When I read the AFL16 paper by Araki, each party's share is adding a correlated randomness, like r1 = (a1b1−x1y1+α)/3 , r2 = (a2b2−x2y2+β)/3, r3 = (a3b3−x3y3+γ)/3 with α+β+γ=0, and use AES to generate that. Also in the ABY3 paper they add the similiar randomness to each party in arithmetic operations. I wonder why this is necessary. I notice that even without that randomness, the protocol is still correct. Did this miss any security guarantee? The full name of the paper is "High-Throughput Semi-Honest Secure Three-Party Computation with an Honest Majority", can anybody help to explain it?
The multiplication is indeed correct without adding the randomness, but it is no longer private (i.e., it leaks information). By adding the correlated randomness, it ensures that what any single party sees during the computation is just uniformly distributed and reveals nothing.