  Quantum communication aims to provide absolutely secure transmission of
secret information. State-of-the-art methods encode symbols into single photons
or coherent light with much less than one photon on average. For long distance
communication, typically a single-mode fiber is used and significant effort has
been devoted already to increase the data carrying capacity of a single optical
line. Here we propose and demonstrate a fundamentally new concept for remote
key establishment. Our method allows high-dimensional alphabets using spatial
degrees of freedom by transmitting information through a light-scrambling
multimode fiber and exploiting the no-cloning theorem. Eavesdropper attacks can
be detected without using randomly switched mutually unbiased bases. We prove
the security against a common class of intercept-resend and beam-splitting
attacks with single-photon Fock states and with weak coherent light. Since it
is optical fiber based, our method allows to naturally extend secure
communication to larger distances. We experimentally demonstrate this new type
of key exchange method by encoding information into a few-photon light pulse
decomposed over guided modes of an easily available multimode fiber.
