Abstract  A fundamental question of complexity theory is the direct product question. A famous example is Yaos XOR-lemma, in which one assumes that some function f is hard on average for small circuits (meaning 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 on independently chosen . All known proofs of this lemma have the property that s < s . In words, the circuit which attempts to compute 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 and is in particular larger than s. We study the question of proving strong direct product question for decision trees and communication protocols.