Secure symmetric ciphers over the real field

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.