Virtually all of modern cryptography relies on unproven assumptions. This is necessary, as the existence of cryptography would have wide ranging implications. In particular, it would hold that $\mathsf{P} \neq \mathsf{NP}$, which is not known to be true. Nevertheless, there clearly is a risk that the assumptions may be wrong. Therefore, an important field of research explores which assumptions are strictly necessary under different circumstances. This thesis contributes to this field by establishing lower bounds on the minimal assumptions in three different areas of cryptography.
We establish that assuming the existence of physically uncloneable functions (PUF), a specific kind of secure hardware, is not by itself sufficient to allow for secure two-party computation protocols without trusted setup. Specifically, we prove that unconditionally secure oblivious transfer can in general not be constructed from PUFs. Secondly, we establish a bound on the potential tightness of security proofs for Schnorr signatures. Essentially, no security proof based on virtually arbitrary non-interactive assumptions defined over an abstract group can be significantly tighter than the known, forking lemma based, proof. Thirdly, for very weak forms of program obfuscation, namely approximate indistinguishability obfuscation, we prove that they cannot exist with statistical security and computational assumptions are therefore necessary. This result holds unless the polynomial hierarchy collapses or one-way functions do not exist.