In spite of their broad popularity, blockchain systems are increasingly facing issues related to large storage overhead and computational cost. A commonly proposed solution to blockchain scalability is the inclusion of so-called light nodes. Light nodes are resource-constrained users who do not routinely download the full block of transaction data; rather, they rely on full nodes to broadcast fraud proofs if any transaction in a block is invalid. While this approach allows resource-constrained nodes to participate in a blockchain, it also exposes them to a so-called Data Availability (DA) attack. A DA attack is described by an event in which an adversarial block producer creates a block containing invalid transactions to be hidden from the rest of the network. Error-correcting codes are a promising solution to DA attacks by forcing the adversary to withhold a larger portion of the block (that holds invalid transactions), thus making adversarial actions easier to be discovered by honest nodes in the network. In a previous work by Al Bassam et al., it was recognized that 2D Reed-Solomon codes are a promising code construction due to their small fraud proof sizes. In this work, we develop new light node sampling strategies, tailored for 2D Reed-Solomon codes, that increase the resilience of the system against DA attacks. We compare our solution to the naive random sampling without replacement and demonstrate significant gains.

High-dimensional quantum key distribution (QKD) is a popular protocol that provides information theoretically secure keys to multiple parties. Two important steps of QKD are 1) the information reconciliation (IR) step, where parties reconcile mismatches in generated keys through classical communication, and 2) the privacy amplification (PA) step, where parties distill their common key into a new secure key that the adversary has little to no information about. In general, these two steps have been abstracted as two distinct problems. In this work, we design our IR protocol to be aware of the PA step and utilize sampling to relax the requirement on the IR step without sacrificing the final key length of the PA step, allowing for more bits generated in key creation utilizing practical decoders. We provide a novel PA-aware LDPC code construction known as Block-MDS QC-LDPC codes that can utilize the relaxed requirement. We demonstrate through simulations that our technique of sampling can provide notable gains in successfully creating secret keys.

In this paper, we present a novel variation of the coded matrix multiplication problem which we refer to as fully private grouped matrix multiplication (FPGMM). In FPGMM, a master wants to compute a group of matrix products between two matrix libraries that can be accessed by all workers while ensuring that any number of prescribed colluding workers learn nothing about which matrix products the master desires, nor the number of matrix products. We present an achievable scheme using a variant of Cross-Subspace Alignment (CSA) codes that offers flexibility in communication and computation cost. Additionally, we demonstrate how our scheme can outperform naive applications of schemes used in a related privacy focused coded matrix multiplication problem.

Light nodes in blockchain systems are known to be vulnerable to data availability (DA) attacks where they accept an invalid block with unavailable portions. Previous works have used LDPC and 2-D Reed Solomon (2D-RS) codes with Merkle Trees to mitigate DA attacks. While these codes have demonstrated improved performance across a variety of metrics such as DA detection probability, they are difficult to apply to blockchains with large blocks due to generally intractable code guarantees for large codelengths (LDPC), large decoding complexity (2D-RS), or large coding fraud proof sizes (2D-RS). We address these issues by proposing the novel Polar Coded Merkle Tree (PCMT) which is a Merkle Tree built from the encoding graphs of polar codes and a specialized polar code construction called Sampling-Efficient Freezing (SEF). We demonstrate that the PCMT with SEF polar codes performs well in detecting DA attacks for large block sizes.

Light nodes in blockchains improve the scalability of the system by storing a small portion of the blockchain ledger. In certain blockchains, light nodes are vulnerable to adata availability(DA) attack where a malicious node makes the light nodes accept an invalid block by hiding the invalid portion of the block from the nodes in the system. Recently, a technique based on LDPC codes called Coded Merkle Tree (CMT) was proposed by Yuet al.that enables light nodes to detect a DA attack by randomly requesting/sampling portions of the block from the malicious node. However, light nodes fail to detect a DA attack with high probability if a malicious node hides a small stopping set of the LDPC code. To mitigate this problem, Yuet al.used random LDPC codes that achieve large minimum stopping set size with high probability. Although effective, these codes are not necessarily optimal for this application, especially at short code lengths, which are relevant for low latency systems, IoT blockchains, etc.. In this paper, we focus on short code lengths and demonstrate that a suitable co-design of specialized LDPC codes and the light node sampling strategy can improve the probability of detection of DA attacks. We consider different adversary models based on their computational capabilities of finding stopping sets in LDPC codes. For a weak adversary model, we devise a new LDPC code construction termed as theentropy-constrainedPEG (EC-PEG) algorithm whichconcentratesstopping sets to a small group of variable nodes. We demonstrate that the EC-PEG algorithm coupled with a greedy sampling strategy improves the probability of detection of DA attacks. For stronger adversary models, we provide a co-design of a sampling strategy calledlinear-programming-sampling(LP-sampling) and an LDPC code construction calledlinear-programming-constrainedPEG (LC-PEG) algorithm. The new co-design demonstrates a higher probability of detection of DA attacks compared to approaches in earlier literature.

In this paper, we introduce the Variable Coded Distributed Batch Matrix Multiplication (VCDBMM) problem which tasks a distributed system to perform batch matrix multiplication where matrices are not necessarily distinct among batch jobs. Most coded matrix-matrix computation work has broadly focused in two directions: matrix partitioning for computing a single computation task and batch processing of multiple distinct computation tasks. While these works provide codes with good straggler resilience and fast decoding for their problem spaces, these codes would not be able to take advantage of the natural redundancy of re-using matrices across batch jobs. Inspired by Cross-Subspace Alignment codes, we develop Flexible Cross-Subspace Alignments (FCSA) codes that are flexible enough to utilize this redundancy. We provide a full characterization of FCSA codes which allow for a wide variety of system complexities including good straggler resilience and fast decoding. We theoretically demonstrate that, under certain practical conditions, FCSA codes are within a factor of two of the optimal solution when it comes to straggler resilience; our simulations demonstrate that our codes achieve even better optimality gaps in practice.

A popular method of improving the throughput of blockchain systems is by running smaller side blockchains that push the hashes of their blocks onto a trusted blockchain. Side blockchains are vulnerable to stalling attacks where a side blockchain node pushes the hash of a block to the trusted blockchain but makes the block unavailable to other side blockchain nodes. Recently, Shenget al. proposed a data availability oracle based on LDPC codes and a data dispersal protocol as a solution to the above problem. While showing improvements, the codes and dispersal protocol were designed disjointly which may not be optimal in terms of the communication cost associated with the oracle. In this paper, we provide a tailored dispersal protocol and specialized LDPC code construction based on the Progressive Edge Growth (PEG) algorithm, called the dispersal-efficient PEG (DE-PEG) algorithm, aimed to reduce the communication cost associated with the new dispersal protocol. Our new code construction reduces the communication cost and, additionally, is less restrictive in terms of system design. Full paper version [1]: https://arxiv.org/pdf/2105.06004.pdf

In this paper, we propose a non-uniform windowed decoder for multi-dimensional spatially-coupled LDPC (MD-SCLDPC) codes over the binary erasure channel. An MD-SC-LDPC code is constructed by connecting together several SC-LDPC codes into one larger code that provides major benefits over a variety of channel models. In general, SC codes allow for lowlatency windowed decoding. While a standard windowed decoder can be naively applied, such an approach does not fully utilize the unique structure of MD-SC-LDPC codes. In this paper, we propose and analyze a novel non-uniform decoder to provide more flexibility between latency and reliability. Our theoretical derivations and empirical results show that our non-uniform decoder greatly improves upon the standard windowed decoder in terms of design flexibility, latency, and complexity.

In certain blockchain systems, light nodes are clients that download only a small portion of the block. Light nodes are vulnerable to data availability (DA) attacks where a malicious node hides an invalid portion of the block from the light nodes. Recently, a technique based on erasure codes called Coded Merkle Tree (CMT) was proposed by Yu et al. that enables light nodes to detect a DA attack with high probability. The CMT is constructed using LDPC codes for fast decoding but can fail to detect a DA attack if a malicious node hides a small stopping set of the code. To combat this, Yu et al. used well-studied techniques to design random LDPC codes with high minimum stopping set size. Although effective, these codes are not necessarily optimal for this application. In this paper, we demonstrate a more specialized LDPC code design to improve the security against DA attacks. We achieve this goal by providing a deterministic LDPC code construction that focuses on concentrating stopping sets to a small group of variable nodes rather than only eliminating stopping sets. We design these codes by modifying the Progressive Edge Growth algorithm into a technique called the entropy-constrained PEG (EC-PEG) algorithm. This new method demonstrates a higher probability of detecting DA attacks and allows for good codes at short lengths.