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1. Two Source Extractors for Asymptotically Optimal Entropy, and (Many) More

   https://doi.org/10.1109/FOCS57990.2023.00075  

   Li, Xin (November 2023, IEEE)

   A long line of work in the past two decades or so established close connections between several different pseudorandom objects and applications, including seeded or seedless non-malleable extractors, two source extractors, (bipartite) Ramsey graphs, privacy amplification protocols with an active adversary, non-malleable codes and many more. These connections essentially show that an asymptotically optimal construction of one central object will lead to asymptotically optimal solutions to all the others. However, despite considerable effort, previous works can get close but still lack one final step to achieve truly asymptotically optimal constructions. In this paper we provide the last missing link, thus simultaneously achieving explicit, asymptotically optimal constructions and solutions for various well studied extractors and applications, that have been the subjects of long lines of research. Our results include: 1. Asymptotically optimal seeded non-malleable extractors, which in turn give two source extractors for asymptotically optimal min-entropy of $$O(\log n)$$, explicit constructions of $$K$$-Ramsey graphs on $$N$$ vertices with $$K=\log^{O(1)} N$$, and truly optimal privacy amplification protocols with an active adversary. 2. Two source non-malleable extractors and affine non-malleable extractors for some linear min-entropy with exponentially small error, which in turn give the first explicit construction of non-malleable codes against $$2$$-split state tampering and affine tampering with constant rate and \emph{exponentially} small error. 3. Explicit extractors for affine sources, sumset sources, interleaved sources, and small space sources that achieve asymptotically optimal min-entropy of $$O(\log n)$ or $$2s+O(\log n)$$ (for space $$s$$ sources). 4. An explicit function that requires strongly linear read once branching programs of size $$2^{n-O(\log n)}$$, which is optimal up to the constant in $$O(\cdot)$$. Previously, even for standard read once branching programs, the best known size lower bound for an explicit function is $$2^{n-O(\log^2 n)}$$. 

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2. Nonmalleable Digital Lockers and Robust Fuzzy Extractors in the Plain Model

   https://doi.org/10.1007/978-3-031-22972-5_13  

   Apon, Daniel; Cachet, Chloe; Fuller, Benjamin; Hall, Peter; Liu, Feng-Hao (January 2023, Advances in Cryptology - Asiacrypt)

   We give the first constructions in the plain model of 1) nonmalleable digital lockers (Canetti and Varia, TCC 2009) and 2) robust fuzzy extractors (Boyen et al., Eurocrypt 2005) that secure sources with entropy below 1/2 of their length. Constructions were previously only known for both primitives assuming random oracles or a common reference string (CRS). Along the way, we define a new primitive called a nonmalleable point function obfuscation with associated data. The associated data is public but protected from all tampering. We use the same paradigm to then extend this to digital lockers. Our constructions achieve nonmalleability over the output point by placing a CRS into the associated data and using an appropriate non-interactive zero-knowledge proof. Tampering is protected against the input point over low-degree polynomials and over any tampering to the output point and associated data. Our constructions achieve virtual black box security. These constructions are then used to create robust fuzzy extractors that can support low-entropy sources in the plain model. By using the geometric structure of a syndrome secure sketch (Dodis et al., SIAM Journal on Computing 2008), the adversary’s tampering function can always be expressed as a low-degree polynomial; thus, the protection provided by the constructed nonmalleable objects suffices. 

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3. Low-Degree Polynomials Extract From Local Sources

   Alrabiah, Omar; Chattopadhyay, Eshan; Goodman, Jesse; Li, Xin; Ribeiro, Joao (January 2022, Leibniz international proceedings in informatics)

   Bojanczyk, Mikolaj; Merelli, Emanuela; Woodruff, David P. (Ed.)

   We continue a line of work on extracting random bits from weak sources that are generated by simple processes. We focus on the model of locally samplable sources, where each bit in the source depends on a small number of (hidden) uniformly random input bits. Also known as local sources, this model was introduced by De and Watson (TOCT 2012) and Viola (SICOMP 2014), and is closely related to sources generated by AC⁰ circuits and bounded-width branching programs. In particular, extractors for local sources also work for sources generated by these classical computational models. Despite being introduced a decade ago, little progress has been made on improving the entropy requirement for extracting from local sources. The current best explicit extractors require entropy n^{1/2}, and follow via a reduction to affine extractors. To start, we prove a barrier showing that one cannot hope to improve this entropy requirement via a black-box reduction of this form. In particular, new techniques are needed. In our main result, we seek to answer whether low-degree polynomials (over 𝔽₂) hold potential for breaking this barrier. We answer this question in the positive, and fully characterize the power of low-degree polynomials as extractors for local sources. More precisely, we show that a random degree r polynomial is a low-error extractor for n-bit local sources with min-entropy Ω(r(nlog n)^{1/r}), and we show that this is tight. Our result leverages several new ingredients, which may be of independent interest. Our existential result relies on a new reduction from local sources to a more structured family, known as local non-oblivious bit-fixing sources. To show its tightness, we prove a "local version" of a structural result by Cohen and Tal (RANDOM 2015), which relies on a new "low-weight" Chevalley-Warning theorem. 

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4. (Continuous) Non-malleable Codes for Partial Functions with Manipulation Detection and Light Updates

   https://doi.org/10.1007/s00145-024-09498-2  

   Kiayias, Aggelos; Liu, Feng-Hao; Tselekounis, Yiannis (April 2024, Journal of Cryptology)

   Abstract Non-malleable codes were introduced by Dziembowski et al. (in: Yao (ed) ICS2010, Tsinghua University Press, 2010), and its main application is the protection of cryptographic devices against tampering attacks on memory. In this work, we initiate a comprehensive study on non-malleable codes for the class of partial functions, that read/write on an arbitrary subset of codeword bits with specific cardinality. We present two constructions: the first one is in the CRS model and allows the adversary to selectively choose the subset of codeword bits, while the latter is in the standard model and adaptively secure. Our constructions are efficient in terms of information rate, while allowing the attacker to access asymptotically almost the entire codeword. In addition, they satisfy a notion which is stronger than non-malleability, that we call non-malleability with manipulation detection, guaranteeing that any modified codeword decodes to either the original message or to$$\bot $$ $\perp$. We show that our primitive implies All-Or-Nothing Transforms (AONTs), and as a result our constructions yield efficient AONTs under standard assumptions (only one-way functions), which, to the best of our knowledge, was an open question until now. Furthermore, we construct a notion of continuous non-malleable codes (CNMC), namely CNMC with light updates, that avoids the full re-encoding process and only uses shuffling and refreshing operations. Finally, we present a number of additional applications of our primitive in tamper resilience. 

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5. Learning Hierarchical Polynomials with Three-Layer Neural Networks

   Wang, Zihao; Nichani, Eshaan; Lee, Jason D (May 2024, International Conference on Learning Representations)

   We study the problem of learning hierarchical polynomials over the standard Gaussian distribution with three-layer neural networks. We specifically consider target functions of the form where is a degree polynomial and is a degree polynomial. This function class generalizes the single-index model, which corresponds to , and is a natural class of functions possessing an underlying hierarchical structure. Our main result shows that for a large subclass of degree polynomials , a three-layer neural network trained via layerwise gradient descent on the square loss learns the target up to vanishing test error in samples and polynomial time. This is a strict improvement over kernel methods, which require samples, as well as existing guarantees for two-layer networks, which require the target function to be low-rank. Our result also generalizes prior works on three-layer neural networks, which were restricted to the case of being a quadratic. When is indeed a quadratic, we achieve the information-theoretically optimal sample complexity , which is an improvement over prior work (Nichani et al., 2023) requiring a sample size of . Our proof proceeds by showing that during the initial stage of training the network performs feature learning to recover the feature with samples. This work demonstrates the ability of three-layer neural networks to learn complex features and as a result, learn a broad class of hierarchical functions. 

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