Lossy Functions Do Not Amplify Well

We consider the problem of amplifying the “lossiness” of functions. We say that an oracle circuit C∗ : {0, 1} → {0, 1}∗ amplifies relative lossiness from `/n to L/m if for every function f : {0, 1} → {0, 1} it holds that 1. If f is injective then so is C . 2. If f has image size of at most 2n−`, then C has image size at most 2m−L. The question is whether such C∗ exists for L/m `/n. This problem arises naturally in the context of cryptographic “lossy functions,” where the relative lossiness is the key parameter. We show that for every circuit C∗ that makes at most t queries to f , the relative lossiness of C is at most L/m ≤ `/n+O(log t)/n. In particular, no black-box method making a polynomial t = poly(n) number of queries can amplify relative lossiness by more than an O(logn)/n additive term. We show that this is tight by giving a simple construction (cascading with some randomization) that achieves such amplification.