet, because they guarantee authenticity, integrity, and non-repudiation, of namely e-mails, software updates, and in the Transport Layer Security (TLS) protocol which is used for secure data transfer, for example. Most signature schemes that are currently in use such as the RSA signature scheme, are considered secure as long as the integer factorization problem or the discrete logarithm (DL) problem are computationally hard. At present, no algorithms have yet been found to solve these problems on conventional computers in polynomial time. However, in 1997, Shor published a polynomial-time algorithm that uses quantum computation to solve the integer factorization and the DL problem. In particular, this means that RSA signatures are considered broken as soon as large-scale quantum computers exist. Due to significant advances in the area of quantum computing, it is reasonable to assume that within 20 years, quantum computers that are able to break the RSA scheme, could exist. In order to maintain authenticity, integrity, and non-repudiation of data, cryptographic schemes that cannot be broken by quantum attacks are required. In addition, these so-called post-quantum secure schemes should be sufficiently efficient to be suitable for all established applications. Furthermore, solutions enabling a timely and secure transition from classical to post-quantum schemes are needed. This thesis contributes to the above-mentioned transition. In this thesis, we present the two lattice-based digital signature schemes TESLA and qTESLA, whereby lattice-based cryptography is one of five approaches to construct post-quantum secure schemes. Furthermore, we prove that our signature schemes are secure as long as the so-called Learning With Errors (LWE) problem is computationally hard to solve. It is presumed that even quantum computers cannot solve the LWE problem in polynomial time. The security of our schemes is proven using security reductions. Since our reductions are tight and explicit, efficient instantiations are possible that provably guarantee a selected security level, as long as the corresponding LWE instance provides a certain hardness level. Since both our reductions (as proven in the quantum random oracle model) and instantiations, take into account quantum attackers, TESLA and qTESLA are considered post-quantum secure. Concurrently, the run-times for generating and verifying signatures of qTESLA are similar (or faster) than those of the RSA scheme. However, key and signature sizes of RSA are smaller than those of qTESLA. In order to protect both the theoretical signature schemes and their implementations against attacks, we analyze possible vulnerabilities against implementation attacks. In particular, cache-side-channel attacks resulting from observing the cache behavior and fault attacks, which recover secret information by actively disrupting the execution of an algorithm are focused. We present effective countermeasures for each implementation attack we found. Our analyses and countermeasures also influence the design and implementation of qTESLA. Although our schemes are considered (post-quantum) secure according to state-of-the-art LWE attacks, cryptanalysis of lattice-based schemes is still a relatively new field of research in comparison to RSA schemes. Hence, there is a lack of confidence in the concrete instantiations and their promised security levels. However, due to developments within the field of quantum computers, a transition to post-quantum secure solutions seems to be more urgently required than ever. To solve this dilemma, we present an approach to combine two schemes, e.g., qTESLA and the RSA signature scheme, so that the combination is secure as long as one of the two combined schemes is secure. We present several of such combiners to construct hybrid signature schemes and hybrid key encapsulation mechanisms to ensure both authenticity and confidentiality in our Public-Key Infrastructure (PKI). Lastly, we also demonstrate how to apply the resulting hybrid schemes in standards such as X.509 or TLS. To summarize, this work presents post-quantum secure candidates which can, using our hybrid schemes, add post-quantum security to the current classical security in our PKI.
Digital signature is one of the most important cryptography primitives. Recently, more and more works have been done to construct signatures over lattice problems to keep them secure in the quantum age. Among them, a ring-based signature scheme named Dilithium is the most efficient one and a candidate in the third round of the National Institute of Standards and Technology's post-quantum cryptography project. To make those schemes work well in large network, we constructed the first ring-based Identity-based signature (IBS) scheme for light-weight authentication. The construction in this paper relies on the transformations introduced by Bellare et al. in Journal of Cryptology (Vol.22, No.1, pp.1–61, 2009) and its security can be proved under the hardness of ring-learning with errors problem in the random oracle model. Due to better trapdoor and polynomial ring setting, our proposed scheme are much better than the previous ones in terms of both computation and communication complexities.
In this paper, we discuss the challenges of quantum computing to traditional cryptography and the latest development of post-quantum cryptography as a coping strategy. As quantum computing advances, cryptographic algorithms that rely on the complexity of classical computers, such as RSA and ECC, are at risk of failing, which has prompted the cryptography community to seek new encryption solutions. Post-quantum cryptography focuses on encryption technology and develops resistance to quantum computing attacks. It studies encryption schemes based on lattice theory, multivariate polynomials, coding theory, and hash functions. In addition, lattice-based cryptography is critical. It builds multiple algorithms, including RLWE and NTRU, focusing on LWE and SIS issues, demonstrating strong potential in a quantum computing environment. In terms of signature algorithms, Dilithium and Falcon lattice-based schemes are the focus of standardization. In the simulation analysis of these algorithms, we evaluated their security, efficiency, and compatibility, laying the theoretical and practical foundation for the cryptosystem transformation in the post-quantum era.