Amir Dembo

Naomi Sagan, Amir Dembo, Matthew Ho et al.

We study a family of processes generated according to sequential probability assignments induced by the LZ78 universal compressor. We characterize entropic and distributional properties such as their entropy and relative entropy rates, finite-state compressibility and log loss of their realizations, and the empirical distributions that they induce. Though not quite stationary, these sources are "almost stationary and ergodic;" similar to stationary and ergodic processes, they satisfy a Shannon-McMillan-Breiman-type property: the normalized log probability of their realizations converges almost surely to their entropy rate. Further, they are locally "almost i.i.d." in the sense that the finite-dimensional empirical distributions of their realizations converge almost surely to a deterministic i.i.d. law. However, unlike stationary ergodic sources, the finite-state compressibility of their realizations is almost surely strictly larger than their entropy rate by a "Jensen gap". We present simulations demonstrating the theoretical results. These sources allow to gauge the performance of sequential probability models, both classical and deep learning-based, on non-Markovian non-stationary data. As such, we apply realizations of the LZ78 source to the study of in-context learning in transformer models.

Amir Dembo, Sreeram Kannan, Ertem Nusret Tas et al.

Nakamoto invented the longest chain protocol, and claimed its security by analyzing the private double-spend attack, a race between the adversary and the honest nodes to grow a longer chain. But is it the worst attack? We answer the question in the affirmative for three classes of longest chain protocols, designed for different consensus models: 1) Nakamoto's original Proof-of-Work protocol; 2) Ouroboros and SnowWhite Proof-of-Stake protocols; 3) Chia Proof-of-Space protocol. As a consequence, exact characterization of the maximum tolerable adversary power is obtained for each protocol as a function of the average block time normalized by the network delay. The security analysis of these protocols is performed in a unified manner by a novel method of reducing all attacks to a race between the adversary and the honest nodes.

Vivek Bagaria, Amir Dembo, Sreeram Kannan et al.

The Nakamoto longest chain protocol is remarkably simple and has been proven to provide security against any adversary with less than 50% of the total hashing power. Proof-of-stake (PoS) protocols are an energy efficient alternative; however existing protocols adopting Nakamoto's longest chain design achieve provable security only by allowing long-term predictability (which have serious security implications). In this paper, we prove that a natural longest chain PoS protocol with similar predictability as Nakamoto's PoW protocol can achieve security against any adversary with less than 1/(1+e) fraction of the total stake. Moreover we propose a new family of longest chain PoS protocols that achieve security against a 50% adversary, while only requiring short-term predictability. Our proofs present a new approach to analyzing the formal security of blockchains, based on a notion of adversary-proof convergence.