skip to main content

Title: On the Security of Proofs of Sequential Work in a Post-Quantum World
A Proof of Sequential Work (PoSW) allows a prover to convince a resource-bounded verifier that the prover invested a substantial amount of sequential time to perform some underlying computation. PoSWs have many applications including time-stamping, blockchain design, and universally verifiable CPU benchmarks. Mahmoody, Moran, and Vadhan (ITCS 2013) gave the first construction of a PoSW in the random oracle model though the construction relied on expensive depth-robust graphs. In a recent breakthrough, Cohen and Pietrzak (EUROCRYPT 2018) gave an efficient PoSW construction that does not require expensive depth-robust graphs. In the classical parallel random oracle model, it is straightforward to argue that any successful PoSW attacker must produce a long ℋ-sequence and that any malicious party running in sequential time T-1 will fail to produce an ℋ-sequence of length T except with negligible probability. In this paper, we prove that any quantum attacker running in sequential time T-1 will fail to produce an ℋ-sequence except with negligible probability - even if the attacker submits a large batch of quantum queries in each round. The proof is substantially more challenging and highlights the power of Zhandry’s recent compressed oracle technique (CRYPTO 2019). We further extend this result to establish post-quantum security of more » a non-interactive PoSW obtained by applying the Fiat-Shamir transform to Cohen and Pietrzak’s efficient construction (EUROCRYPT 2018). « less
; ;
Tessaro, Stefano
Award ID(s):
1755708 2047272 1910659
Publication Date:
Journal Name:
2nd Conference on Information-Theoretic Cryptography (ITC 2021)
Page Range or eLocation-ID:
Sponsoring Org:
National Science Foundation
More Like this
  1. Zero-knowledge succinct arguments of knowledge (zkSNARKs) enable efficient privacy-preserving proofs of membership for general NP languages. Our focus in this work is on post-quantum zkSNARKs, with a focus on minimizing proof size. Currently, there is a 1000x gap in the proof size between the best pre-quantum constructions and the best post-quantum ones. Here, we develop and implement new lattice-based zkSNARKs in the designated-verifier preprocessing model. With our construction, after an initial preprocessing step, a proof for an NP relation of size 2^20 is just over 16 KB. Our proofs are 10.3x shorter than previous post-quantum zkSNARKs for general NP languages.more »Compared to previous lattice-based zkSNARKs (also in the designated-verifier preprocessing model), we obtain a 42x reduction in proof size and a 60x reduction in the prover's running time, all while achieving a much higher level of soundness. Compared to the shortest pre-quantum zkSNARKs by Groth (Eurocrypt 2016), the proof size in our lattice-based construction is 131x longer, but both the prover and the verifier are faster (by 1.2x and 2.8x, respectively). Our construction follows the general blueprint of Bitansky et al. (TCC 2013) and Boneh et al. (Eurocrypt 2017) of combining a linear probabilistically checkable proof (linear PCP) together with a linear-only vector encryption scheme. We develop a concretely-efficient lattice-based instantiation of this compiler by considering quadratic extension fields of moderate characteristic and using linear-only vector encryption over rank-2 module lattices.« less
  2. 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 themore »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.« less
  3. Public key quantum money can be seen as a version of the quantum no-cloning theorem that holds even when the quantum states can be verified by the adversary. In this work, we investigate quantum lightning where no-cloning holds even when the adversary herself gener- ates the quantum state to be cloned. We then study quantum money and quantum lightning, showing the following results: – We demonstrate the usefulness of quantum lightning beyond quan- tum money by showing several potential applications, such as gen- erating random strings with a proof of entropy, to completely decen- tralized cryptocurrency without a block-chain, wheremore »transactions is instant and local. – We give Either/Or results for quantum money/lightning, showing that either signatures/hash functions/commitment schemes meet very strong recently proposed notions of security, or they yield quan- tum money or lightning. Given the difficulty in constructing public key quantum money, this suggests that natural schemes do attain strong security guarantees. – We show that instantiating the quantum money scheme of Aaron- son and Christiano [STOC’12] with indistinguishability obfuscation that is secure against quantum computers yields a secure quantum money scheme. This construction can be seen as an instance of our Either/Or result for signatures, giving the first separation between two security notions for signatures from the literature. – Finally, we give a plausible construction for quantum lightning, which we prove secure under an assumption related to the multi- collision resistance of degree-2 hash functions. Our construction is inspired by our Either/Or result for hash functions, and yields the first plausible standard model instantiation of a non-collapsing col- lision resistant hash function. This improves on a result of Unruh [Eurocrypt’16] which is relative to a quantum oracle.« less
  4. 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), theymore »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 𝒪((N^2 log log N)/log N). By contrast, the dMHF scrypt achieves maximal CMC Ω(N^2) - though the CMC of scrypt would be reduced to just 𝒪(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 Ω(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 ε > 0 we show how to construct a k-restricted dynamic graph with k=Ω(N^(1-ε)) that provably achieves maximum cumulative pebbling cost Ω(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.« less
  5. We consider the problem of efficiently simulating random quantum states and random unitary operators, in a manner which is convincing to unbounded adversaries with black-box oracle access. This problem has previously only been considered for restricted adversaries. Against adversaries with an a priori bound on the number of queries, it is well-known that t-designs suffice. Against polynomial-time adversaries, one can use pseudorandom states (PRS) and pseudorandom unitaries (PRU), as defined in a recent work of Ji, Liu, and Song; unfortunately, no provably secure construction is known for PRUs. In our setting, we are concerned with unbounded adversaries. Nonetheless, we aremore »able to give stateful quantum algorithms which simulate the ideal object in both settings of interest. In the case of Haar-random states, our simulator is polynomial-time, has negligible error, and can also simulate verification and reflection through the simulated state. This yields an immediate application to quantum money: a money scheme which is information-theoretically unforgeable and untraceable. In the case of Haar-random unitaries, our simulator takes polynomial space, but simulates both forward and inverse access with zero error. These results can be seen as the first significant steps in developing a theory of lazy sampling for random quantum objects.« less