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Two RSA-based Cryptosystems

A. Telveenus

TL;DR

The paper addresses integrating primitive $m^{th}$ roots of unity within rings with zero divisors to create RSA-based cryptosystems. It builds a mathematical foundation around halidon rings and a ring-DFT framework, then develops two cryptosystems: RSA-DFT, which uses a hidden $oldsymbol{ extomega}$ and DFT/inverse-DFT for encryption/decryption, and RSA-HGR, which encodes messages via Halidon Group Rings. The main contributions are the formal characterization of halidon rings, the ring-DFT construct, and concrete RSA-based schemes with worked examples. The work suggests a pathway to alternative cryptosystems whose security hinges on ring-theoretic root-of-unity computations and the hardness of factoring, with potential future extensions via cyclotomic polynomials.

Abstract

The cryptosystem RSA is a very popular cryptosystem in the study of Cryptography. In this article, we explore how the idea of a primitive mth root of unity in a ring can be integrated into the Discrete Fourier Transform, leading to the development of new cryptosystems known as RSA-DFT and RSA-HGR.

Two RSA-based Cryptosystems

TL;DR

The paper addresses integrating primitive roots of unity within rings with zero divisors to create RSA-based cryptosystems. It builds a mathematical foundation around halidon rings and a ring-DFT framework, then develops two cryptosystems: RSA-DFT, which uses a hidden and DFT/inverse-DFT for encryption/decryption, and RSA-HGR, which encodes messages via Halidon Group Rings. The main contributions are the formal characterization of halidon rings, the ring-DFT construct, and concrete RSA-based schemes with worked examples. The work suggests a pathway to alternative cryptosystems whose security hinges on ring-theoretic root-of-unity computations and the hardness of factoring, with potential future extensions via cyclotomic polynomials.

Abstract

The cryptosystem RSA is a very popular cryptosystem in the study of Cryptography. In this article, we explore how the idea of a primitive mth root of unity in a ring can be integrated into the Discrete Fourier Transform, leading to the development of new cryptosystems known as RSA-DFT and RSA-HGR.
Paper Structure (6 sections, 15 theorems, 18 equations)

This paper contains 6 sections, 15 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Discrete Fourier Transforms
  4. RSA-DFT Cryptosystem
  5. RSA-HGR Cryptosystem
  6. Conclusion

Key Result

Theorem 1

(A. Telveenus ath) A finite commutative ring $R$ with unity is a halidon ring with index $m$ if and only if there is a primitive $m^{th}$ root of unity $\omega$ such that $m$, $\omega^{d}-1 \in U(R)$; the unit group of $R$ for all divisors $d$ of $m$ and $d<m$.

Theorems & Definitions (32)

  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Lemma 4
  • proof
  • Definition 1
  • Example 1
  • Proposition 5
  • proof
  • Theorem 6
  • ...and 22 more