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}^m → {0,1}* amplifies relative lossiness from ℓ/n to L/m if for every function f:{0,1}ⁿ → {0,1}ⁿ it holds that 1 If f is injective then so is C^f. 2 If f has image size of at most 2 ^n-ℓ, then C^f has image size at most 2^m-L. The question is whether such C* exists for L/m ≫ ℓ/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^f is at most L/m ≤ ℓ/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.