  Adiabatic heating of solar wind electrons at the Earth's bow shock and its
foreshock region produces transversely anisotropic hot electrons that, in turn,
generate intense high-frequency whistler-mode waves. These waves are often
detected by spacecraft as narrow-band, electromagnetic emissions in the
frequency range of [0.1,0.5] of the local electron gyrofrequency. Resonant
interactions between these waves and electrons may cause electron acceleration
and pitch-angle scattering, which can be important for creating the electron
population that seeds shock drift acceleration. The high intensity and
coherence of the observed whistler-mode waves prohibit the use of quasi-linear
theory to describe their interaction with electrons. In this paper, we aim to
develop a new theoretical approach to describe this interaction, that
incorporates nonlinear resonant interactions, gradients of the background
density and magnetic field, and the fine structure of the waveforms that
usually consist of short, intense wave-packet trains. This is the first of two
accompanying papers. It outlines a probabilistic approach to describe the
wave-particle interaction. We demonstrate how the wave-packet size affects
electron nonlinear resonance at the bow shock and foreshock regions, and how to
evaluate electron distribution dynamics in such a system that is frequented by
short, intense whistler-mode wave-packets. In the second paper, this
probabilistic approach is merged with a mapping technique, which allows us to
model systems containing short and long wave-packets.
