An efficient and secure RSA--like cryptosystem exploiting R\'edei rational functions over conics

TL;DR

This paper introduces a new RSA-like cryptosystem using Rédéi rational functions over conics, achieving faster decryption and enhanced security in broadcast scenarios compared to traditional RSA.

Contribution

The paper presents a novel RSA-like scheme based on conic isomorphisms and Rédéi functions, improving efficiency and security over existing RSA and curve-based schemes.

Findings

Decryption is twice as fast as RSA.

Requires fewer modular inversions than other RSA-like schemes.

Offers comparable security to RSA in one-to-one communication and increased security in broadcast applications.

Abstract

We define an isomorphism between the group of points of a conic and the set of integers modulo a prime equipped with a non-standard product. This product can be efficiently evaluated through the use of R\'edei rational functions. We then exploit the isomorphism to construct a novel RSA-like scheme. We compare our scheme with classic RSA and with RSA-like schemes based on the cubic or conic equation. The decryption operation of the proposed scheme turns to be two times faster than RSA, and involves the lowest number of modular inversions with respect to other RSA-like schemes based on curves. Our solution offers the same security as RSA in a one-to-one communication and more security in broadcast applications.