Algebraic Reductions of Knowledge.

We introduce reductions of knowledge , a generalization of arguments of knowledge, which reduce checking knowledge of a witness in one relation to checking knowledge of a witness in another (simpler) relation. Reductions of knowledge unify a growing class of modern techniques as well as provide a compositional framework to modularly reason about individual steps in complex arguments of knowledge. As a demonstration, we simplify and unify recursive arguments over linear algebraic statements by decomposing them as a sequence of reductions of knowledge. To do so, we develop the tensor reduction of knowledge , which generalizes the central reductive step common to many recursive arguments. Underlying the tensor reduction of knowledge is a new information-theoretic reduction, which, for any modules U , U 1 , and U 2 such that U ≅ U 1 ⊗ U 2 , reduces the task of evaluating a homomorphism in U to evaluating a homomorphism in U 1 and evaluating a homomorphism in U 2 .