Static-Memory-Hard Functions and Nonlinear Space-Time Tradeoffs via Pebbling

TL;DR

This paper introduces a new static-memory-hardness model for cryptographic functions, develops pebbling-based constructions, and explores nonlinear time-space tradeoffs, advancing understanding of memory-hard functions and their graph-theoretic foundations.

Contribution

It proposes a formal definition of static-memory-hard functions, introduces new pebbling games and graph constructions, and models nonlinear time-space tradeoffs in cryptography.

Findings

New static-memory-hardness measure incorporating static memory.

Pebbling-based constructions for static-memory-hard functions.

Demonstration of nonlinear time-space tradeoffs affecting strategies.

Abstract

Pebble games were originally formulated to study time-space tradeoffs in computation, modeled by games played on directed acyclic graphs (DAGs). Close connections between pebbling and cryptography have been known for decades. A series of recent research starting with (Alwen and Serbinenko, STOC 2015) has deepened our understanding of the notion of memory-hardness in cryptography --- a useful property of hash functions for deterring large-scale password-cracking attacks --- and has shown memory-hardness to have intricate connections with the theory of graph pebbling. In this work, we improve upon two main limitations of existing models of memory-hardness. First, existing measures of memory-hardness only account for dynamic (i.e., runtime) memory usage, and do not consider static memory usage. We propose a new definition of static-memory-hard function (SHF) which takes into account static memory usage and allows the formalization of larger memory requirements for efficient functions, than in the dynamic setting (where memory usage is inherently bounded by runtime). We then give two SHF constructions based on pebbling; to prove static-memory-hardness, we define a new pebble game ("black-magic pebble game"), and new graph constructions with optimal complexity under our proposed measure. Secondly, existing memory-hardness models implicitly consider linear tradeoffs between the costs of time and space. We propose a new model to capture nonlinear time-space trade-offs and prove that nonlinear tradeoffs can in fact cause adversaries to employ different strategies from linear tradeoffs. Finally, as an additional contribution of independent interest, we present the first asymptotically tight graph construction that achieves the best possible space complexity up to $\log{\log{n}}$-factors for an existing memory-hardness measure called cumulative complexity in the sequential pebbling model.