Linear and almost-linear perfect hashing
I think there is a constraint that the domain of the hashing function must be {0...n-1}, where n means the number of elements to hash. Only defining in this way , the reduction from 3SUM to Conv3SUM works. 

The is an almost-linearity function in wiki. And it is easy to figure out a counter example for the existence of perfect linear hashing function.

Consider the following questions:

Let A = {1,2,3,4}, B={0,1,2,3}


how many triples (x,y,z) satisfy x+y=z, where x,y,z are in A and distinct！
what if x,y,z are in B


The answer of the first question is 


1+1=2
1+2=3
1+3=4
2+2=4


But there are only two triples in the second question. You can find out a bijective function, satisfying that linearity property, mapping A onto B .