Impossibility of strong kdm security with auxiliary input

We show that a strong notion of KDM security cannot be obtained by any encryption scheme in the auxiliary input setting, assuming Learning With Errors (LWE) and one-way permutations. The notion of security we deal with guarantees that for any (possibly inefficient) function f, it is computationally hard to distinguish between an encryption of $$\mathbf {0}$$ and an encryption of $$f(\mathsf {pk}, z)$$, where $$\mathsf {pk} $$ is the public key and z is the auxiliary input. Furthermore, we show that this holds even when restricted to bounded-length auxiliary input where z is much shorter than $$\mathsf {pk} $$ under the additional assumption that (non-leveled) fully homomorphic encryption exists.