Relaxed Locally Correctable Codes in Computationally Bounded Channels

Blocki, Jeremiah; Gandikota, Venkata; Grigorescu, Elena; Zhou, Samson

doi:10.1109/tit.2021.3076396

Cited by 11 publications

(11 citation statements)

References 47 publications

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“…δr nodes the graph G contains an edge (x, y) with x ∈ X and y ∈ Y . We remark that the construction of computationally relaxed locally correctable codes [BGGZ21] relies on a family of δ-local exanders which is a strictly stronger property than depth-robustness -δ-local expanders are also automatically -extreme depth-robust [ABP18].…”

Section: Our Contributionsmentioning

confidence: 99%

“…, v d such that (v i , v i+1 ) ∈ E for each i < d and v i ∈ V \ S for each i ≤ d. As an example the complete DAG K N = (V = [N ], E = {(i, j) : 1 ≤ i < j ≤ n} has the property that it is (e, d)-depth-robust for any integers e, d such that e + d ≤ N . Depth-robust graphs have found many applications in cryptography including the design of dataindependent Memory-Hard Functions (e.g., [AB16,ABP17]), Proofs of Space [DFKP15], Proofs of Replication [Pie19,Fis19] and Computationally Relaxed Locally Correctable Codes [BGGZ21]. In many of these applications it is desirable to construct depth-robust graphs with low-indegree (e.g., indeg(G) = O(1) or indeg(G) = O(log N )) and we also require that the graphs are locally navigable, i.e., given any node v ∈ V = [N ] there is an efficient algorithm GetParents(v) which returns the set {u : (u, v) ∈ E} containing all of v's parent nodes in time O(polylog N ).…”

Section: Introductionmentioning

confidence: 99%

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On Explicit Constructions of Extremely Depth Robust Graphs

Blocki¹,

Cinkoske²,

Lee³

et al. 2021

Preprint

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A directed acyclic graph G = (V, E) is said to be (e, d)-depth robust if for every subset S ⊆ V of |S| ≤ e nodes the graph G − S still contains a directed path of length d. If the graph is (e, d)-depth-robust for any e, d such that e + d ≤ (1 − )|V | then the graph is said to be -extreme depth-robust. In the field of cryptography, (extremely) depth-robust graphs with low indegree have found numerous applications including the design of side-channel resistant Memory-Hard Functions, Proofs of Space and Replication and in the design of Computationally Relaxed Locally Correctable Codes. In these applications, it is desirable to ensure the graphs are locally navigable, i.e., there is an efficient algorithm GetParents running in time polylog |V | which takes as input a node v ∈ V and returns the set of v's parents. We give the first explicit construction of locally navigable -extreme depth-robust graphs with indegree O(log |V |). Previous constructions of -extreme depth-robust graphs either had indegree ω(log 2 |V |) or were not explicit.

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Section: Our Contributionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On Explicit Constructions of Extremely Depth Robust Graphs

Blocki¹,

Cinkoske²,

Lee³

et al. 2021

Preprint

Self Cite

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“…Similar techniques for using green/red nodes to check for local consistency were also used in [BGGZ19]. Note that correctness of the fixed labeling ℓ i for an input x is not required, i.e.…”

Section: H-sequences For Quantum Attacksmentioning

confidence: 99%

On the Security of Proofs of Sequential Work in a Post-Quantum World

Blocki¹,

Lee²,

Zhou³

2020

Preprint

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A proof of sequential work allows a prover to convince a resource bounded verifier that the prover invested a substantial amount of sequential time to perform some underlying computation. Proofs of sequential work have many applications including time-stamping, blockchain design, and universally verifiable CPU benchmarks. Mahmoody, Moran and Vadhan (ITCS 2013) gave the first construction of proofs of sequential work in the random oracle model though the construction relied on expensive depth-robust graphs. In a recent breakthrough, Cohen and Pietrzak (EUROCRYPT 2018) gave a more efficient construction that does not require depth-robust graphs. In each of these constructions, the prover commits to a labeling of a directed acyclic graph G with N nodes and the verifier audits the prover by checking that a small subset of labels are locally consistent, e.g., Lv = H(Lv 1 , . . . , Lv δ ), where v1, . . . , v δ denote the parents of node v. Provided that the graph G has certain structural properties (e.g., depth-robustness), the prover must produce a long H-sequence to pass the audit with non-negligible probability. An Hsequence x0, x1 . . . xT has the property that H(xi) is a substring of xi+1 for each i, i.e., we can find strings ai, bi such that xi+1 = ai • H(xi) • bi.In the parallel random oracle model, it is straightforward to argue that any attacker running in sequential time T − 1 will fail to produce an H-sequence of length T except with negligible probability -even if the attacker submits large batches of random oracle queries in each round. In this paper, we introduce the parallel quantum random oracle model and 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).

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“…Recently, Blocki et al [BGGZ19] study RLDCs and RLCCs on adversarial but computationally bounded channels in an effort to reduce these tradeoffs. They obtain RLDCs and RLCCs over the binary alphabet, with constant information rate, and poly-logarithmic locality.…”

Section: Related Workmentioning

confidence: 99%

Relaxed Locally Correctable Codes in Computationally Bounded Channels

Blocki¹,

Gandikota

Grigorescu³

et al. 2019

2019 IEEE International Symposium on Information Theory (ISIT)

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Constructions of locally decodable codes (LDCs) have one of two undesirable properties: low rate or high locality (polynomial in the length of the message). In settings where the encoder/decoder have already exchanged cryptographic keys and the channel is a probabilistic polynomial time (PPT) algorithm, it is possible to circumvent these barriers and design LDCs with constant rate and small locality. However, the assumption that the encoder/decoder have exchanged cryptographic keys is often prohibitive. We thus consider the problem of designing explicit and efficient LDCs in settings where the channel is slightly more constrained than the encoder/decoder with respect to some resource e.g., space or (sequential) time.Given an explicit function f that the channel cannot compute, we show how the encoder can transmit a random secret key to the local decoder using f (·) and a random oracle H(·). This allows bootstrap from the private key LDC construction of Ostrovsky, Pandey and Sahai (ICALP, 2007), thereby answering an open question posed by Guruswami and Smith (FOCS 2010) of whether such bootstrapping techniques may apply to LDCs in weaker channel models than just PPT algorithms. Specifically, in the random oracle model we show how to construct explicit constant rate LDCs with optimal locality of polylog in the security parameter against various resource constrained channels. √ log n log log n [KMRS17]. Our codes are robust against constant fraction of corruptions, and are non-adaptive i.e. all the queries made by the decoder are independent of each other, a property which may be desired when it is beneficial to submit all local decoding queries in a single batch.Our constructions stand at the intersection of coding theory and cryptography, using well-known tools and techniques from cryptography to provide notions of (information theoretic) randomness and security for communication protocols between sender/receiver. To prove the security of our constructions, we introduce a two-phase distinguisher hybrid argument, which may be of independent interest to apply to other coding theoretic problems in these resource bounded channel models. Preliminaries NotationWe use the notation [n] to represent the set {1, 2, . . . , n}. For any x, y ∈ Σ n , let HAM(x) denote the Hamming weight of x, i.e. the number of non-zero coordinates of x. Let HAM(x, y) = HAM(x − y) 1 Note that small alphabet sizes are attractive for practical channels designed to transmit bits efficiently. 2 In this paper we use the security parameter κ in an asymptotic sense e.g., for any attacker running in time poly(κ) there is a negligible function negl(κ) upper bounding the probability that the attacker succeeds. In particular, the function negl(κ) = 2 − log 1+ε κ) is negligible, but does not provide κ-bits of concrete security i.e., any attacker running in time t succeeds with probability at most t2 −κ .3 [OPS07] gives a construction with locality f (κ) = ω(log κ) "one-time" private LDCs, but the construction needs to be modified if we want security again...

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Relaxed Locally Correctable Codes in Computationally Bounded Channels

Cited by 11 publications

References 47 publications

On Explicit Constructions of Extremely Depth Robust Graphs

On Explicit Constructions of Extremely Depth Robust Graphs

On the Security of Proofs of Sequential Work in a Post-Quantum World

Relaxed Locally Correctable Codes in Computationally Bounded Channels

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