Post-quantum encryption algorithms of high-degree 3-variable polynomial congruences: BS cryptosystems and BS key generation
Nicholas J. Daras
TL;DR
The paper proposes post-quantum encryption algorithms based on the Beal-Schur three-variable congruence $x^{p}+y^{q}\equiv z^{r} \pmod{\mathcal{N}}$ in finite fields. It provides a constructive proof of Beal's conjecture in the integers and a Fourier-analytic local result showing nontrivial finite-field solutions exist for large primes under divisibility conditions among $p,q,r$, enabling BS cryptosystems and key-generation schemes with an infinite parameter supply. The main contributions include three BS encryption schemes, two BS key-generation schemes, and theoretical guarantees on solution existence that underpin both security and flexibility. This work links Diophantine problem hardness to practical post-quantum cryptography, offering scalable parameter choices and novel cryptographic primitives. The results have potential impact on cryptosystem design by leveraging high-degree multivariate congruences to achieve post-quantum resilience with configurable key structures.
Abstract
We will construct post-quantum encryption algorithms based on three-variable polynomial Beal-Schur congruence. After giving a proof of Beal's conjecture and citing some applications of it to selected cases where the discrete logarithm and some of its generalizations are unsolvable problems, we will investigate the formulation and validity of an appropriate version of the Beal's conjecture on finite fields of integers. In contrast to the infinite case, we will show that the corresponding Beal-Schur congruence equation $x^{p}+y^{q}\equiv z^{r} (mod \mathcal{N})$ has non-trivial solutions into the finite field $\mathbb{Z}_{\mathcal{N}} $, for all sufficiently large primes $\mathcal{N}$ that do not divide the product $xyz$, under certain mutual divisibility conditions of the exponents $p$, $q$ and $r$. We will apply this result to generate the so-called BS cryptosystems, i.e., simple and secure post-quantum encryption algorithms based on the Beal-Schur congruence equation, as well as new cryptographic key generation methods, whose post-quantum algorithmic encryption security relies on having an infinite number of options for the parameters $p$, $q$, $r$, $\mathcal{N}$.