  We propose a generalization of an RSA-like scheme based on R\'edei rational
functions over the Pell hyperbola. Instead of a modulus which is a product of
two primes, we define the scheme on a multi-factor modulus, i.e. on a product
of more than two primes. This results in a scheme with a decryption which is
quadratically faster, in the number of primes factoring the modulus, than the
original RSA, while preserving a better security. The scheme reaches its best
efficiency advantage over RSA for high security levels, since in these cases
the modulus can contain more primes. Compared to the analog schemes based on
elliptic curves, as the KMOV cryptosystem, the proposed scheme is more
efficient. Furthermore a variation of the scheme with larger ciphertext size
does not suffer of impossible group operation attacks, as it happens for
schemes based on elliptic curves.
