Given two MAC schemes $\prod_1 = (keyGen_1, S_1, V_1)$ and $\prod_2=(keyGen_2, S_2, V_2)$.
$\prod_3$ runs $keyGen$ from $\prod_1$ and $\prod_2$, respectively, to obtain $(k_1, k_2)$. $\prod_3$, where $S_3 = ((k_1,k_2), (m_1,m_2))$ then runs $S_1(k_1,m_1)\rightarrow t_1$ and $S_2(k_2,m_2)\rightarrow t_2$. And obtain $t_3 := t_1||t_2$. Would $\prod_3$ be a secure MAC?
And the follow up is when $t_3:= t_1 \oplus t_2 $, would this also be a secure MAC?
My guess is that the concatenation is secure because attackers would have no way of knowing how $t_1$ and $t_2$ is generated. For the second one, my intuition tells me that adversary can somehow swap the messages and produce a valid (m,t) pair?