arXiv subjects

Christoph Hofmeister

Publications and source records attributed to Christoph Hofmeister.

14 recordsLinked to original sources

arXiv2026

Random Access Expectation in DNA Storage and Fountain Codes

Motivated by DNA data storage, we study the expected number of coded symbols drawn from a linear code until a desired information symbol can be decoded - the random access expectation. We focus on generator matrices with a type of symmetry, conjectured in prior work to be optimal, which we call fully symmetric. We point out an equivalence between binary fully symmetric codes and LT codes. Using this observation, we analyze the random access expectation of binary fully symmetric codes under a peeling decoder, in the large blocklength limit. Under these assumptions, the random access expectation, normalized by the number of information symbols, is at least $π/4 \approx 0.7854$, while a value of $\approx 0.7869$ is achievable.

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arXiv2026

On the Extension of Private Distributed Matrix Multiplication Schemes to the Grid Partition

We consider polynomial codes for private distributed matrix multiplication (PDMM/SDMM). Existing codes for PDMM are either specialized for the outer product partitioning (OPP), or inner product partitioning (IPP), or are valid for the more general grid partitioning (GP). We design extension operations that can be applied to a large class of OPP code designs to extend them to the GP case. Applying them to existing codes improves upon the state-of-the-art for certain parameters. Additionally, we show that the GP schemes resulting from extension fulfill additional combinatorial constraints, potentially limiting their performance. We illustrate this point by presenting a new GP scheme that does not adhere to these constraints and outperforms the state-of-the-art for a range of parameters.

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arXiv2025

CAT and DOG: Improved Codes for Private Distributed Matrix Multiplication

We present novel constructions of polynomial codes for private distributed matrix multiplication (PDMM/SDMM) using outer product partitioning (OPP). We extend the degree table framework from the literature to cyclic-addition degree tables (CATs). By using roots of unity as evaluation points, we enable modulo-addition in the table. Based on CATs, we present an explicit construction, called CATx, that requires fewer workers than existing schemes in the low-privacy regime. Additionally, we present new families of schemes based on conventional degree tables, called GASPrs and DOGrs, that outperform the state-of-the-art for a wide range of parameters.

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arXiv2025

Perfect Privacy for Discriminator-Based Byzantine-Resilient Federated Learning

Federated learning (FL) shows great promise in large-scale machine learning but introduces new privacy and security challenges. We propose ByITFL and LoByITFL, two novel FL schemes that enhance resilience against Byzantine users while keeping the users' data private from eavesdroppers. To ensure privacy and Byzantine resilience, our schemes build on having a small representative dataset available to the federator and crafting a discriminator function allowing the mitigation of corrupt users' contributions. ByITFL employs Lagrange coded computing and re-randomization, making it the first Byzantine-resilient FL scheme with perfect Information-Theoretic (IT) privacy, though at the cost of a significant communication overhead. LoByITFL, on the other hand, achieves Byzantine resilience and IT privacy at a significantly reduced communication cost, but requires a Trusted Third Party, used only in a one-time initialization phase before training. We provide theoretical guarantees on privacy and Byzantine resilience, along with convergence guarantees and experimental results validating our findings.

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arXiv2025

LoByITFL: Low Communication Secure and Private Federated Learning

Privacy of the clients' data and security against Byzantine clients are key challenges in Federated Learning (FL). Existing solutions to joint privacy and security incur sacrifices on the privacy guarantee. We introduce LoByITFL, the first communication-efficient information-theoretically private and secure FL scheme that makes no sacrifices on the privacy guarantees while ensuring security against Byzantine adversaries. The key components are a small and representative dataset available to the federator, a careful modification of the FLTrust algorithm, and the one-time use of a trusted third party during an initialization period. We provide theoretical guarantees on the privacy and Byzantine resilience, as well as experimental results showing the convergence of LoByITFL.

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arXiv2025

Byzantine-Resilient Secure Aggregation for Federated Learning Without Privacy Compromises

Federated learning (FL) shows great promise in large scale machine learning, but brings new risks in terms of privacy and security. We propose ByITFL, a novel scheme for FL that provides resilience against Byzantine users while keeping the users' data private from the federator and private from other users. The scheme builds on the preexisting non-private FLTrust scheme, which tolerates malicious users through trust scores (TS) that attenuate or amplify the users' gradients. The trust scores are based on the ReLU function, which we approximate by a polynomial. The distributed and privacy-preserving computation in ByITFL is designed using a combination of Lagrange coded computing, verifiable secret sharing and re-randomization steps. ByITFL is the first Byzantine resilient scheme for FL with full information-theoretic privacy.

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arXiv2025

Achieving DNA Labeling Capacity with Minimum Labels through Extremal de Bruijn Subgraphs

DNA labeling is a tool in molecular biology and biotechnology to visualize, detect, and study DNA at the molecular level. In this process, a DNA molecule is labeled by a set of specific patterns, referred to as labels, and is then imaged. The resulting image is modeled as an $(\ell+1)$-ary sequence, where $\ell$ is the number of labels, in which any non-zero symbol indicates the appearance of the corresponding label in the DNA molecule. The labeling capacity refers to the maximum information rate that can be achieved by the labeling process for any given set of labels. The main goal of this paper is to study the minimum number of labels of the same length required to achieve the maximum labeling capacity of 2 for DNA sequences or $\log_2q$ for an arbitrary alphabet of size $q$. The solution to this problem requires the study of path unique subgraphs of the de Bruijn graph with the largest number of edges. We provide upper and lower bounds on this value. We draw new connections to existing literature that let us prove an asymptotic result as the label length tends to infinity.

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arXiv2024

Private Aggregation in Hierarchical Wireless Federated Learning with Partial and Full Collusion

In federated learning, a federator coordinates the training of a model, e.g., a neural network, on privately owned data held by several participating clients. The gradient descent algorithm, a well-known and popular iterative optimization procedure, is run to train the model. Every client computes partial gradients based on their local data and sends them to the federator, which aggregates the results and updates the model. Privacy of the clients' data is a major concern. In fact, it is shown that observing the partial gradients can be enough to reveal the clients' data. Existing literature focuses on private aggregation schemes that tackle the privacy problem in federated learning in settings where all users are connected to each other and to the federator. In this paper, we consider a hierarchical wireless system architecture in which the clients are connected to base stations; the base stations are connected to the federator either directly or through relays. We examine settings with and without relays, and derive fundamental limits on the communication cost under information-theoretic privacy with different collusion assumptions. We introduce suitable private aggregation schemes tailored for these settings whose communication costs are multiplicative factors away from the derived bounds.

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arXiv2024

Scalable and Reliable Over-the-Air Federated Edge Learning

Federated edge learning (FEEL) has emerged as a core paradigm for large-scale optimization. However, FEEL still suffers from a communication bottleneck due to the transmission of high-dimensional model updates from the clients to the federator. Over-the-air computation (AirComp) leverages the additive property of multiple-access channels by aggregating the clients' updates over the channel to save communication resources. While analog uncoded transmission can benefit from the increased signal-to-noise ratio (SNR) due to the simultaneous transmission of many clients, potential errors may severely harm the learning process for small SNRs. To alleviate this problem, channel coding approaches were recently proposed for AirComp in FEEL. However, their error-correction capability degrades with an increasing number of clients. We propose a digital lattice-based code construction with constant error-correction capabilities in the number of clients, and compare to nested-lattice codes, well-known for their optimal rate and power efficiency in the point-to-point AWGN channel.

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arXiv2024

Interactive Byzantine-Resilient Gradient Coding for General Data Assignments

We tackle the problem of Byzantine errors in distributed gradient descent within the Byzantine-resilient gradient coding framework. Our proposed solution can recover the exact full gradient in the presence of $s$ malicious workers with a data replication factor of only $s+1$. It generalizes previous solutions to any data assignment scheme that has a regular replication over all data samples. The scheme detects malicious workers through additional interactive communication and a small number of local computations at the main node, leveraging group-wise comparisons between workers with a provably optimal grouping strategy. The scheme requires at most $s$ interactive rounds that incur a total communication cost logarithmic in the number of data samples.

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arXiv2024

Byzantine-Resilient Gradient Coding through Local Gradient Computations

We consider gradient coding in the presence of an adversary controlling so-called malicious workers trying to corrupt the computations. Previous works propose the use of MDS codes to treat the responses from malicious workers as errors and correct them using the error-correction properties of the code. This comes at the expense of increasing the replication, i.e., the number of workers each partial gradient is computed by. In this work, we propose a way to reduce the replication to $s+1$ instead of $2s+1$ in the presence of $s$ malicious workers. Our method detects erroneous inputs from the malicious workers, transforming them into erasures. This comes at the expense of $s$ additional local computations at the main node and additional rounds of light communication between the main node and the workers. We define a general framework and give fundamental limits for fractional repetition data allocations. Our scheme is optimal in terms of replication and local computation and incurs a communication cost that is asymptotically, in the size of the dataset, a multiplicative factor away from the derived bound. We furthermore show how additional redundancy can be exploited to reduce the number of local computations and communication cost, or, alternatively, tolerate straggling workers.

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arXiv2023

Trading Communication for Computation in Byzantine-Resilient Gradient Coding

We consider gradient coding in the presence of an adversary, controlling so-called malicious workers trying to corrupt the computations. Previous works propose the use of MDS codes to treat the inputs of the malicious workers as errors and correct them using the error-correction properties of the code. This comes at the expense of increasing the replication, i.e., the number of workers each partial gradient is computed by. In this work, we reduce replication by proposing a method that detects the erroneous inputs from the malicious workers, hence transforming them into erasures. For $s$ malicious workers, our solution can reduce the replication to $s+1$ instead of $2s+1$ for each partial gradient at the expense of only $s$ additional computations at the main node and additional rounds of light communication between the main node and the workers. We give fundamental limits of the general framework for fractional repetition data allocation. Our scheme is optimal in terms of replication and local computation but incurs a communication cost that is asymptotically, in the size of the dataset, a multiplicative factor away from the derived bound.

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arXiv2022

Nested Gradient Codes for Straggler Mitigation in Distributed Machine Learning

We consider distributed learning in the presence of slow and unresponsive worker nodes, referred to as stragglers. In order to mitigate the effect of stragglers, gradient coding redundantly assigns partial computations to the worker such that the overall result can be recovered from only the non-straggling workers. Gradient codes are designed to tolerate a fixed number of stragglers. Since the number of stragglers in practice is random and unknown a priori, tolerating a fixed number of stragglers can yield a sub-optimal computation load and can result in higher latency. We propose a gradient coding scheme that can tolerate a flexible number of stragglers by carefully concatenating gradient codes for different straggler tolerance. By proper task scheduling and small additional signaling, our scheme adapts the computation load of the workers to the actual number of stragglers. We analyze the latency of our proposed scheme and show that it has a significantly lower latency than gradient codes.

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arXiv2022

Secure Private and Adaptive Matrix Multiplication Beyond the Singleton Bound

We consider the problem of designing secure and private codes for distributed matrix-matrix multiplication. A master server owns two private matrices and hires worker nodes to help compute their product. The matrices should remain information-theoretically private from the workers. Some of the workers are malicious and return corrupted results to the master. We design a framework for security against malicious workers in private matrix-matrix multiplication. The main idea is a careful use of Freivalds' algorithm to detect erroneous matrix multiplications. Our main goal is to apply this security framework to schemes with adaptive rates. Adaptive schemes divide the workers into clusters and thus provide flexibility in trading decoding complexity for efficiency. Our new scheme, SRPM3, provides a computationally efficient security check per cluster that detects the presence of one or more malicious workers with high probability. An additional per worker check is used to identify the malicious nodes. SRPM3 can tolerate the presence of an arbitrary number of malicious workers. We provide theoretical guarantees on the complexity of the security checks and simulation results on both, the missed detection rate as well as on the time needed for the integrity check.

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