Memory-hard functions (MHF) are functions whose evaluation cost is dominated by memory cost. MHFs are egalitarian, in the sense that evaluating them on dedicated hardware (like FPGAs or ASICs) is not much cheaper than on off-the-shelf hardware (like x86 CPUs). MHFs have interesting cryptographic applications, most notably to password hashing and securing blockchains.
Alwen and Serbinenko [STOC’15] define the cumulative memory complexity (cmc) of a function as the sum (over all time-steps) of the amount of memory required to compute the function. They advocate that a good MHF must have high cmc. Unlike previous notions, cmc takes into account that dedicated hardware might exploit amortization and parallelism. Still, cmc has been critizised as insufficient, as it fails to capture possible time-memory trade-offs; as memory cost doesn’t scale linearly, functions with the same cmc could still have very different actual hardware cost.
In this work we address this problem, and introduce the notion of sustained-memory complexity, which requires that any algorithm evaluating the function must use a large amount of memory for many steps. We construct functions (in the parallel random oracle model) whose sustained-memory complexity is almost optimal: our function can be evaluated using n steps and O(n/log(n)) memory, in each step making one query to the (fixed-input length) random oracle, while any algorithm that can make arbitrary many parallel queries to the random oracle, still needs Ω(n/log(n)) memory for Ω(n) steps.
As has been done for various notions (including cmc) before, we reduce the task of constructing an MHFs with high sustained-memory complexity to proving pebbling lower bounds on DAGs. Our main technical contribution is the construction is a family of DAGs on n nodes with constant indegree with high “sustained-space complexity”, meaning that any parallel black-pebbling strategy requires Ω(n/log(n)) pebbles for at least Ω(n) steps.
Along the way we construct a family of maximally “depth-robust” DAGs with maximum indegree O(logn) , improving upon the construction of Mahmoody et al. [ITCS’13] which had maximum indegree O(log2n⋅polylog(logn)) .
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Data-Independent Memory Hard Functions: New Attacks and Stronger Constructions
Memory-hard functions (MHFs) are a key cryptographic primitive underlying the design of moderately expensive password hashing algorithms and egalitarian proofs of work. Over the past few years several increasingly stringent goals for an MHF have been proposed including the requirement that the MHF have high sequential space-time (ST) complexity, parallel space-time complexity, amortized area-time (aAT) complexity and sustained space complexity. Data-Independent Memory Hard Functions (iMHFs) are of special interest in the context of password hashing as they naturally resist side-channel attacks. iMHFs can be specified using a directed acyclic graph (DAG) $G$ with $N=2^n$ nodes and low indegree and the complexity of the iMHF can be analyzed using a pebbling game. Recently, Alwen et al. [CCS'17] constructed an DAG called DRSample which has aAT complexity at least $\Omega\left( N^2/\log N\right)$. Asymptotically DRSample outperformed all prior iMHF constructions including Argon2i, winner of the password hashing competition (aAT cost $\mathcal{O}\left(N^{1.767}\right)$), though the constants in these bounds are poorly understood. We show that the the greedy pebbling strategy of Boneh et al. [ASIACRYPT'16] is particularly effective against DRSample e.g., the aAT cost is $\mathcal{O}\left( N^2/\log N\right)$. In fact, our empirical analysis {\em reverses} the prior conclusion of Alwen et al. that DRSample provides stronger resistance to known pebbling attacks for practical values of $N \leq 2^{24}$. We construct a new iMHF candidate (DRSample+BRG) by using the bit-reversal graph to extend DRSample. We then prove that the construction is asymptotically optimal under every MHF criteria, and we empirically demonstrate that our iMHF provides the best resistance to {\em known} pebbling attacks. For example, we show that any parallel pebbling attack either has aAT cost $\omega(N^2)$ or requires at least $\Omega(N)$ steps with $\Omega(N/\log N)$ pebbles on the DAG. This makes our construction the first practical iMHF with a strong sustained space-complexity guarantee and immediately implies that any parallel pebbling has aAT complexity $\Omega(N^2/\log N)$. We also prove that any sequential pebbling (including the greedy pebbling attack) has aAT cost $\Omega\left( N^2\right)$ and, if a plausible conjecture holds, any parallel pebbling has aAT cost $\Omega(N^2 \log \log N/\log N)$ --- the best possible bound for an iMHF.
We implement our new iMHF and demonstrate that it is just as fast as Argon2. Along the way we propose a simple modification to the Argon2 round function which increases an attacker's aAT cost by nearly an order of magnitude without increasing running time on a CPU. Finally, we give a pebbling reduction which proves that in the parallel random oracle model (PROM) the cost of evaluating an iMHF like Argon2i or DRSample+BRG is given by the pebbling cost of the underlying DAG. Prior pebbling reductions assumed that the iMHF round function concatenates input labels before hashing and did not apply to practical iMHFs such as Argon2i, DRSample or DRSample+BRG where input labels are instead XORed together.
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- Award ID(s):
- 1755708
- NSF-PAR ID:
- 10092895
- Date Published:
- Journal Name:
- Crypto
- ISSN:
- 1790-2703
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Memory Hard Functions (MHFs) have been proposed as an answer to the growing inequality between the computational speed of general purpose CPUs and Application Specific Integrated Circuits (ASICs). MHFs have seen widespread applications including password hashing, key stretching and proofs of work. Several metrics have been proposed to quantify the “memory hardness” of a function. Cumulative memory complexity (CMC) [8] (or amortized Area × Time complexity [4]) attempts to quantify the cost to acquire/build the hardware to evaluate the function — after normalizing the time it takes to evaluate the function. By contrast, bandwidth hardness [30] attempts to quantify the amortized energy costs of evaluating this function on hardware — which in turn is largely dominated by the number of cache misses. Ideally, a good MHF would be both bandwidth hard and have high cumulative memory complexity. While the cumulative memory complexity of leading MHF candidates is well understood, little is known about the bandwidth hardness of many prominent MHF candidates. Our contributions are as follows: First, we provide the first reduction proving that, in the parallel random oracle model, the bandwidth hardness of a Data-Independent Memory Hard Function (iMHF) is described by the red-blue pebbling cost of the directed acyclic graph (DAG) associated with that iMHF. Second, we show that the goals of designing an MHF with high CMC/bandwidth hardness are well aligned. In particular, we prove that any function with high CMC also has relatively high energy costs. This result leads to the first unconditional lower bound on the energy cost of scrypt in the parallel random oracle model. Third, we analyze the bandwidth hardness of several prominent iMHF candidates such as Argon2i [11], winner of the password hashing competition, aATSample and DRSample [4] — the first practical iMHF with essentially asymptotically optimal CMC. We show Argon2i, aATSample and DRSample are maximally bandwidth hard under appropriate cache size. Finally, we show that the problem of finding a red-blue pebbling with minimum energy cost is NP-hard.more » « less
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Dodis, Y. (Ed.)Memory-hard functions (MHFs) are a useful cryptographic primitive which can be used to design egalitarian proof of work puzzles and to protect low entropy secrets like passwords against brute-force attackers. Intuitively, a memory-hard function is a function whose evaluation costs are dominated by memory costs even if the attacker uses specialized hardware (FPGAs/ASICs), and several cost metrics have been proposed to quantify this intuition. For example, space-time cost looks at the product of running time and the maximum space usage over the entire execution of an algorithm. Alwen and Serbinenko (STOC 2015) observed that the space-time cost of evaluating a function multiple times may not scale linearly in the number of instances being evaluated and introduced the stricter requirement that a memory-hard function has high cumulative memory complexity (CMC) to ensure that an attacker’s amortized space-time costs remain large even if the attacker evaluates the function on multiple different inputs in parallel. Alwen et al. (EUROCRYPT 2018) observed that the notion of CMC still gives the attacker undesirable flexibility in selecting space-time tradeoffs e.g., while the MHF Scrypt has maximal CMC Ω(N^2), an attacker could evaluate the function with constant O(1) memory in time O(N^2). Alwen et al. introduced an even stricter notion of Sustained Space complexity and designed an MHF which has s=Ω(N/logN) sustained complexity t=Ω(N) i.e., any algorithm evaluating the function in the parallel random oracle model must have at least t=Ω(N) steps where the memory usage is at least Ω(N/logN). In this work, we use dynamic pebbling games and dynamic graphs to explore tradeoffs between sustained space complexity and cumulative memory complexity for data-dependent memory-hard functions such as Argon2id and Scrypt. We design our own dynamic graph (dMHF) with the property that any dynamic pebbling strategy either (1) has Ω(N) rounds with Ω(N) space, or (2) has CMC Ω(N^{3−ϵ})—substantially larger than N^2. For Argon2id we show that any dynamic pebbling strategy either(1) has Ω(N) rounds with Ω(N^{1−ϵ}) space, or (2) has CMC ω(N^2). We also present a dynamic version of DRSample (Alwen et al. 2017) for which any dynamic pebbling strategy either (1) has Ω(N) rounds with Ω(N/log N) space, or (2) has CMC Ω(N^3/log N).more » « less
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Kiltz, E. (Ed.)The classical (parallel) black pebbling game is a useful abstraction which allows us to analyze the resources (space, space-time, cumulative space) necessary to evaluate a function f with a static data-dependency graph G. Of particular interest in the field of cryptography are data-independent memory-hard functions fG,H which are defined by a directed acyclic graph (DAG) G and a cryptographic hash function H. The pebbling complexity of the graph G characterizes the amortized cost of evaluating fG,H multiple times as well as the total cost to run a brute-force preimage attack over a fixed domain X, i.e., given y∈{0,1}∗ find x∈X such that fG,H(x)=y. While a classical attacker will need to evaluate the function fG,H at least m=|X| times a quantum attacker running Grover’s algorithm only requires O(m−−√) blackbox calls to a quantum circuit CG,H evaluating the function fG,H. Thus, to analyze the cost of a quantum attack it is crucial to understand the space-time cost (equivalently width times depth) of the quantum circuit CG,H. We first observe that a legal black pebbling strategy for the graph G does not necessarily imply the existence of a quantum circuit with comparable complexity—in contrast to the classical setting where any efficient pebbling strategy for G corresponds to an algorithm with comparable complexity for evaluating fG,H. Motivated by this observation we introduce a new parallel reversible pebbling game which captures additional restrictions imposed by the No-Deletion Theorem in Quantum Computing. We apply our new reversible pebbling game to analyze the reversible space-time complexity of several important graphs: Line Graphs, Argon2i-A, Argon2i-B, and DRSample. Specifically, (1) we show that a line graph of size N has reversible space-time complexity at most O(N^{1+2/√logN}). (2) We show that any (e, d)-reducible DAG has reversible space-time complexity at most O(Ne+dN2^d). In particular, this implies that the reversible space-time complexity of Argon2i-A and Argon2i-B are at most O(N^2 loglogN/√logN) and O(N^2/(log N)^{1/3}), respectively. (3) We show that the reversible space-time complexity of DRSample is at most O((N^2loglog N)/log N). We also study the cumulative pebbling cost of reversible pebblings extending a (non-reversible) pebbling attack of Alwen and Blocki on depth-reducible graphs.more » « less
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Memory-hard functions (MHF) are functions whose evaluation cost is dominated by memory cost. MHFs are egalitarian, in the sense that evaluating them on dedicated hardware (like FPGAs or ASICs) is not much cheaper than on off-the-shelf hardware (like x86 CPUs). MHFs have interesting cryptographic applications, most notably to password hashing and securing blockchains. Alwen and Serbinenko [STOC'15] define the cumulative memory complexity (cmc) of a function as the sum (over all time-steps) of the amount of memory required to compute the function. They advocate that a good MHF must have high cmc. Unlike previous notions, cmc takes into account that dedicated hardware might exploit amortization and parallelism. Still, cmc has been critizised as insufficient, as it fails to capture possible time-memory trade-offs, as memory cost doesn't scale linearly, functions with the same cmc could still have very different actual hardware cost. In this work we address this problem, and introduce the notion of sustained-memory complexity, which requires that any algorithm evaluating the function must use a large amount of memory for many steps. We construct functions (in the parallel random oracle model) whose sustained-memory complexity is almost optimal: our function can be evaluated using n steps and O(n/log(n)) memory, in each step making one query to the (fixed-input length) random oracle, while any algorithm that can make arbitrary many parallel queries to the random oracle, still needs Ω(n/log(n)) memory for Ω(n) steps. Our main technical contribution is the construction is a family of DAGs on n nodes with constant indegree with high "sustained-space complexity", meaning that any parallel black-pebbling strategy requires Ω(n/log(n)) pebbles for at least Ω(n) steps.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1007/978-3-030-26951-7_20
