Secure symmetric ciphers over the real field

TL;DR

This paper explores the possibility of creating secure symmetric key ciphers over the real numbers, extending previous work and enhancing security through multiple encryptions and entropy-based key uncertainty measures.

Contribution

It demonstrates the feasibility of real-number-based symmetric ciphers and extends existing designs to improve security using composite encryption and entropy analysis.

Findings

Security can be enhanced with multiple encryptions.

Feasibility of secure ciphers over real numbers is demonstrated.

Entropy measures indicate increased key uncertainty.

Abstract

Most cryptosystems are defined over finite algebraic structures where arithmetic operations are performed modulo natural numbers. This applies to private key as well as to public key ciphers. No secure cryptosystems defined over the field of real numbers are known. In this work, we demonstrate the feasibility of constructing secure symmetric key ciphers defined over the field of real numbers. We consider the security of ciphers introduced in a previous work and based on solving linear and non-linear equations numerically. We complement the design of those ciphers to satisfy the requirements of secure systems and, consequently, extend them into composite ciphers with multiple encryptions. We show security enhancements by estimating the uncertainty in finding the keys using a measure based on Shannon's entropy function.