Cryptology ePrint Archive: Report 2018/205

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

Thaddeus Dryja and Quanquan C. Liu and Sunoo Park

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.

Definitions of memory-hardness are not yet unified in this somewhat nascent field, however, and the guarantees proven are with respect to a range of proposed definitions.

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 loglogn-factors for an existing memory-hardness measure called cumulative complexity in the sequential pebbling model.

Category / Keywords: memory-hard functions, pebbling, space-time trade-offs

Date: received 20 Feb 2018, last revised 11 Mar 2018

Contact author: quanquan at mit edu

Available format(s): PDF | BibTeX Citation

Version: 20180312:023118 (All versions of this report)

Short URL:

Discussion forum: Show discussion | Start new discussion

[ Cryptology ePrint archive ]