A fundamental problem in designing secure multi-party protocols is how to deal with adaptive adversaries i.e., adversaries that may choose the corrupted parties during the course of the computation, in a setting where the channels are insecure and secure communication is achieved by cryptographic primitives based on the computational limitations of the adversary. It turns out that the power of an adaptive adversary is greatly a ected by the amount of information gathered upon the corruption of a party. This amount of information models the extent to which uncorrupted parties are trusted to carry out instructions that cannot be externally veri ed, such as erasing records of past con gurations. It has been shown that if the parties are trusted to erase such records, then adaptively secure computation can be carried out using known primitives. However, this total trust in parties may be unrealistic in many scenarios. An important question, open since 1986, is whether adaptively secure mu...