  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.
