Relaxed Locally Correctable Codes in Computationally Bounded Channels*
Error-correcting codes that admit {\em local} decoding and correcting algorithms have been the focus of much recent research due to their numerous theoretical and practical applications. An important goal is to obtain the best possible tradeoffs between the number of queries the algorithm makes to its oracle (the {\em locality} of the task), and the amount of redundancy in the encoding (the {\em information rate}). In Hamming's classical adversarial channel model, the current tradeoffs are dramatic, allowing either small locality, but superpolynomial blocklength, or small blocklength, but high locality. However, in the computationally bounded, adversarial channel model, proposed by Lipton (STACS 1994), constructions of locally decodable codes suddenly exhibit small locality and small blocklength, but these constructions require strong trusted setup assumptions e.g., Ostrovsky, Pandey and Sahai (ICALP 2007) construct private locally decodable codes in the setting where the sender and receiver already share a symmetric key. We study variants of locally decodable and locally correctable codes in computationally bounded, adversarial channels, in a setting with no public-key or private-key cryptographic setup. The only setup assumption we require is the selection of the {\em public} parameters (seed) for a collision-resistant hash function. Specifically, we provide constructions of {\em relaxed locally correctable} and more »
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NSF-PAR ID:
10092900
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Relaxed Locally Correctable Codes in Computationally Bounded Channels
Page Range or eLocation-ID:
2414 to 2418
1. We formally introduce, define, and construct {\em memory-hard puzzles}. Intuitively, for a difficulty parameter $t$, a cryptographic puzzle is memory-hard if any parallel random access machine (PRAM) algorithm with small'' cumulative memory complexity ($\ll t^2$) cannot solve the puzzle; moreover, such puzzles should be both easy'' to generate and be solvable by a sequential RAM algorithm running in time $t$. Our definitions and constructions of memory-hard puzzles are in the standard model, assuming the existence of indistinguishability obfuscation (\iO) and one-way functions (OWFs), and additionally assuming the existence of a {\em memory-hard language}. Intuitively, a language is memory-hard if it is undecidable by any PRAM algorithm with small'' cumulative memory complexity, while a sequential RAM algorithm running in time $t$ can decide the language. Our definitions and constructions of memory-hard objects are the first such definitions and constructions in the standard model without relying on idealized assumptions (such as random oracles). We give two applications which highlight the utility of memory-hard puzzles. For our first application, we give a construction of a (one-time) {\em memory-hard function} (MHF) in the standard model, using memory-hard puzzles and additionally assuming \iO and OWFs. For our second application, we show any cryptographic puzzle (\eg,more »
2. We construct locally decodable codes (LDCs) to correct insertion-deletion errors in the setting where the sender and receiver share a secret key or where the channel is resource-bounded. Our constructions rely on a so-called Hamming-to-InsDel'' compiler (Ostrovsky and Paskin-Cherniavsky, ITS '15 \& Block et al., FSTTCS '20), which compiles any locally decodable Hamming code into a locally decodable code resilient to insertion-deletion (InsDel) errors. While the compilers were designed for the classical coding setting, we show that the compilers still work in a secret key or resource-bounded setting. Applying our results to the private key Hamming LDC of Ostrovsky, Pandey, and Sahai (ICALP '07), we obtain a private key InsDel LDC with constant rate and polylogarithmic locality. Applying our results to the construction of Blocki, Kulkarni, and Zhou (ITC '20), we obtain similar results for resource-bounded channels; i.e., a channel where computation is constrained by resources such as space or time.