A functional formulation and partial solution is given of the non-abelian eikonal problem associated with the exchange of non-interacting, charged or colored bosons between a pair of fermions, in the large $s$/small $t$ limit. A simple, functional ``contiguity" prescription is devised for extracting those terms which exponentiate, and appear to generate the leading, high-energy behavior of each perturbative order of this simplest non-abelian eikonal function; the lowest non-trivial order agrees with the corresponding SU(N) perturbative amplitude, while higher-order contributions to this eikonal generate an ``effective Reggeization" of the exchanged bosons, resembling previous results for the perturbative amplitude. One exact and several approximate examples are given, including an application to self-energy radiative corrections. In particular, for this class of graphs and to all orders in the coupling, we calculate the leading-log eikonal for SU(2). Based on this result, we conjecture the form of the eikonal scattering amplitude for SU(N).

19 pages, latex