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1. Sustained Space and Cumulative Complexity Trade-Offs for Data-Dependent Memory-Hard Functions

   Blocki, Jeremiah ; Holman, Blake ( October 2022 , Dodis, Y., Shrimpton, T. (eds) Advances in Cryptology – CRYPTO 2022)

   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). 

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2. Bandwidth-Hard Functions: Reductions and Lower Bounds

   https://doi.org/10.1145/3243734.3243773  

   Blocki, Jeremiah ; Ren, Ling ; Zhou, Samson ( October 2018 , 2018 ACM SIGSAC Conference on Computer and Communications Security (CCS ’18))

   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. 

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3. Sustained Space Complexity

   https://doi.org/10.1007/978-3-319-78375-8_4  

   Alwen, Joël ; Blocki, Jeremiah ; Pietrzak, Krzysztof ( May 2018 , Advances in Cryptology – EUROCRYPT 2018)

   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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4. Approximating Cumulative Pebbling Cost Is Unique Games Hard

   https://doi.org/10.4230/LIPIcs.ITCS.2020.13  

   Blocki, J ; Lee, S ; Zhou, S. ( January 2020 , Leibniz international proceedings in informatics)

   null (Ed.)

   The cumulative pebbling complexity of a directed acyclic graph G is defined as cc(G) = min_P ∑_i |P_i|, where the minimum is taken over all legal (parallel) black pebblings of G and |P_i| denotes the number of pebbles on the graph during round i. Intuitively, cc(G) captures the amortized Space-Time complexity of pebbling m copies of G in parallel. The cumulative pebbling complexity of a graph G is of particular interest in the field of cryptography as cc(G) is tightly related to the amortized Area-Time complexity of the Data-Independent Memory-Hard Function (iMHF) f_{G,H} [Joël Alwen and Vladimir Serbinenko, 2015] defined using a constant indegree directed acyclic graph (DAG) G and a random oracle H(⋅). A secure iMHF should have amortized Space-Time complexity as high as possible, e.g., to deter brute-force password attacker who wants to find x such that f_{G,H}(x) = h. Thus, to analyze the (in)security of a candidate iMHF f_{G,H}, it is crucial to estimate the value cc(G) but currently, upper and lower bounds for leading iMHF candidates differ by several orders of magnitude. Blocki and Zhou recently showed that it is NP-Hard to compute cc(G), but their techniques do not even rule out an efficient (1+ε)-approximation algorithm for any constant ε>0. We show that for any constant c > 0, it is Unique Games hard to approximate cc(G) to within a factor of c. Along the way, we show the hardness of approximation of the DAG Vertex Deletion problem on DAGs of constant indegree. Namely, we show that for any k,ε >0 and given a DAG G with N nodes and constant indegree, it is Unique Games hard to distinguish between the case that G is (e_1, d_1)-reducible with e_1=N^{1/(1+2 ε)}/k and d_1=k N^{2 ε/(1+2 ε)}, and the case that G is (e_2, d_2)-depth-robust with e_2 = (1-ε)k e_1 and d_2= 0.9 N^{(1+ε)/(1+2 ε)}, which may be of independent interest. Our result generalizes a result of Svensson who proved an analogous result for DAGs with indegree 𝒪(N). 

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5. On the Computational Complexity of Minimal Cumulative Cost Graph Pebbling

   Jeremiah Blocki, Samson Zhou ( February 2018 , Financial Cryptography and Data Security (FC 2018))

   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. 

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