  We give a general reduction of lengths-of-proofs lower bounds for constant
depth Frege systems in DeMorgan language augmented by a connective counting
modulo a prime $p$ (the so called $AC^0[p]$ Frege systems) to computational
complexity lower bounds for search tasks involving search trees branching upon
values of maps on the vector space of low degree polynomials over the finite
filed with $p$ elements.
