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

New Directions in Multivariate Public Key Cryptography

Most public key cryptosystems used in practice are based on integer factorization or discrete logarithms (in finite fields or elliptic curves). However, these systems suffer from two potential drawbacks. First, they must use large keys to maintain security, resulting in decreased efficiency. Second, if large enough quantum computers can be built, Shor\u27s algorithm will render them completely insecure. Multivariate public key cryptosystems (MPKC) are one possible alternative. MPKC makes use of the fact that solving multivariate polynomial systems over a finite field is an NP-complete problem, for which it is not known whether there is a polynomial algorithm on quantum computers. The main goal of this work is to show how to use new mathematical structures, specifically polynomial identities from algebraic geometry, to construct new multivariate public key cryptosystems. We begin with a basic overview of MPKC and present several significant cryptosystems that have been proposed. We also examine in detail some of the most powerful attacks against MPKCs. We propose a new framework for constructing multivariate public key cryptosystems and consider several strategies for constructing polynomial identities that can be utilized by the framework. In particular, we have discovered several new families of polynomial identities. Finally, we propose our new cryptosystem and give parameters for which it is secure against known attacks on MPKCs

PUBLIC KEY CRYPTOGRAPHY USING PERMUTATION P-POLYNOMIALS OVER FINITE FIELDS

In this paper we propose an efficient multivariate public key cryptosystem based on permutation p-polynomials over finite fields. We first characterize a class of permutation p-polynomials over finite fields $F_{q^{m}}$ and then construct a trapdoor function using this class of permutation p-polynomials. The complexity of encryption in our public key cryptosystem is $O(m^{3})$ multiplication which is equivalent to other multivariate public key cryptosystems. However the decryption is much faster than other multivariate public key cryptosystems. In decryption we need $O(m^{2})$ left cyclic shifts and $O(m^{2})$ xor operations

Poly-Dragon: An efficient Multivariate Public Key Cryptosystem

In this paper we propose an efficient multivariate public key cryptosystem. Public key of our cryptosystem contains polynomials of total degree three in plaintext and ciphertext variables, two in plaintext variables and one in ciphertext variables. However, it is possible to reduce the public key size by writing it as two sets of quadratic multivariate polynomials. The complexity of encryption in our public key cryptosystem is $O(n^{3})$, where $n$ is bit size, which is equivalent to other multivariate public key cryptosystems. For decryption we need only four exponentiations in the binary field. Our Public key algorithm is bijective and can be used for encryption as well as for signatures

Proposal of a Signature Scheme based on STS Trapdoor

A New digital signature scheme based on Stepwise Triangular Scheme (STS) is proposed. The proposed trapdoor has resolved the vulnerability of STS and secure against both Gröbner Bases and Rank Attacks. In addition, as a basic trapdoor, it is more efficient than the existing systems. With the efficient implementation, the Multivariate Public Key Cryptosystems (MPKC) signature public key has the signature longer than the message by less than 25 %, for example

[[alternative]]Computing and Crypto Applications of Discrete Algebraic Structures

計畫編號：NSC93-2115-M032-008研究期間：200408~200507研究經費：398,000[[sponsorship]]行政院國家科學委員

Nonlinear Piece In Hand Perturbation Vector Method for Enhancing Security of Multivariate Public Key Cryptosystems

Abstract. The piece in hand (PH) is a general scheme which is applicable to any reasonable type of multivariate public key cryptosystems for the purpose of enhancing their security. In this paper, we propose a new class PH method called NLPHPV (NonLinear Piece in Hand Perturbation Vector) method. Although our NLPHPV uses similar perturbation vectors as is used for the previously known internal perturbation method, this new method can avoid redundant repetitions in decryption process. With properly chosen parameter sizes, NLPHPV achieves an observable gain in security from the original multivariate public key cryptosystem. We demonstrate these by both theoretical analyses and computer simulations against major known attacks and provides the concrete sizes of security parameters, with which we even expect the grater security against potential quantum attacks

Enhanced STS using Check Equation --Extended Version of the Signature scheme proposed in the PQCrypt2010--

We propose solutions to the problems which has been left in the Enhanced STS, which was proposed in the PQCrypto 2010. Enhanced STS signature scheme is dened as the public key with the Complementary STS structure, in which two STS public keys are symmetrically joined together. Or, the complementary STS is the public key where simply two STS public keys are joined together, without the protection with Check Equation. We discuss the following issues left in the Enhanced STS, which was prosented in the PQCrypt2010: (i) We implied that there may exist a way to cryptanalyze the Complementary STS structure. Although it has been proposed that the system be protected by Check Equations [35][37], in order to cope with an unknown attack, we did not show the concrete procedure. We show the actual procedure to cryptanalyze it and forge a signature. (ii) We assumed that the Check Equation should be changed every time a document is signed. This practice is not always allowed. We improved this matter. The Check Equation which was proposed in the PQCrypto 2010 dened the valid life as a function of the number of times the documents are signed, because the secret key of Check Equation is analyzed by collecting valid signatures. Now we propose a new method of integrating the Check Equation into the secret key and eliminate the risk of the hidden information drawn from the existing signature

CyclicRainbow - A multivariate Signature Scheme with a Partially Cyclic Public Key based on Rainbow

Multivariate Cryptography is one of the alternatives to guarantee the security of communication in the post-quantum world. One major drawback of such schemes is the huge size of their keys. In \cite{PB10} Petzoldt et al. proposed a way how to reduce the public key size of the UOV scheme by a large factor. In this paper we extend this idea to the Rainbow signature scheme of Ding and Schmidt \cite{DS05}. By our construction it is possible to reduce he size of the public key by up to 62 \verb!%!