Towards proving strong direct product theorems

A fundamental question of complexity theory is the direct product question. A famous example is Yao's XOR-lemma, in which one assumes that some function f is hard on average for small circuits (mean-ing that every circuit of some fixed size s which attempts to compute f is wrong on a non-negligible fraction of the inputs) and concludes that every circuit of size s′ only has a small advantage over guessing randomly when computing f^⊕k (x ₁,..., x_k) = f(x₁)⊕...⊕f(x _k) on independently chosen x₁,..., x_k. All known proofs of this lemma have the property that s′ < s. In words, the circuit which attempts to compute f^⊕k is smaller than the circuit which attempts to compute f on a single input! This paper addresses the issue of proving strong direct product assertions, that is, ones in which s′ ≈ ks and is in particular larger than s. We study the question of proving strong direct product question for decision trees and communication protocols.