Ragnar Freij-Hollanti

Charul Rajput, B. Sundar Rajan, Ragnar Freij-Hollanti et al.

Function-correcting codes (FCCs) are designed to provide error protection for the value of a function computed on the data. Existing work typically focuses solely on protecting the function value and not the underlying data. In this work, we propose a general framework that offers protection for both the data and the function values. Since protecting the data inherently contributes to protecting the function value, we focus on scenarios where the function value requires stronger protection than the data itself. We first introduce a more general approach and a framework for function-correcting codes that incorporates data protection along with protection of function values. A two-step construction procedure for such codes is proposed, and bounds on the optimal redundancy of general FCCs with data protection are reported. Using these results, we exhibit examples that show that data protection can be added to existing FCCs without increasing redundancy. Using our two-step construction procedure, we present explicit constructions of FCCs with data protection for specific families of functions, such as locally bounded functions and the Hamming weight function. We associate a graph called minimum-distance graph to a code and use it to show that perfect codes and maximum distance separable (MDS) codes cannot provide additional protection to function values over and above the amount of protection for data for any function. Then we focus on linear FCCs and provide some results for linear functions, leveraging their inherent structural properties. To the best of our knowledge, this is the first instance of FCCs with a linear structure. Finally, we generalize the Plotkin and Hamming bounds well known in classical error-correcting coding theory to FCCs with data protection.

Charul Rajput, B. Sundar Rajan, Ragnar Freij-Hollanti et al.

Function-correcting codes with data protection simultaneously protect both the data and a function of the data at distinct error-correction levels. When the function receives strictly stronger protection than the data, such a code is called a strict function-correcting code with data protection. While prior work showed that perfect and MDS codes cannot serve as strict function-correcting codes, which codes can serve this role, and how to construct them, has remained open. In this paper, we address the existence and construction of strict function-correcting codes for linear codes through three main contributions. First, using the $α$-distance graph framework from our prior work, we establish a graph-theoretic existence condition under which a code can serve as a strict function-correcting code. For linear codes, we prove this distance graph is isomorphic to a Cayley graph, which implies the connected components are cosets of the subcode generated by low-weight codewords. This transforms the existence problem into a subcode generation problem. Second, a classical result of Simonis shows any linear code can be transformed into one with the same parameters whose basis consists entirely of minimum-weight codewords. We develop a converse construction: under certain conditions on the weight distribution, a linear code can be transformed into a new code with the same parameters but fewer independent minimum-weight codewords, thereby producing codes suitable for use as strict function-correcting codes. As a source of codes satisfying these conditions, we introduce chain codes, an infinite family of linear codes generated by their minimum-weight codewords. Third, we present an independent construction of strict function-correcting codes from narrow-sense BCH codes with designed distance three, by proving the minimum-weight codewords of such codes are contained in a proper subcode.

Delio Jaramillo-Velez, Charul Rajput, Ragnar Freij-Hollanti et al.

In federated learning, multiple parties train models locally and share their parameters with a central server, which aggregates them to update a global model. To address the risk of exposing sensitive data through local models, secure aggregation via secure multiparty computation has been proposed to enhance privacy. At the same time, perfect privacy can only be achieved by a uniform distribution of the masked local models to be aggregated. This raises a problem when working with real valued data, as there is no measure on the reals that is invariant under the masking operation, and hence information leakage is bound to occur. Shifting the data to a finite field circumvents this problem, but as a downside runs into an inherent accuracy complexity tradeoff issue due to fixed point modular arithmetic as opposed to floating point numbers that can simultaneously handle numbers of varying magnitudes. In this paper, a novel secure parameter aggregation method is proposed that employs the torus rather than a finite field. This approach guarantees perfect privacy for each party's data by utilizing the uniform distribution on the torus, while avoiding accuracy losses. Experimental results show that the new protocol performs similarly to the model without secure aggregation while maintaining perfect privacy. Compared to the finite field secure aggregation, the torus-based protocol can in some cases significantly outperform it in terms of model accuracy and cosine similarity, hence making it a safer choice.

Julian Renner, Sven Puchinger, Antonia Wachter-Zeh et al.

We define and analyze low-rank parity-check (LRPC) codes over extension rings of the finite chain ring $\mathbb{Z}_{p^r}$, where $p$ is a prime and $r$ is a positive integer. LRPC codes have originally been proposed by Gaborit et al.(2013) over finite fields for cryptographic applications. The adaption to finite rings is inspired by a recent paper by Kamche et al. (2019), which constructed Gabidulin codes over finite principle ideal rings with applications to space-time codes and network coding. We give a decoding algorithm based on simple linear-algebraic operations. Further, we derive an upper bound on the failure probability of the decoder. The upper bound is valid for errors whose rank is equal to the free rank.