  Resonant photopion production with the cosmic microwave background predicts a
suppression of extragalactic protons above the famous Greisen-Zatsepin-Kuzmin
cutoff at about E_{GZK} ~ 5 x 10^10 GeV. Current cosmic ray data measured by
the AGASA and HiRes Collaborations do not unambiguously confirm the GZK cutoff
and leave a window for speculations about the origin and chemical composition
of the highest energy cosmic rays. In this work we analyze the possibility of
strongly interacting neutrino primaries and derive model-independent
quantitative requirements on the neutrino-nucleon inelastic cross section for a
viable explanation of the cosmic ray data. Search results on weakly interacting
cosmic particles from the AGASA and RICE experiments are taken into account
simultaneously. Using a flexible parameterization of the inelastic
neutrino-nucleon cross section we find that a combined fit of the data does not
favor the Standard Model neutrino-nucleon inelastic cross section, but
requires, at 90 % confidence level, a steep increase within one energy decade
around E_{GZK} by four orders of magnitude. We illustrate such an enhancement
within some extensions of the Standard Model. The impact of new cosmic ray data
or cosmic neutrino search results on this scenario, notably from the Pierre
Auger Observatory soon, can be immediately evaluated within our approach.
