Computationally Data-Independent Memory Hard Functions

Memory hard functions (MHFs) are an important cryptographic primitive that are used to design egalitarian proofs of work and in the construction of moderately expensive key-derivation functions resistant to brute-force attacks. Broadly speaking, MHFs can be divided into two categories: data-dependent memory hard functions (dMHFs) and data-independent memory hard functions (iMHFs). iMHFs are resistant to certain side-channel attacks as the memory access pattern induced by the honest evaluation algorithm is independent of the potentially sensitive input e.g., password. While dMHFs are potentially vulnerable to side-channel attacks (the induced memory access pattern might leak useful information to a brute-force attacker), they can achieve higher cumulative memory complexity (CMC) in comparison than an iMHF. In particular, any iMHF that can be evaluated in N steps on a sequential machine has CMC at most ?((N^2 log log N)/log N). By contrast, the dMHF scrypt achieves maximal CMC ?(N^2) - though the CMC of scrypt would be reduced to just ?(N) after a side-channel attack. In this paper, we introduce the notion of computationally data-independent memory hard functions (ciMHFs). Intuitively, we require that memory access pattern induced by the (randomized) ciMHF evaluation algorithm appears to be independent from the standpoint of a computationally bounded eavesdropping attacker - even if the attacker selects the initial input. We then ask whether it is possible to circumvent known upper bound for iMHFs and build a ciMHF with CMC ?(N^2). Surprisingly, we answer the question in the affirmative when the ciMHF evaluation algorithm is executed on a two-tiered memory architecture (RAM/Cache). We introduce the notion of a k-restricted dynamic graph to quantify the continuum between unrestricted dMHFs (k=n) and iMHFs (k=1). For any ? > 0 we show how to construct a k-restricted dynamic graph with k=?(N^(1-?)) that provably achieves maximum cumulative pebbling cost ?(N^2). We can use k-restricted dynamic graphs to build a ciMHF provided that cache is large enough to hold k hash outputs and the dynamic graph satisfies a certain property that we call "amenable to shuffling". In particular, we prove that the induced memory access pattern is indistinguishable to a polynomial time attacker who can monitor the locations of read/write requests to RAM, but not cache. We also show that when k=o(N^(1/log log N))then any k-restricted graph with constant indegree has cumulative pebbling cost o(N^2). Our results almost completely characterize the spectrum of k-restricted dynamic graphs

Moderately Hard Functions: Definition, Instantiations, and Applications

Several cryptographic schemes and applications are based on functions that are both reasonably efficient to compute and moderately hard to invert, including client puzzles for Denial-of-Service protection, password protection via salted hashes, or recent proof-of-work blockchain systems. Despite their wide use, a definition of this concept has not yet been distilled and formalized explicitly. Instead, either the applications are proven directly based on the assumptions underlying the function, or some property of the function is proven, but the security of the application is argued only informally. The goal of this work is to provide a (universal) definition that decouples the efforts of designing new moderately hard functions and of building protocols based on them, serving as an interface between the two. On a technical level, beyond the mentioned definitions, we instantiate the model for four different notions of hardness. We extend the work of Alwen and Serbinenko (STOC 2015) by providing a general tool for proving security for the first notion of memory-hard functions that allows for provably secure applications. The tool allows us to recover all of the graph-theoretic techniques developed for proving security under the older, non-composable, notion of security used by Alwen and Serbinenko. As an application of our definition of moderately hard functions, we prove the security of two different schemes for proofs of effort (PoE). We also formalize and instantiate the concept of a non-interactive proof of effort (niPoE), in which the proof is not bound to a particular communication context but rather any bit-string chosen by the prover

Foundations, Properties, and Security Applications of Puzzles: A Survey

Cryptographic algorithms have been used not only to create robust ciphertexts but also to generate cryptograms that, contrary to the classic goal of cryptography, are meant to be broken. These cryptograms, generally called puzzles, require the use of a certain amount of resources to be solved, hence introducing a cost that is often regarded as a time delay---though it could involve other metrics as well, such as bandwidth. These powerful features have made puzzles the core of many security protocols, acquiring increasing importance in the IT security landscape. The concept of a puzzle has subsequently been extended to other types of schemes that do not use cryptographic functions, such as CAPTCHAs, which are used to discriminate humans from machines. Overall, puzzles have experienced a renewed interest with the advent of Bitcoin, which uses a CPU-intensive puzzle as proof of work. In this paper, we provide a comprehensive study of the most important puzzle construction schemes available in the literature, categorizing them according to several attributes, such as resource type, verification type, and applications. We have redefined the term puzzle by collecting and integrating the scattered notions used in different works, to cover all the existing applications. Moreover, we provide an overview of the possible applications, identifying key requirements and different design approaches. Finally, we highlight the features and limitations of each approach, providing a useful guide for the future development of new puzzle schemes.Comment: This article has been accepted for publication in ACM Computing Survey

On the Cryptographic Hardness of Local Search

We show new hardness results for the class of Polynomial Local Search problems (PLS): - Hardness of PLS based on a falsifiable assumption on bilinear groups introduced by Kalai, Paneth, and Yang (STOC 2019), and the Exponential Time Hypothesis for randomized algorithms. Previous standard model constructions relied on non-falsifiable and non-standard assumptions. - Hardness of PLS relative to random oracles. The construction is essentially different than previous constructions, and in particular is unconditionally secure. The construction also demonstrates the hardness of parallelizing local search. The core observation behind the results is that the unique proofs property of incrementally-verifiable computations previously used to demonstrate hardness in PLS can be traded with a simple incremental completeness property