We introduce a new bounded theory RS 1 2 , and show that the functions which are Σ b 1 -representable in it are precisely random functions which can be computed in polynomial time. Concretely, we pass through the class of oracle functions over strings POR, which is introduced in Section 2, together with RS 1 2 . Then, we show that functions computed by poly-time PTMs are arithmetically characterized by a class of probabilistic bounded formulas: in Section 3.2, we prove that the class of poly-time oracle functions is equivalent to that of functions which are Σ b 1 -representable in RS 1 2 , and in Section 4 we establish the converse.