Based on the RSA algorithm template, a \emph{cryptosystem with public keys and public operations} is defined, which complements the scheme with the property that encryption is performed by public keys and public operations, and decryption is performed by secret keys and secret operations. Removing the use of keys yields \emph{cryptosystems with public operations}. The sets of operations meant in this context are non-classical arithmetics. Asymmetric encryption without keys is assumed to be possible from the fact of the existence of operational equations, i.e. equations where the operations are unknown, but the numerical arguments are known ($12\uunderast x 3 \uadd y 45 = 67$, for example). However, more research is needed to establish the feasibility of encryption by public operations followed by decryption by secret ones. Asymmetric schemes with public keys and public operations, but equipped with non-classical arithmetics, will be called \emph{public-key cryptosystems with non-classical arithmetics}.