Private and Resource-Bounded Locally Decodable Codes for Insertions and Deletions

Abstract

We revisit computationally relaxed locally decodable codes (crLDCs) (Blocki et al., Trans. Inf. Theory '21) and give two new constructions. Our first construction is a Hamming crLDC that is conceptually simpler than prior constructions, leveraging digital signature schemes and an appropriately chosen Hamming code. Our second construction is an extension of our Hamming crLDC to handle insertion-deletion (InsDel) errors, yielding an InsDel crLDC. This extension crucially relies on the noisy binary search techniques of Block et al. (FSTTCS '20) to handle InsDel errors. Both crLDC constructions have binary codeword alphabets, are resilient to a constant fraction of Hamming and InsDel errors, respectively, and under suitable parameter choices have poly-logarithmic locality and encoding length linear in the message length and polynomial in the security parameter. These parameters compare favorably to prior constructions in the poly-logarithmic locality regime.

On Relaxed Locally Decodable Codes for Hamming and Insertion-Deletion Errors

“…As our second contribution, we introduce and study the notion of RLDCs correcting insertions and deletions, namely Insdel RLDCs. The non-relaxed variants of Insdel LDCs were first introduced in [OPC15], and were further studied in [BBG + 20,CLZ20,BB21]. Local decoding in the Insdel setting is motivated in DNA storage [YGM17], and in particular [BSB + 21] show recent advances in bio-technological aspects of random access to data in these precise settings.…”

“…[BKZ20] extended the construction from [OPS07] to settings where the sender/decoder do not share randomness, but the adversarial channel is resource bounded. [BB21] applied the [BBG + 20] compiler to the private key Hamming LDC of [OPS07] (resp. resource bounded LDCs of [BKZ20]) to obtain private key Insdel LDCs (resp.…”

“…LDCs in these channels under Hamming error were studied in [OPS07, HO08, HOSW11, HOW15, BGGZ21, BKZ20]. [BB21] applied the [BBG + 20] compiler to the Hamming LDC of [BKZ20] to obtain a constant rate Insdel LDCs with polylog(n) locality for resource bounded channels. The work of [CLZ20] proposes the notion of locally decodable codes with randomized encoding, in both the Hamming and edit distance regimes, and in the setting where the channel is oblivious to the encoded message, or the encoder and decoder share randomness.…”