Delving into Potential Asymmetric Cryptographic Algorithms for the Post-Quantum Era

In this paper we will look at how quantum computers solve problems used in modern cryptographic systems, and what post-quantum cryptographic systems exist. We focus on the scheme, that developed in the direction of multivariate cryptography. Our paper presents a symmetric cryptosystem in a finite field utilizing a matrix $\boldsymbol{A}_{m \times n}$ and its right inverse $B_{n x m}$ for $\boldsymbol{m}\gt \boldsymbol{n}$. This property of matrices, particularly evident when the dimensions are equal $m=n$, is actively used in AES. This utilization involves matrix transformations integral to the SubBytes() and InvSubBytes() functions. Moreover, a similar design principle is employed in the O’zDSt 1105:2006 data encryption algorithm, where it constitutes essential components of the AralashHolat() and TesAralashHolat() transformations. Additionally, we introduce a one-way function based on a special matrix with a zero determinant in a finite field, a concept not currently existing in asymmetric cryptography within our scheme. This approach fortifies defenses, diversifies cryptosystems, and addresses current vulnerabilities. We delve into matrix based asymmetric algorithms in finite fields, seeking to fortify our digital realm against the quantum unknown. This research promises postquantum security, aligns with existing cryptographic abstractions, and fosters agility in an evolving landscape.