Paper 2025/1191

A Polynomial Public-Key Cryptosystem Based on Jacobian-Preserving Composition

Abstract

We propose a public-key cryptosystem based on Jacobian-preserving polynomial compositions, utilizing algebraically invertible polynomial maps with hard-to-invert composition. The construction utilizes polynomial maps over $\mathbb{Z}_p$, where $p$ is a prime number, with Jacobian determinant equal to 1 to ensure invertibility. The public key function $H : \mathbb{Z}_p^n \to \mathbb{Z}_p^n$ is defined as the composition of invertible polynomial maps $f_1, f_2, \dots, f_k$, each with Jacobian determinant 1, while the private key consists of the individual components used in the composition. Although inverting the composition is possible, inverting without the knowledge of the factors is computationally infeasible. This system incorporates both triangular and affine polynomial maps. We discuss the construction, provide formal correctness proofs, analyze hardness assumptions, and present a Python-based prototype with benchmark results.

Note: This paper presents an original construction of a public-key cryptosystem based on Jacobian-preserving polynomial compositions. It includes formal proofs, worked examples, security analysis, and a Python implementation with benchmarks. In future versions, I plan to continue developing the system, including generalizations beyond tame automorphisms.