Extending the ElGamal Cryptosystem to the Third Group of Units of $\Z_{n}$

TL;DR

The paper addresses extending ElGamal to the third group of units $U^{3}(\mathbb{Z}_{n})$ under cyclicity, aiming to enhance security. It constructs a bridge to a additive cyclic model via the isomorphism $f(a)=\log_{g_{3}}\log_{g_{2}}\log_{g_{1}}a \bmod n$ using generators $g_{1},g_{2},g_{3}$, mapping $U^{3}(\mathbb{Z}_{n})$ to $\mathbb{Z}_{\varphi(\varphi(\varphi(n)))}$ and defining the group operations accordingly. The paper then formulates the ElGamal scheme over $U^{3}(\mathbb{Z}_{n})$ with standard key generation, encryption, and decryption steps, illustrated by a concrete example ($n=81$). A numerical efficiency and security analysis against the $U^{2}$ extension shows significantly higher per-iteration cost (roughly 30×), supporting stronger security while acknowledging increased computation, backed by simulated results. The approach offers a new cryptographic platform leveraging deeper unit-group structure in modular rings for potential practical security gains.

Abstract

In this paper, we extend the ElGamal cryptosystem to the third group of units of the ring $\Z_{n}$, which we prove to be more secure than the previous extensions. We describe the arithmetic needed in the new setting. We also provide some numerical simulations that shows the security and efficiency of our proposed cryptosystem.

Figures (1)

• Figure 1: Time comparison between iterations done on algorithms of $U^{2}(\mathbb{Z}_{n})$ and $U^{3}(\mathbb{Z}_{m})$

Theorems & Definitions (8)

• Lemma 12
• Lemma 13
• Theorem 3.1
• Example 14
• Theorem 3.2
• Definition 15
• Theorem 4.1
• Example 19