  We study the multifractal spectrum of the current in the two-dimensional
random resistor network at the percolation threshold. We consider two ways of
applying the voltage difference: (i) two parallel bars, and (ii) two points.
Our numerical results suggest that in the infinite system limit, the
probability distribution behaves for small current i as P(i) ~ 1/i. As a
consequence, the moments of i of order q less than q_c=0 do not exist and all
current of value below the most probable one have the fractal dimension of the
backbone. The backbone can thus be described in terms of only (i) blobs of
fractal dimension d_B and (ii) high current carrying bonds of fractal dimension
going from $1/\nu$ to d_B.
