  Chebyshev polynomials have been recently proposed for designing public-key
systems. Indeed, they enjoy some nice chaotic properties, which seem to be
suitable for use in Cryptography. Moreover, they satisfy a semi-group property,
which makes possible implementing a trapdoor mechanism. In this paper we study
a public key cryptosystem based on such polynomials, which provides both
encryption and digital signature. The cryptosystem works on real numbers and is
quite efficient. Unfortunately, from our analysis it comes up that it is not
secure. We describe an attack which permits to recover the corresponding
plaintext from a given ciphertext. The same attack can be applied to produce
forgeries if the cryptosystem is used for signing messages. Then, we point out
that also other primitives, a Diffie-Hellman like key agreement scheme and an
authentication scheme, designed along the same lines of the cryptosystem, are
not secure due to the aforementioned attack. We close the paper by discussing
the issues and the possibilities of constructing public key cryptosystems on
real numbers.
