MQ Signature and Proxy Signature Schemes with Exact Security Based on UOV Signature

Multivariate public key cryptography which relies on MQ (Multivariate Quadratic) problems is one of the main approaches to guarantee the security of communication in the post-quantum world. In this paper, we propose a combined MQ signature scheme based on the yet unbroken UOV (Unbalanced Oil and Vinegar) signature if parameters are properly chosen. Our scheme can not only reduce the public key size of the UOV signature, but also provide more tighter bound of security against chosen-message attack in the random oracle model. On the other hand, we propose a proxy signature scheme based on our proposed combined signature scheme. Additionally, we give a strict security proof for our proxy signature scheme. Finally, we present experiments for all of our proposed schemes and the baseline schemes. Comparisons with related schemes show that our work has some advantages on performance along with more strict security

Post quantum proxy signature scheme based on the multivariate public key cryptographic signature

Proxy signature is a very useful technique which allows the original signer to delegate the signing capability to a proxy signer to perform the signing operation. It finds wide applications especially in the distributed environment where the entities such as the wireless sensors are short of computational power and needed to be convinced to the authenticity of the server. Due to less proxy signature schemes in the post-quantum cryptography aspect, in this article, we investigate the proxy signature in the post-quantum setting so that it can resist against the potential attacks from the quantum adversaries. A general multivariate public key cryptographic proxy scheme based on a multivariate public key cryptographic signature scheme is proposed, and a heuristic security proof is given for our general construction. We show that the construction can reach Existential Unforgeability under an Adaptive Chosen Message Attack with Proxy Key Exposure assuming that the underlying signature is Existential Unforgeability under an Adaptive Chosen Message Attack. We then use our general scheme to construct practical proxy signature schemes for three well-known and promising multivariate public key cryptographic signature schemes. We implement our schemes and compare with several previous constructions to show our efficiency advantage, which further indicates the potential application prospect in the distributed network environment

Selecting and Reducing Key Sizes for Multivariate Cryptography

Cryptographic techniques are essential for the security of communication in modern society. As more and more business processes are performed via the Internet, the need for efficient cryptographic solutions will further increase in the future. Today, nearly all cryptographic schemes used in practice are based on the two problems of factoring large integers and solving discrete logarithms. However, schemes based on these problems will become insecure when large enough quantum computers are built. The reason for this is Shor's algorithm, which solves number theoretic problems such as integer factorization and discrete logarithms in polynomial time on a quantum computer. Therefore one needs alternatives to those classical public key schemes. Besides lattice, code and hash based cryptosystems, multivariate cryptography seems to be a candidate for this. Additional to their (believed) resistance against quantum computer attacks, multivariate schemes are very fast and require only modest computational resources, which makes them attractive for the use on low cost devices such as RFID chips and smart cards. However, there remain some open problems to be solved, such as the unclear parameter choice of multivariate schemes, the large key sizes and the lack of more advanced multivariate schemes like signatures with special properties and key exchange protocols. In this dissertation we address two of these open questions in the area of multivariate cryptography. In the first part we consider the question of the parameter choice of multivariate schemes. We start with the security model of Lenstra and Verheul, which, on the basis of certain assumptions like the development of the computing environment and the budget of an attacker, proposes security levels for now and the near future. Based on this model we study the known attacks against multivariate schemes in general and the Rainbow signature scheme in particular and use this analysis to propose secure parameter sets for these schemes for the years 2012 - 2050. In the second part of this dissertation we present an approach to reduce the public key size of certain multivariate signature schemes such as UOV and Rainbow. We achieve the reduction by inserting a structured matrix into the coefficient matrix of the public key, which enables us to store the public key in an efficient way. We propose several improved versions of UOV and Rainbow which reduce the size of the public key by factors of 8 and 3 respectively. Using the results of the first part, we show that using structured public keys does not weaken the security of the underlying schemes against known attacks. Furthermore we show how the structure of the public key can be used to speed up the verification process of the schemes. Hereby we get a speed up of factors of 6 for UOV and 2 for Rainbow. Finally we show how to apply our techniques to the QUAD stream cipher. By doing so we can increase the data throughput of QUAD by a factor of 7

Signing Information in the Quantum Era

Signatures are primarily used as a mark of authenticity, to demonstrate that the sender of a message is who they claim to be. In the current digital age, signatures underpin trust in the vast majority of information that we exchange, particularly on public networks such as the internet. However, schemes for signing digital information which are based on assumptions of computational complexity are facing challenges from advances in mathematics, the capability of computers, and the advent of the quantum era. Here we present a review of digital signature schemes, looking at their origins and where they are under threat. Next, we introduce post-quantum digital schemes, which are being developed with the specific intent of mitigating against threats from quantum algorithms whilst still relying on digital processes and infrastructure. Finally, we review schemes for signing information carried on quantum channels, which promise provable security metrics. Signatures were invented as a practical means of authenticating communications and it is important that the practicality of novel signature schemes is considered carefully, which is kept as a common theme of interest throughout this review

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

Short Solutions to Nonlinear Systems of Equations

This paper presents a new hard problem for use in cryptography, called Short Solutions to Nonlinear Equations (SSNE). This problem generalizes the Multivariate Quadratic (MQ) problem by requiring the solution be short; as well as the Short Integer Solutions (SIS) problem by requiring the underlying system of equations be nonlinear. The joint requirement causes common solving strategies such as lattice reduction or Gröbner basis algorithms to fail, and as a result SSNE admits shorter representations of equally hard problems. We show that SSNE can be used as the basis for a provably secure hash function. Despite failing to find public key cryptosystems relying on SSNE, we remain hopeful about that possibility

On new multivariate cryptosystems based on hidden Eulerian equations over finite fields

We propose new multivariate cryptosystems over $n$-dimensional vector space over a finite field $F_q$ based on idea of hidden discrete logarithm problem for ${F^*}_q$. These cryptosystems are based on hidden eulerian equations $x^{\alpha}=a$, $(\alpha, q-1)=1$. The method is based on the idea of Eulerian transformations, which allow us to use asymmetric algorithms based on families of nonlinear multiplicatively injective maps of prescribed polynomial density and flexible degree

On two windows multivariate cryptosystem depending on random parameters

The concept of multivariate bijective map of an affine space Kn over commutative Ring K was already used in Cryptography. We consider the idea of nonbijective multivariate polynomial map Fn of Kn into Kn represented as ''partially invertible decomposition'' F(1)nF(2)n…F(k)n, k=k(n), such that knowledge on the decomposition and given value u=F(v) allow to restore a special part v′ of reimage v. We combine an idea of ''oil and vinegar signatures cryptosystem'' with the idea of linguistic graph based map with partially invertible decomposition to introduce a new cryptosystem. The decomposition will be induced by pseudorandom walk on the linguistic graph and its special quotient (homomorphic image). We estimate the complexity of such general algorithm in case of special family of graphs with quotients, where both graphs form known families of Extremal Graph Theory. The map created by key holder (Alice) corresponds to pseudorandom sequence of ring elements. The postquantum version of the algorithm can be obtained simply by the usage of random strings instead of pseudorandom

Developments in multivariate post quantum cryptography.

Ever since Shor\u27s algorithm was introduced in 1994, cryptographers have been working to develop cryptosystems that can resist known quantum computer attacks. This push for quantum attack resistant schemes is known as post quantum cryptography. Specifically, my contributions to post quantum cryptography has been to the family of schemes known as Multivariate Public Key Cryptography (MPKC), which is a very attractive candidate for digital signature standardization in the post quantum collective for a wide variety of applications. In this document I will be providing all necessary background to fully understand MPKC and post quantum cryptography as a whole. Then, I will walk through the contributions I provided in my publications relating to differential security proofs for HFEv and HFEv−, key recovery attack for all parameters of HFEm, and my newly proposed multivariate encryption scheme, HFERP