Homomorphic Polynomial Public Key Cryptography for Quantum-secure Digital Signature

In their 2022 study, Kuang et al. introduced the Multivariable Polynomial Public Key (MPPK) cryptography, a quantum-safe public key cryptosystem leveraging the mutual inversion relationship between multiplication and division. MPPK employs multiplication for key pair construction and division for decryption, generating public multivariate polynomials. Kuang and Perepechaenko expanded the cryptosystem into the Homomorphic Polynomial Public Key (HPPK), transforming product polynomials over large hidden rings using homomorphic encryption through modular multiplications. Initially designed for key encapsulation mechanism (KEM), HPPK ensures security through homomorphic encryption of public polynomials over concealed rings. This paper extends its application to a digital signature scheme. The framework of HPPK KEM can not be directly applied to the digital signatures dues to the different nature of verification procedure compared to decryption procedure. Thus, in order to use the core ideas of the HPPK KEM scheme in the framework of digital signatures, the authors introduce an extension of the Barrett reduction algorithm. This extension transforms modular multiplications over hidden rings into divisions in the verification equation, conducted over a prime field. The extended algorithm non-linearly embeds the signature into public polynomial coefficients, employing the floor function of big integer divisions. This innovative approach overcomes vulnerabilities associated with linear relationships of earlier MPPK DS schemes. The security analysis reveals exponential complexity for both private key recovery and forged signature attacks, taking into account that the bit length of the rings is twice that of the prime field size. The effectiveness of the proposed Homomorphic Polynomial Public Key Digital Signature (HPPK DS) scheme is illustrated through a practical toy example, showcasing its intricate functionality and enhanced security features

Envisioning the Future of Cyber Security in Post-Quantum Era: A Survey on PQ Standardization, Applications, Challenges and Opportunities

The rise of quantum computers exposes vulnerabilities in current public key cryptographic protocols, necessitating the development of secure post-quantum (PQ) schemes. Hence, we conduct a comprehensive study on various PQ approaches, covering the constructional design, structural vulnerabilities, and offer security assessments, implementation evaluations, and a particular focus on side-channel attacks. We analyze global standardization processes, evaluate their metrics in relation to real-world applications, and primarily focus on standardized PQ schemes, selected additional signature competition candidates, and PQ-secure cutting-edge schemes beyond standardization. Finally, we present visions and potential future directions for a seamless transition to the PQ era

Quantum Resistant Authenticated Key Exchange for OPC UA using Hybrid X.509 Certificates

While the current progress in quantum computing opens new opportunities in a wide range of scientific fields, it poses a serious threat to today?s asymmetric cryptography. New quantum resistant primitives are already available but under active investigation. To avoid the risk of deploying immature schemes we combine them with well-established classical primitives to hybrid schemes, thus hedging our bets. Because quantum resistant primitives have higher resource requirements, the transition to them will affect resource constrained IoT devices in particular. We propose two modifications for the authenticated key establishment process of the industrial machine-to-machine communication protocol OPC UA to make it quantum resistant. Our first variant is based on Kyber for the establishment of shared secrets and uses either Falcon or Dilithium for digital signatures in combination with classical RSA. The second variant is solely based on Kyber in combination with classical RSA. We modify existing opensource software (open62541, mbedTLS) to integrate our two proposed variants and perform various performance measurement

Architectures for Code-based Post-Quantum Cryptography

L'abstract è presente nell'allegato / the abstract is in the attachmen

Universal Gaussian Elimination Hardware for Cryptographic Purposes

In this paper, we investigate the possibility of performing Gaussian elimination for arbitrary binary matrices on hardware. In particular, we presented a generic approach for hardware-based Gaussian elimination, which is able to process both non-singular and singular matrices. Previous works on hardware-based Gaussian elimination can only process non-singular ones. However, a plethora of cryptosystems, for instance, quantum-safe key encapsulation mechanisms based on rank-metric codes, ROLLO and RQC, which are among NIST post-quantum cryptography standardization round-2 candidates, require performing Gaussian elimination for random matrices regardless of the singularity. We accordingly implemented an optimized and parameterized Gaussian eliminator for (singular) matrices over binary fields, making the intense computation of linear algebra feasible and efficient on hardware. To the best of our knowledge, this work solves for the first time eliminating a singular matrix on reconfigurable hardware and also describes the a generic hardware architecture for rank-code based cryptographic schemes. The experimental results suggest hardware-based Gaussian elimination can be done in linear time regardless of the matrix type

Post-Quantum Cryptography: Riemann Primitives and Chrysalis

The Chrysalis project is a proposed method for post-quantum cryptography using the Riemann sphere. To this end, Riemann primitives are introduced in addition to a novel implementation of this new method. Chrysalis itself is the first cryptographic scheme to rely on Holomorphic Learning with Errors, which is a complex form of Learning with Errors relying on the Gauss Circle Problem within the Riemann sphere. The principle security reduction proposed by this novel cryptographic scheme applies complex analysis of a Riemannian manifold along with tangent bundles relative to a disjoint union of subsets based upon a maximal element. A surjective function allows the mapping of multivariate integrals onto subspaces. The proposed NP-Hard problem for security reduction is the non-commutative Grothendieck problem. The reduction of this problem is achieved by applying bilinear matrices in terms of the holomorphic vector bundle such that coordinate systems are intersected via surjective functions between each holomorphic expression. The result is an arbitrarily selected set of points within constraints of bilinear matrix inequalities approximate to the non-commutative problem. This is achieved by applying the quadratic form of bilinear matrices to a linear matrix inequality.Comment: Originally available on ResearchGate and now archive