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1. Relaxed Locally Correctable Codes in Computationally Bounded Channels*

   https://doi.org/10.1109/ISIT.2019.8849322  

   Blocki, Jeremiah ; Gandikota, Venkata ; Grigorescu, Elena ; Zhou, Samson ( July 2019 , 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 locallymore »correctable} and {\em relaxed locally decodable codes} over the binary alphabet, with constant information rate, and poly-logarithmic locality. Our constructions, which compare favorably with their classical analogues in the computationally unbounded Hamming channel, crucially employ {\em collision-resistant hash functions} and {\em local expander graphs}, extending ideas from recent cryptographic constructions of memory-hard functions.« less
2. Local Algorithms for Sparse Spanning Graphs

   https://doi.org/10.1007/s00453-019-00612-6  

   Levi, Reut ; Ron, Dana ; Rubinfeld, Ronitt ( January 2020 , Algorithmica)

   Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. We consider a relaxed version of this problem in the setting of local algorithms. The relaxation is that the constructed subgraph is a sparse spanning subgraph containing at most (1+ϵ)n edges (where n is the number of vertices and ϵ is a given approximation/sparsity parameter). In the local setting, the goal is to quickly determine whether a given edge e belongs to such a subgraph, without constructing the whole subgraph, but rather by inspecting (querying) the local neighborhood of e. The challenge is to maintain consistency. That is, to provide answers concerning different edges according to the same spanning subgraph. We first show that for general bounded-degree graphs, the query complexity of any such algorithm must be Ω(n−−√). This lower bound holds for constant-degree graphs that have high expansion. Next we design an algorithm for (bounded-degree) graphs with high expansion, obtaining a result that roughly matches the lower bound. We then turn to study graphs that exclude a fixed minor (and are hence non-expanding). We design an algorithm for such graphs, which may have an unbounded maximum degree. The query complexity of thismore »algorithm is poly(1/ϵ,h) (independent of n and the maximum degree), where h is the number of vertices in the excluded minor. Though our two algorithms are designed for very different types of graphs (and have very different complexities), on a high-level there are several similarities, and we highlight both the similarities and the differences.« less
3. Explicit Abelian Lifts and Quantum LDPC Codes

   https://doi.org/10.4230/LIPIcs.ITCS.2022.88  

   Jeronimo, Fernando Granha ( January 2022 , Leibniz international proceedings in informatics)

   Braverman, Mark (Ed.)

   For an abelian group H acting on the set [𝓁], an (H,𝓁)-lift of a graph G₀ is a graph obtained by replacing each vertex by 𝓁 copies, and each edge by a matching corresponding to the action of an element of H. Expanding graphs obtained via abelian lifts, form a key ingredient in the recent breakthrough constructions of quantum LDPC codes, (implicitly) in the fiber bundle codes by Hastings, Haah and O'Donnell [STOC 2021] achieving distance Ω̃(N^{3/5}), and in those by Panteleev and Kalachev [IEEE Trans. Inf. Theory 2021] of distance Ω(N/log(N)). However, both these constructions are non-explicit. In particular, the latter relies on a randomized construction of expander graphs via abelian lifts by Agarwal et al. [SIAM J. Discrete Math 2019]. In this work, we show the following explicit constructions of expanders obtained via abelian lifts. For every (transitive) abelian group H ⩽ Sym(𝓁), constant degree d ≥ 3 and ε > 0, we construct explicit d-regular expander graphs G obtained from an (H,𝓁)-lift of a (suitable) base n-vertex expander G₀ with the following parameters: ii) λ(G) ≤ 2√{d-1} + ε, for any lift size 𝓁 ≤ 2^{n^{δ}} where δ = δ(d,ε), iii) λ(G) ≤ ε ⋅ d, formore »any lift size 𝓁 ≤ 2^{n^{δ₀}} for a fixed δ₀ > 0, when d ≥ d₀(ε), or iv) λ(G) ≤ Õ(√d), for lift size "exactly" 𝓁 = 2^{Θ(n)}. As corollaries, we obtain explicit quantum lifted product codes of Panteleev and Kalachev of almost linear distance (and also in a wide range of parameters) and explicit classical quasi-cyclic LDPC codes with wide range of circulant sizes. Items (i) and (ii) above are obtained by extending the techniques of Mohanty, O'Donnell and Paredes [STOC 2020] for 2-lifts to much larger abelian lift sizes (as a byproduct simplifying their construction). This is done by providing a new encoding of special walks arising in the trace power method, carefully "compressing" depth-first search traversals. Result (iii) is via a simpler proof of Agarwal et al. [SIAM J. Discrete Math 2019] at the expense of polylog factors in the expansion.« less
4. Non-malleability Against Polynomial Tampering

   Ball, Marshall ; Chattopadhyay, Eshan ; Liao, Jyun-Jie ; Malkin, Tal ; Tan, Li-Yang. ( July 2020 , CRYPTO)

   Micciancio, Daniele ; Ristenpart, Thomas. (Ed.)

   We present the first explicit construction of a non-malleable code that can handle tampering functions that are bounded-degree polynomials. Prior to our work, this was only known for degree-1 polynomials (affine tampering functions), due to Chattopad- hyay and Li (STOC 2017). As a direct corollary, we obtain an explicit non-malleable code that is secure against tampering by bounded-size arithmetic circuits. We show applications of our non-malleable code in constructing non-malleable se- cret sharing schemes that are robust against bounded-degree polynomial tampering. In fact our result is stronger: we can handle adversaries that can adaptively choose the polynomial tampering function based on initial leakage of a bounded number of shares. Our results are derived from explicit constructions of seedless non-malleable ex- tractors that can handle bounded-degree polynomial tampering functions. Prior to our work, no such result was known even for degree-2 (quadratic) polynomials.
5. Weighted Additive Spanners

   https://doi.org/10.1007/978-3-030-60440-0_32  

   Ahmed, R ; Bodwin, G ; Darabi, F ; Kobourov, S ; Spence, R ( October 2020 , 46th International Workshop on Graph-Theoretic Concepts in Computer Science (WG))

   A spanner of a graph G is a subgraph H that approximately preserves shortest path distances in G. Spanners are commonly applied to compress computation on metric spaces corresponding to weighted input graphs. Classic spanner constructions can seamlessly handle edge weights, so long as error is measured multiplicatively. In this work, we investigate whether one can similarly extend constructions of spanners with purely additive error to weighted graphs. These extensions are not immediate, due to a key lemma about the size of shortest path neighborhoods that fails for weighted graphs. Despite this, we recover a suitable amortized version, which lets us prove direct extensions of classic +2 and +4 unweighted spanners (both all-pairs and pairwise) to +2W and +4W weighted spanners, where W is the maximum edge weight. Specifically, we show that a weighted graph G contains all-pairs (pairwise) +2W and +4W weighted spanners of size O(n3/2) and O(n7/5) (O(np1/3) and O(np2/7)) respectively. For a technical reason, the +6 unweighted spanner becomes a +8W weighted spanner; closing this error gap is an interesting remaining open problem. That is, we show that G contains all-pairs (pairwise) +8W weighted spanners of size O(n4/3) (O(np1/4)).