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1. Formalizing Algorithmic Bounds in the Query Model in EasyCrypt

   Stoughton, Alley; Chen, Carol; Gaboardi, Marco; Qu, Weihao (August 2022, 13th International Conference on Interactive Theorem Proving (ITP 2022))

   Andronick, June; de Moura, Leonardo (Ed.)

   We use the EasyCrypt proof assistant to formalize the adversarial approach to proving lower bounds for computational problems in the query model. This is done using a lower bound game between an algorithm and adversary, in which the adversary answers the algorithm’s queries in a way that makes the algorithm issue at least the desired number of queries. A complementary upper bound game is used for proving upper bounds of algorithms; here the adversary incrementally and adaptively realizes an algorithm’s input. We prove a natural connection between the lower and upper bound games, and apply our framework to three computational problems, including searching in an ordered list and comparison-based sorting, giving evidence for the generality of our notion of algorithm and the usefulness of our framework. 

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2. Practical Settlement Bounds for Proof-of-Work Blockchains

   https://doi.org/10.1145/3548606.3559368  

   Gazi, Peter; Ren, Ling; Russell, Alexander (November 2022, Proceedings of the 2022 ACM SIGSAC Conference on Computer and Communications Security)

   Nakamoto proof-of-work ledger consensus currently underlies the majority of deployed cryptocurrencies and smart-contract blockchains. While a long and fruitful line of work has succeeded to identify its exact security region---that is, the set of parametrizations under which it possesses asymptotic security---the existing theory does not provide concrete settlement time guarantees that are tight enough to inform practice. In this work we provide a new approach for obtaining concrete and practical settlement time guarantees suitable for reasoning about deployed systems. We give an efficient method for computing explicit upper bounds on settlement time as a function of primary system parameters: honest and adversarial computational power and a bound on network delays. We implement this computational method and provide a comprehensive sample of concrete bounds for several settings of interest. We also analyze a well-known attack strategy to provide lower bounds on the settlement times. For Bitcoin, for example, our upper and lower bounds are within 90 seconds of each other for 1-hour settlement assuming 10 second network delays and a 10% adversary. In comparison, the best prior result has a gap of 2 hours in the upper and lower bounds with the same parameters. 

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3. Local Enumeration and Majority Lower Bounds

   https://doi.org/10.4230/LIPIcs.CCC.2024.17  

   Gurumukhani, Mohit; Paturi, Ramamohan; Pudlák, Pavel; Saks, Michael; Talebanfard, Navid (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Santhanam, Rahul (Ed.)

   Depth-3 circuit lower bounds and k-SAT algorithms are intimately related; the state-of-the-art Σ^k_3-circuit lower bound (Or-And-Or circuits with bottom fan-in at most k) and the k-SAT algorithm of Paturi, Pudlák, Saks, and Zane (J. ACM'05) are based on the same combinatorial theorem regarding k-CNFs. In this paper we define a problem which reveals new interactions between the two, and suggests a concrete approach to significantly stronger circuit lower bounds and improved k-SAT algorithms. For a natural number k and a parameter t, we consider the Enum(k, t) problem defined as follows: given an n-variable k-CNF and an initial assignment α, output all satisfying assignments at Hamming distance t(n) of α, assuming that there are no satisfying assignments of Hamming distance less than t(n) of α. We observe that an upper bound b(n, k, t) on the complexity of Enum(k, t) simultaneously implies depth-3 circuit lower bounds and k-SAT algorithms: - Depth-3 circuits: Any Σ^k_3 circuit computing the Majority function has size at least binom(n,n/2)/b(n, k, n/2). - k-SAT: There exists an algorithm solving k-SAT in time O(∑_{t=1}^{n/2}b(n, k, t)). A simple construction shows that b(n, k, n/2) ≥ 2^{(1 - O(log(k)/k))n}. Thus, matching upper bounds for b(n, k, n/2) would imply a Σ^k_3-circuit lower bound of 2^Ω(log(k)n/k) and a k-SAT upper bound of 2^{(1 - Ω(log(k)/k))n}. The former yields an unrestricted depth-3 lower bound of 2^ω(√n) solving a long standing open problem, and the latter breaks the Super Strong Exponential Time Hypothesis. In this paper, we propose a randomized algorithm for Enum(k, t) and introduce new ideas to analyze it. We demonstrate the power of our ideas by considering the first non-trivial instance of the problem, i.e., Enum(3, n/2). We show that the expected running time of our algorithm is 1.598ⁿ, substantially improving on the trivial bound of 3^{n/2} ≃ 1.732ⁿ. This already improves Σ^3_3 lower bounds for Majority function to 1.251ⁿ. The previous bound was 1.154ⁿ which follows from the work of Håstad, Jukna, and Pudlák (Comput. Complex.'95). By restricting ourselves to monotone CNFs, Enum(k, t) immediately becomes a hypergraph Turán problem. Therefore our techniques might be of independent interest in extremal combinatorics. 

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4. Fragile Complexity of Comparison-Based Algorithms

   https://doi.org/10.4230/LIPIcs.ESA.2019.2  

   Afshani, P.; Fagerberg, R.; Hammer, D.; Jacob, R.; Kostitsyna, I.; Meyer, U.; Penschuck, M.; Sitchinava, N. (September 2019, Proceedings of the 27th European Symposium on Algorithms)

   We initiate a study of algorithms with a focus on the computational complexity of individual elements, and introduce the fragile complexity of comparison-based algorithms as the maximal number of comparisons any individual element takes part in. We give a number of upper and lower bounds on the fragile complexity for fundamental problems, including Minimum, Selection, Sorting and Heap Construction. The results include both deterministic and randomized upper and lower bounds, and demonstrate a separation between the two settings for a number of problems. The depth of a comparator network is a straight-forward upper bound on the worst case fragile complexity of the corresponding fragile algorithm. We prove that fragile complexity is a different and strictly easier property than the depth of comparator networks, in the sense that for some problems a fragile complexity equal to the best network depth can be achieved with less total work and that with randomization, even a lower fragile complexity is possible. 

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5. Achievable Rates for Low-Complexity Posterior Matching over the Binary Symmetric Channel

   https://doi.org/10.1109/TIT.2023.3322095  

   Antonini, Amaael; Gimelshein, Rita; Wesel, Richard (October 2023, IEEE Transactions on Information Theory)

   Horstein, Burnashev, Shayevitz and Feder, Naghshvar et al . and others have studied sequential transmission of a k-bit message over the binary symmetric channel (BSC) with full, noiseless feedback using posterior matching. Yang et al . provide an improved lower bound on the achievable rate using martingale analysis that relies on the small-enough difference (SED) partitioning introduced by Naghshvar et al . SED requires a relatively complex encoder and decoder. To reduce complexity, this paper replaces SED with relaxed constraints that admit the small enough absolute difference (SEAD) partitioning rule. The main analytical results show that achievable-rate bounds higher than those found by Yang et al . [2] are possible even under the new constraints, which are less restrictive than SED. The new analysis does not use martingale theory for the confirmation phase and applies a surrogate channel technique to tighten the results. An initial systematic transmission further increases the achievable rate bound. The simplified encoder associated with SEAD has a complexity below order O ( K 2 ) and allows simulations for message sizes of at least 1000 bits. For example, simulations achieve 99% of of the channel’s 0.50-bit capacity with an average block size of 200 bits for a target codeword error rate of 10 -3. 

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