What cryptography primitives can help me (generate unforgeable tag)?
Assuming that all the $m$ values are the same, then about the only thing to do with the precise verification procedure you specified is to set:

$$t_i = c \cdot {a'_i}^{-1}$$

for some secret constant $c$ (which needs to be the same for every tag).

This would appear to work (and any value of $c$ would work)... until some $a'_j$ value is leaked; then, the attacker could rederive $c$, and thus recover all the $a'_i$ values from the corresponding $t_i$ values.