Cryptographic reductions

One of the main tools of cryptography are reductions. The simplest way to prove the security of a complex scheme based on the security of its constituents is to prove that if there exists an adversary that breaks the complex scheme then we can construct an adversary that breaks at least one of the constituent primitives. Since the security of the primitive is taken as a premise, this gives a contradiction and thus the scheme must be secure. However, since we want the scheme to work regardless of the actual primitives being used, we cannot make any assumptions about them other than them being efficient. This gives rise to so-called black-box reductions – we are assumed to be presented with a method of implementing the original primitive (the black box) and we can use it, but we have no idea on how it is constructed.