Multivariate Public Key Cryptosystems Research Articles

Progress in Multivariate Cryptography: Systematic Review, Challenges, and Research Directions

Multivariate Public Key Cryptosystem (MPKC) seem to be promising toward future digital security even in the presence of quantum adversaries. MPKCs derive their security from the difficulty of solving a random system of multivariate polynomial equations over a finite field, which is known to be an NP-hard problem. This article aims at presenting a comprehensive survey that covers multivariate public key encryption and signature schemes specifically targeting toward security, efficiency, and parameter choice. The survey starts by giving an overview of the existing security challenges which include structural attacks such as MinRank attack, differential attack , and finding Gröbner basis for direct attack , and so on. Additionally, it discusses the necessary algorithms for the implementation of the multivariate schemes. This study also compares the promising multivariate encryption and signature schemes. The critical open challenges that are reviewed in this survey will serve as a single comprehensive source of information on multivariate encryption and signature schemes and a ready reference for researchers working in this rising area of public key cryptography.

Solving the Problem of Blockwise Isomorphism of Polynomials with Circulant Matrices

The problem of Isomorphism of Polynomials (IP problem) is known to be important to study the security of multivariate public key cryptosystems, one of the major candidates of post-quantum cryptography, against key recovery attacks. In these years, several schemes based on the IP problem itself or its generalization have been proposed. At PQCrypto 2020, Santoso introduced a generalization of the problem of Isomorphism of Polynomials, called the problem of Blockwise Isomorphism of Polynomials (BIP problem), and proposed a new Diffie-Hellman type encryption scheme based on this problem with Circulant matrices (BIPC problem). Quite recently, Ikematsu et al. proposed an attack called the linear stack attack to recover an equivalent key of Santoso's encryption scheme. While this attack reduced the security of the scheme, it does not contribute to solving the BIPC problem itself. In the present paper, we describe how to solve the BIPC problem directly by simplifying the BIPC problem due to the conjugation property of circulant matrices. In fact, we experimentally solved the BIPC problem with the parameter, which has 256 bit security by Santoso's security analysis and has 72.7bit security against the linear stack attack, by about 10 minutes.

Selection Strategy of F4-Style Algorithm to Solve MQ Problems Related to MPKC

Multivariate public-key cryptosystems are potential candidates for post-quantum cryptography. The security of multivariate public-key cryptosystems relies on the hardness of solving a system of multivariate quadratic polynomial equations. Faugère’s F4 algorithm is one of the solution techniques based on the theory of Gröbner bases and selects critical pairs to compose the Macaulay matrix. Reducing the matrix size is essential. Previous research has not fully examined how many critical pairs it takes to reduce to zero when echelonizing the Macaulay matrix in rows. Ito et al. (2021) proposed a new critical-pair selection strategy for solving multivariate quadratic problems associated with encryption schemes. Instead, this paper extends their selection strategy for solving the problems associated with digital signature schemes. Using the OpenF4 library, we compare the software performance between the integrated F4-style algorithm of the proposed methods and the original F4-style algorithm. Our experimental results demonstrate that the proposed methods can reduce the processing time of the F4-style algorithm by up to a factor of about seven under certain specific parameters. Moreover, we compute the minimum number of critical pairs to reduce to zero and propose their extrapolation outside our experimental scope for further research.

Multivariate Broadcast Encryption with Group Key Algorithm for Secured IoT

The expanding and ubiquitous availability of the Internet of Things (IoT) have changed everyone’s life easier and more convenient. Same time it also offers a number of issues, such as effectiveness, security, and excessive power consumption, which constitute a danger to intelligent IoT-based apps. Group managing is primarily used for transmitting and multi-pathing communications that are secured with a general group key and it can only be decrypted by an authorized group member. A centralized trustworthy system, which is in charge of key distribution and upgrades, is used to maintain group keys. To provide longitudinal access controls, Software Defined Network (SDN) based security controllers are employed for group administration services. Cloud service providers provide a variety of security features. There are just a few software security answers available. In the proposed system, a hybrid protocols were used in SDN and it embeds edge system to improve the security in the group communication. Tree-based algorithms compared with Group Key Establishment (GKE) and Multivariate public key cryptosystem with Broadcast Encryption in the proposed system. When all factors are considered, Broadcast Encryption (BE) appears to become the most logical solution to the issue. BE enables an initiator to send encrypted messages to a large set of recipients in a efficient and productive way, meanwhile assuring that the data can only be decrypted by defining characteristic. The proposed method improves the security, efficiency of the system and reduces the power consumption and minimizes the cost.

A multivariate identity-based broadcast encryption with applications to the internet of things

<p style='text-indent:20px;'>When Kevin Ashton proposed the catchword 'Internet of Things' in 1999, little did he know that technology will become an indispensable part of human lives in just two decades. In short, the Internet of Things (IoT), is a catch-all terminology used to describe devices connected to the internet. These devices can share and receive data as well as provide instructions over a network. By design itself, the IoT system requires multicasting data and information to a set of designated devices, securely. Taking everything into account, Broadcast Encryption (BE) seems to be the natural choice to address the problem. BE allows an originator to broadcast ciphertexts to a big group of receivers in a well-organized and competent way, while ensuring that only designated people can decrypt the data. In this work, we put forward the first Identity-Based Broadcast Encryption scheme based on multivariate polynomials that achieves post-quantum security. Multivariate public key cryptosystems (MPKC), touted as one of the most promising post-quantum cryptography candidates, forms the foundation on which our scheme relies upon, which allows it to be very cost-effective and faster when implemented. In addition, it also provides resistance to collusion attack, and as a consequence our scheme can be utilized to form an efficient and robust IoT system.</p>

Blockchain-Envisioned Provably Secure Multivariate Identity-Based Multi-Signature Scheme for Internet of Vehicles Environment

The deployed vehicles in an Internet of Vehicles (IoV) can take intelligent decisions by means of exchanging the real-time traffic-related information between the vehicles and IoV infrastructures. This further reduces the probability of the traffic jams and accidents. However, the insecure (public) communication among the various entities in IoV makes various security threats and attacks that can be launched by passive/active adversaries present in the network. In view of this context, there is a need of an efficient cryptographic primitive which can produce single compact signature. A multi-signature scheme (MSS) empowers a collection of signers to conjointly sign a given message using a single compact signature that can be verified by any verifier. Herein, we put forward a new identity-based multivariate MSS, namely <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">MV-MSS</monospace> , which is built on top of the intractability of multivariate-quadratic (MQ) problem. The fact is that multivariate public key cryptosystem provides fast, post-quantum safe and efficient primitives, which makes it the front runner candidate among the post-quantum cryptographic candidates. <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">MV-MSS</monospace> is proven to be secure in the existential unforgeability under chosen-message and chosen identity attack model if solving the MQ problem is NP-hard. We then incorporate the designed <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">MV-MSS</monospace> in IoV application where the leader (cluster head) selected from a group of vehicles in a dynamic cluster forms the multi-signatures on the messages securely received from its member vehicles. Later, the messages along with their multi-signatures are forwarded to the nearby road-side unit (RSU) of the cluster head, which are then forwarded to a cloud server in the blockchain center maintained by a Peer-to-Peer (P2P) cloud servers network. In this way, the messages and their signatures considered as transactions are put in blocks and added into a public blockchain with the help of consensus algorithm. A comparative study among the proposed <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">MV-MSS</monospace> and other existing schemes shows that <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">MV-MSS</monospace> is efficient and secure as compared to other schemes. Finally, a blockchain implementation through simulation study has been performed to show its practical use in IoV application.

A Multivariate-Based Provably Secure Certificateless Signature Scheme With Applications To The Internet Of Medical Things

Abstract Over the last few years, Internet of Medical Things (IoMT) has completely transformed the healthcare industry. It is bringing out the most notable, and unprecedented impacts on human health, and has totally changed the way we look at the healthcare industry. The healthcare sector all around the globe are leapfrogging, and adopting the technology, helping in transforming drastically in a very short span of time. However, as more and more number of medical devices are being connected to IoMT, security issues like ensuring authenticity and integrity of the transmitted data are also on the rise. In view of the context, there is a need of an efficient cryptographic primitive that can address these issues in a viable manner. A signature scheme seems to be the natural choice to mitigate the security concerns. But, traditional signature schemes, both public-key-infrastructure-based and Identity-based, have their own disadvantages, which makes them unsuitable for IoMT networks. Thus, to address the security issues and problems like certificate management and key escrow, herein, we put forward the first multivariate-based certificateless signature scheme, namely, Multivariate Certificateless Signature (Mul-CLS), which is built on top of the intractability of multivariate-quadratic (MQ) problem. The fact that multivariate public key cryptosystem provides fast, post-quantum safe and efficient primitives makes it a front-runner candidate among the other post-quantum cryptography candidates. Our scheme Mul-CLS provides existential unforgeability against chosen message and chosen identity Super Type I and Super Type II adversary if solving the MQ problem is NP-hard. In addition to that, our proposed Mul-CLS presents itself as a robust and cost-friendly cryptographic building block for building IoMT networks.

Practical complexities of probabilistic algorithms for solving Boolean polynomial systems

Solving a polynomial system over a finite field is an NP-complete problem of fundamental importance in both pure and applied mathematics. In particular, the security of the so-called multivariate public-key cryptosystems, such as HFE of Patarin and UOV of Kipnis et al., is based on the postulated hardness of solving quadratic polynomial systems over a finite field. Lokshtanov et al. (2017) were the first to introduce a probabilistic algorithm that, in the worst-case, solves a Boolean polynomial system in time O∗(2δn), for some δ∈(0,1) depending only on the degree of the system, thus beating the brute-force complexity O∗(2n). Later, Björklund et al. (2019) and then Dinur (2021) improved this method and devised probabilistic algorithms with a smaller exponent coefficient δ.We survey the theory behind these probabilistic algorithms, and we illustrate the results that we obtained by implementing them in C. In particular, for random quadratic Boolean systems, we estimate the practical complexities of the algorithms and their probabilities of success as their parameters change.

Quantum Algorithm for Boolean Equation Solving and Quantum Algebraic Attack on Cryptosystems

This paper presents a quantum algorithm to decide whether a Boolean equation system F has a solution and to compute one if F does have solutions with any given success probability. The runtime complexity of the algorithm is polynomial in the size of F and the condition number of certain Macaulay matrix associated with F. As a consequence, the authors give a polynomial-time quantum algorithm for solving Boolean equation systems if their condition numbers are polynomial in the size of F. The authors apply the proposed quantum algorithm to the cryptanalysis of several important cryptosystems: The stream cipher Trivum, the block cipher AES, the hash function SHA-3/Keccak, the multivariate public key cryptosystems, and show that they are secure under quantum algebraic attack only if the corresponding condition numbers are large. This leads to a new criterion for designing such cryptosystems which are safe against the attack of quantum computers: The corresponding condition number.

A Public Key Cryptosystem Using a Group of Permutation Polynomials

Abstract In this paper we propose an efficient multivariate encryption scheme based on permutation polynomials over finite fields. We single out a commutative group ℒ(q, m) of permutation polynomials over the finite field F q m. We construct a trapdoor function for the cryptosystem using polynomials in ℒ(2, m), where m =2 k for some k ≥ 0. The complexity of encryption in our public key cryptosystem is O(m 3) multiplications which is equivalent to other multivariate public key cryptosystems. For decryption only left cyclic shifts, permutation of bits and xor operations are used. It uses at most 5m 2+3m – 4 left cyclic shifts, 5m 2 +3m + 4 xor operations and 7 permutations on bits for decryption.

Certificateless Broadcast Multisignature Scheme Based on MPKC

Broadcast multisignature allows multiple signers to sign the same message, which can be used in many areas, such as electronic contract signing, educational administration management system and grade management system. At present, the security of most broadcast multisignature schemes mainly depends on the intractability of large integer factoring (LIF) or discrete logarithm (DL) problem. Thus, broadcast multisignature schemes will suffer from the potential threat of the quantum computing attacks. Hence, it is an important problem how to solve the quantum computing attacks in traditional broadcast multisignature. In this paper, we construct the first certificateless broadcast multisignature scheme based on multivariate public key cryptosystem (MPKC-CLBMSS), whose security is based on the hardness of the isomorphism of polynomials (IP) problem. MPKC-CLBMSS not only solves the problem of quantum computing attacks, but also avoids the key escrow issue in IB-PKC along with the certificate management problem in traditional PKI. In MPKC-CLBMSS, the signature length is as same as that of the partial signature, regardless of the number of signers; the verification time of signature is as same as for a partial signature. MPKC-CLBMSS has higher computational efficiency than the existing broadcast multisignature scheme. Moreover, the security proof shows that MPKC-CLBMSS satisfies the unforgeability in the random oracle model.

MPKC-based Threshold Proxy Signcryption Scheme

The threshold proxy signcryption can implement signature and encryption simultaneously in one logical step, and can be used to realize the decentralized protection of the group signature key, so it is an efficient technology for network security. Currently, most of the existing threshold proxy signcryption schemes are designed based on the traditional public key cryptosystems, and their security mainly depends on the difficulty of the large integer decomposition and the discrete logarithm. However, the traditional public key cryptosystems cannot resist the quantum computer attack, which makes the existing threshold proxy signcryption schemes based on traditional public key cryptosystems insecure against quantum attacks. Motivated by these concerns, we proposed a threshold proxy signcryption scheme based on Multivariate Public Key Cryptosystem (MPKC) which is one of the quantum attack-resistent public key algorithms. Under the premise of satisfying the threshold signcryption requirements of the threshold proxy, our scheme can not only realize the flexible participation of the proxy signcrypters but also resist the quantum computing attack. Finally, based on the assumption of Multivariate Quadratic (MQ) problem and Isomorphism Polynomial (IP) problem, the proof of the confidentiality and the unforgeability of the proposed scheme under the random oracle model is given.

Key Recovery Attacks on Multivariate Public Key Cryptosystems Derived from Quadratic Forms over an Extension Field

One of major ideas to design a multivariate public key cryptosystem (MPKC) is to generate its quadratic forms by a polynomial map over an extension field. In fact, Matsumoto-Imai's scheme (1988), HFE (Patarin, 1996), MFE (Wang et al., 2006) and multi-HFE (Chen et al., 2008) are constructed in this way and Sflash (Akkar et al., 2003), Quartz (Patarin et al., 2001), Gui (Petzoldt et al, 2015) are variants of these schemes. An advantage of such extension field type MPKCs is to reduce the numbers of variables and equations to be solved in the decryption process. In the present paper, we study the security of MPKCs whose quadratic forms are derived from a “quadratic” map over an extension field and propose a new attack on such MPKCs. Our attack recovers partial information of the secret affine maps in polynomial time when the field is of odd characteristic. Once such partial information is recovered, the attacker can find the plain-text for a given cipher-text by solving a system of quadratic equations over the extension field whose numbers of variables and equations are same to those of the system of quadratic equations used in the decryption process.

Current State of Multivariate Cryptography

A review of the current state of multivariate public-key cryptosystems compares and contrasts the most promising multivariate schemes in digital signatures and public-key encryption as well as their security.

Multivariate Cryptography Based on Clipped Hopfield Neural Network.

Designing secure and efficient multivariate public key cryptosystems [multivariate cryptography (MVC)] to strengthen the security of RSA and ECC in conventional and quantum computational environment continues to be a challenging research in recent years. In this paper, we will describe multivariate public key cryptosystems based on extended Clipped Hopfield Neural Network (CHNN) and implement it using the MVC (CHNN-MVC) framework operated in space. The Diffie-Hellman key exchange algorithm is extended into the matrix field, which illustrates the feasibility of its new applications in both classic and postquantum cryptography. The efficiency and security of our proposed new public key cryptosystem CHNN-MVC are simulated and found to be NP-hard. The proposed algorithm will strengthen multivariate public key cryptosystems and allows hardware realization practicality.

Software watermarking scheme based on multivariate public key cryptosystem

Software watermarking is an efficient tool to verify the copyright of software. Public key cryptosystem-based watermarking is widely researched. However, the popular public key cryptosystem is not secure under quantum algorithm. This paper proposes a novel software watermarking scheme based on multivariate public key cryptosystem. The copyright information generated by copyright holder is transformed into copyright numbers using multivariate quadratic polynomial equations inspired by multivariate public key cryptosystem (MPKC). Every polynomial is embedded into the host program independently. Based on the security performance of MPKC, the robustness and invisibility of the proposed scheme is significantly improved in comparison with the RSA-based watermarking method.

On tameness of Matsumoto-Imai central maps in three variables over the finite field $\mathbb F_2$

Triangular transformation method (TTM) is one of the multivariate public key cryptosystems (MPKC) based on the intractability of tame decomposition problem. In TTM, a special class of tame automorphisms are used to construct encryption schemes. However, because of the specificity of such tame automorphisms, it is important to evaluate the computational complexity of the tame decomposition problem for secure use of MPKC. In this paper, as the first step for security evaluations, we focus on Matsumoto-Imai cryptosystems. We shall prove that the Matsumoto-Imai central maps in three variables over $\mathbb{F}_{2}$ is tame, and we describe the tame decompositions of the Matsumoto-Imai central maps.

Cryptanalysis of the Multivariate Signature Scheme Proposed in PQCrypto 2013

In PQCrypto 2013, Yasuda, Takagi and Sakurai proposed a new signature scheme as one of multivariate public key cryptosystems (MPKCs). This scheme (called YTS) is based on the fact that there are two isometry classes of non-degenerate quadratic forms on a vector space with a prescribed dimension. The advantage of YTS is its efficiency. In fact, its signature generation is eight or nine times faster than Rainbow of similar size. For the security, it is known that the direct attack, the IP attack and the min-rank attack are applicable on YTS, and the running times are exponential time for the first and the second attacks and sub-exponential time for the third attack. In the present paper, we give a new attack on YTS whose approach is to use the diagonalization of matrices. Our attack works in polynomial time and it actually recovers equivalent secret keys of YTS having 140-bits security against min-rank attack in around fifteen seconds.

Proposal of the Multivariate Public Key Cryptosystem Relying on the Difficulty of Factoring a Product of Two Large Prime Numbers

Proposal of the Multivariate Public Key Cryptosystem Relying on the Difficulty of Factoring a Product of Two Large Prime Numbers

A security analysis of uniformly-layered rainbow defined over non-commutative rings

Abstract Hashimoto and Sakurai proposed a signature scheme (HS scheme), whose security is based on the difficulty of integer factorization. In this paper, we redefine HS scheme as a signature scheme in Multivariate Public Key Cryptosystems (MPKC). MPKC are public key cryptosystems whose security is based on the difficulty of solving multivariate quadratic equations, and candidates for post-quantum cryptography. In this paper, we analyze the security of the extended HS scheme using technique of security analysis for MPKC. Furthermore, based on the security analysis of the extended HS scheme, we estimate secure parameters of the extended HS scheme.