  We study a parity-conserving single-spin beam asymmetry of elastic
electron-proton scattering induced by an absorptive part of the two-photon
exchange amplitude. It is demonstrated that excitation of inelastic hadronic
intermediate states by the consecutive exchange of two photons leads to
logarithmic and double-logarithmic enhancement due to contributions of hard
collinear quasi-real photons. The asymmetry at small electron scattering angles
is expressed in terms of the total photoproduction cross section on the proton,
and is predicted to reach the magnitude of 20-30 parts per million. At these
conditions and fixed 4-momentum transfers, the asymmetry is rising
logarithmically with increasing electron beam energy, following the high-energy
diffractive behavior of total photoproduction cross section on the proton.
