In this talk, we present recent results on the Erdös distinct distances problem over prime fields. More precisely, we will show that a Cartesian product set \(A^d \subseteq (F_p)^d\) determines almost \(\lvert A \rvert^2\) distinct distances if \(\lvert A\rvert\) is not too large. We also mention a recent result on expanding polynomials over Fp, namely, we will prove that for a quadratic polynomial \(f \in F_p[x, y, z]\), which is not of the form \(g(h(x) + k(y) + l(z))\), we have \(\lvert f(A×B×C)\rvert >> N^{3/2}\) for any sets \(A,B,C \subseteq F_p\) with \(N=\|A\| = \|B\| = \|C\| ≪ p^{2/3}\). Some applications of these results on sum-product estimates will be provided. This is joint work with Le Anh Vinh and Frank de Zeeuw.