We introduce new techniques for proving lower bounds on the running time of randomized algorithms for asynchronous agreement against powerful adversaries. In particular, we define a strongly adaptive adversary that is computationally unbounded and has a limited ability to corrupt a dynamic subset of processors by erasing their memories. We demonstrate that the randomized agreement algorithms designed by Ben-Or and Bracha to tolerate crash or Byzantine failures in the asynchronous setting extend to defeat a strongly adaptive adversary. These algorithms have essentially perfect correctness and termination, but at the expense of exponential running time. In the case of the strongly adaptive adversary, we show that this dismally slow running time is inherent: we prove that any algorithm with essentially perfect correctness and termination against the strongly adaptive adversary must have exponential running time. We additionally interpret this result as yielding an enhanced understanding of the tools needed to simultaneously achieving perfect correctness and termination as well as fast running time for randomized algorithms tolerating crash or Byzantine failures.

In the continual memory leakage model, security against attackers who can repeatedly obtain leakage is achieved by periodically updating the secret key. This is an appealing model which captures a wide class of side-channel attacks, but all previous constructions in this model provide only a very minimal amount of leakage tolerance during secret key updates. Since key updates may happen frequently, improving security guarantees against attackers who obtain leakage during these updates is an important problem. In this work, we present the first cryptographic primitives which are secure against a super-logarithmic amount of leakage during secret key updates. We present signature and public key encryption schemes in the standard model which can tolerate a constant fraction of the secret key to be leaked between updates as well as a constant fraction of the secret key and update randomness to be leaked during updates. Our signature scheme also allows us to leak a constant fraction of the entire secret state during signing. Before this work, it was unknown how to tolerate super-logarithmic leakage during updates even in the random oracle model. We rely on subgroup decision assumptions in composite order bilinear groups.