“Independent and identically distributed” errors do not accurately capture the noisy behavior of real-world data storage and information transmission technologies. Motivated by this, we study channels withinput-correlatedsynchronization errors, meaning that the distribution of synchronization errors (such as deletions and insertions) applied to thei-th inputximay depend on the whole input stringx. We begin by identifying conditions on the input-correlated synchronization channel under which the channel’s information capacity is achieved by a stationary ergodic input source and is equal to its coding capacity. These conditions capture a wide class of channels, including channels with correlated errors observed in DNA-based data storage systems and their multi-trace versions, and generalize prior work. To showcase the usefulness of the general capacity theorem above, we combine it with techniques of Pernice-Li-Wootters (ISIT 2022) and Brakensiek-Li-Spang (FOCS 2020) to obtain explicit capacity-achieving codes for multi-trace channels withrunlength-dependent deletions, motivated by error patterns observed in DNA-based data storage systems.

Non-malleable codes are fundamental objects at the intersection of cryptography and coding theory. These codes provide security guarantees even in settings where error correction and detection are impossible, and have found applications to several other cryptographic tasks. One of the strongest and most well-studied adversarial tampering models is 2-split-state tampering. Here, a codeword is split into two parts which are stored in physically distant servers, and the adversary can then independently tamper with each part using arbitrary functions. This model can be naturally extended to the secret sharing setting with several parties by having the adversary independently tamper with each share. Previous works on non-malleable coding and secret sharing in the split-state tampering model only considered the encoding of classical messages. Furthermore, until recent work by Aggarwal, Boddu, and Jain (IEEE Trans. Inf. Theory 2024 & arXiv 2022), adversaries with quantum capabilities and shared entanglement had not been considered, and it is a priori not clear whether previous schemes remain secure in this model. In this work, we introduce the notions of split-state non-malleable codes and secret sharing schemes for quantum messages secure against quantum adversaries with shared entanglement. Then, we present explicit constructions of such schemes that achieve low-error non-malleability. More precisely, for some constant$c\gt 0$, we construct efficiently encodable and decodable split-state non-malleable codes and secret sharing schemes for quantum messages preserving entanglement with external systems and achieving security against quantum adversaries having shared entanglement with codeword length n, any message length at most$n^{c}$, and error$\varepsilon =2^{-{n^{c}}}$. In the easier setting of average-case non-malleability, we achieve efficient non-malleable coding with rate close to$1/11$.

We study list-recoverability of random linear codes over small fields, both from errors and from erasures. We consider codes of rate ε-close to capacity, and aim to bound the dependence of the output list sizeLon ε, the input list size ℓ, and the alphabet sizeq. Prior to our work, the best upper bound wasL=qO(ℓ/ε)(Zyablov and Pinsker, Prob. Per. Inf. 1981). Previous work has identified cases in whichlinearcodes provably perform worse than non-linear codes with respect to list-recovery. While there exist non-linear codes that achieveL=O(ℓ/ε), we know thatL≥ ℓΩ(1/ε)is necessary for list recovery from erasures over fields of small characteristic, and for list recovery from errors over large alphabets. We show that in other relevant regimes there is no significant price to pay for linearity, in the sense that we get the correct dependence on the gap-to-capacity ε and go beyond the Zyablov– Pinsker bound for the first time. Specifically, whenqis constant and ε approaches zero, • For list-recovery from erasures overprime fields, we show thatL≤C1/ε. By prior work, such a result cannot be obtained for low-characteristic fields. • For list-recovery from errors over arbitrary fields, we prove thatL≤C2/ε. Above,C1andC2depend on the decoding radius, input list size, and field size. We provide concrete bounds on the constants above, and the upper bounds onLimprove upon the Zyablov– Pinsker bound wheneverq≤ 2(1/ε)cfor some small universal constantc> 0.

Seeded extractors are fundamental objects in pseudorandomness and cryptography, and a deep line of work has designed polynomial-time seeded extractors with nearly-optimal parameters. However, existing constructions of seeded extractors with short seed length and large output length run in time Ω(nlog(1/ε)) and often slower, where n is the input source length and ε is the error of the extractor. Since cryptographic applications of extractors require ε to be small, the resulting runtime makes these extractors impractical. Motivated by this, we explore constructions of strong seeded extractors with short seeds computable in nearly-linear timeO(nlogcn), for any error ε. We show that an appropriate combination of modern condensers and classical approaches for constructing seeded extractors for high min-entropy sources yields such extractors. More precisely, we obtain strong extractors forn-bit sources with any min-entropykand any target error ε with seed lengthd=O(log(n/ε)) and output lengthm= (1 − η)kfor an arbitrarily small constant η > 0, running in nearly-linear time. When k or ε are very small, our construction requires a reasonable one-time preprocessing step. These extractors directly yield privacy amplification protocols with nearly-linear time complexity (possibly after a one-time preprocessing step), large output length, and low communication complexity. As a second contribution, we give an instantiation of Trevisan’s extractor that can be evaluated in truly linear time in the RAM model, as long as the number of output bits is at mostn/log(1/ε) polylog(n). Previous fast implementations of Trevisan’s extractor ran in Õ(n) time in this setting.

BACKGROUND: Pathogenic infections pose a significant threat to global health, affecting millions of people every year and presenting substantial challenges to healthcare systems worldwide. Efficient and timely testing plays a critical role in disease control and transmission prevention. Group testing is a well-established method for reducing the number of tests needed to screen large populations when the disease prevalence is low. However, it does not fully utilize the quantitative information provided by qPCR methods, nor is it able to accommodate a wide range of pathogen loads. RESULTS: To address these issues, we introduce a novel adaptive semi-quantitative group testing (SQGT) scheme to efficiently screen populations via two-stage qPCR testing. The SQGT method quantizes cycle threshold (Ct) values into multiple bins, leveraging the information from the first stage of screening to improve the detection sensitivity. Dynamic Ct threshold adjustments mitigate dilution effects and enhance test accuracy. Comparisons with traditional binary outcome GT methods show that SQGT reduces the number of tests by 24% on the only complete real-world qPCR group testing dataset from Israel, while maintaining a negligible false negative rate. CONCLUSION: In conclusion, our adaptive SQGT approach, utilizing qPCR data and dynamic threshold adjustments, offers a promising solution for efficient population screening. With a reduction in the number of tests and minimal false negatives, SQGT holds potential to enhance disease control and testing strategies on a global scale.

Abstract. We prove that the minimum distance problem ([Formula: see text]) on linear codes over any fixed finite field and parameterized by the input distance bound is [Formula: see text]-hard to approximate within any constant factor. We also prove analogous results for the parameterized shortest vector problem ([Formula: see text]) on integer lattices. Specifically, we prove that the [Formula: see text] in the [Formula: see text] norm is [Formula: see text]-hard to approximate within any constant factor for any fixed [Formula: see text] and [Formula: see text]-hard to approximate within a factor approaching 2 for [Formula: see text]. (We show hardness under randomized reductions in each case.) These results answer the main questions left open (and explicitly posed) by Bhattacharyya et al. [ J. ACM, 68 (2021), 16] on the complexity of the parameterized [Formula: see text] and [Formula: see text]. For the [Formula: see text], they established similar hardness for binary linear codes and left the case of general fields open. For the [Formula: see text] in [Formula: see text] norms with [Formula: see text], they showed inapproximability within some constant factor (depending on [Formula: see text]) and left open showing such hardness for arbitrary constant factors. They also left open showing [Formula: see text]-hardness even of the exact SVP in the [Formula: see text] norm.

We prove that the Minimum Distance Problem (MDP) on linear codes over any fixed finite field and parameterized by the input distance bound is W[1]-hard to approximate within any constant factor. We also prove analogous results for the parameterized Shortest Vector Problem (SVP) on integer lattices. Specifically, we prove that SVP in the ℓp norm is W[1]-hard to approximate within any constant factor for any fixed p >1 and W[1]-hard to approximate within a factor approaching 2 for p=1. (We show hardness under randomized reductions in each case.)

Abstract. Construction of error-correcting codes achieving a designated minimum distance parameter is a central problem in coding theory. In this work, we study a very simple construction of binary linear codes that correct a given number of errors [Formula: see text]. Moreover, we design a simple, nearly optimal syndrome decoder for the code as well. The running time of the decoder is only logarithmic in the block length of the code and nearly linear in the number of errors [Formula: see text]. This decoder can be applied to exact for-all sparse recovery over any field, improving upon previous results with the same number of measurements. Furthermore, computation of the syndrome from a received word can be done in nearly linear time in the block length. We also demonstrate an application of these techniques in nonadaptive group testing and construct simple explicit measurement schemes with [Formula: see text] tests and [Formula: see text] recovery time for identifying up to [Formula: see text] defectives in a population of size [Formula: see text].

We consider the problem of designing low-redundancy codes in settings where one must correct deletions in conjunction with substitutions or adjacent transpositions; a combination of errors that is usually observed in DNA-based data storage. One of the most basic versions of this problem was settled more than 50 years ago by Levenshtein, who proved that binary Varshamov-Tenengolts codes correct one arbitrary edit error, i.e., one deletion or one substitution, with nearly optimal redundancy. However, this approach fails to extend to many simple and natural variations of the binary single-edit error setting. In this work, we make progress on the code design problem above in three such variations: 1) We construct linear-time encodable and decodable length-$n$non-binary codes correcting a single edit error with nearly optimal redundancy$\log n+O(\log \log n)$, providing an alternative simpler proof of a result by Cai et al. (IEEE Trans. Inf. Theory 2021). This is achieved by employing what we call weighted VT sketches, a new notion that may be of independent interest. 2) We show the existence of a binary code correcting one deletion or one adjacent transposition with nearly optimal redundancy$\log n+O(\log \log n)$. 3) We construct linear-time encodable and list-decodable binary codes with list-size 2 for one deletion and one substitution with redundancy$4\log n+O(\log \log n)$. This matches the Gilbert-Varshamov existential bound up to an$O(\log \log n)$additive term.

We consider the problem of designing low-redundancy codes in settings where one must correct deletions in conjunction with substitutions or adjacent transpositions; a combination of errors that is usually observed in DNA-based data storage. One of the most basic versions of this problem was settled more than 50 years ago by Levenshtein, who proved that binary Varshamov-Tenengolts codes correct one arbitrary edit error, i.e., one deletion or one substitution, with nearly optimal redundancy. However, this approach fails to extend to many simple and natural variations of the binary single-edit error setting. In this work, we make progress on the code design problem above in three such variations: - We construct linear-time encodable and decodable length-n non-binary codes correcting a single edit error with nearly optimal redundancy log n+O(log log n), providing an alternative simpler proof of a result by Cai, Chee, Gabrys, Kiah, and Nguyen (IEEE Trans. Inf. Theory 2021). This is achieved by employing what we call weighted VT sketches, a new notion that may be of independent interest. - We show the existence of a binary code correcting one deletion or one adjacent transposition with nearly optimal redundancy log n+O(log log n). - We construct linear-time encodable and list-decodable binary codes with list-size 2 for one deletion and one substitution with redundancy 4log n+O(log log n). This matches the existential bound up to an O(log log n) additive term.

We extend the classical problem of privacy amplification to a setting where the active adversary, Eve, is also allowed tofully corruptthe internal memory (which includes the shared randomness, and local randomness tape) of one of the honest parties, Alice and Bob, before the execution of the protocol. We require that either one of Alice or Bob detects tampering, or they agree on a shared key that is indistinguishable from the uniform distribution to Eve. We obtain the following results: 1) we give a privacy amplification protocol via low-error non-malleable two-source extractors with one source having low min-entropy. In particular, this implies the existence of such (non-efficient) protocols; 2) we show that even slight improvements to the state-of-the-art explicit non-malleable two-source extractors would lead to explicit low-error, low min-entropy two-source extractors, thereby resolving a long-standing open question. This suggests that obtaining (information-theoretically secure)explicitnon-malleable two-source extractors for (1) might be hard; 3) we present explicit constructions of low-error, low min-entropy non-malleable two-source extractors in the CRS model of (Garg, Kalai, Khurana, Eurocrypt 2020), assuming either the quasi-polynomial hardness of DDH or the existence of nearly-optimal collision-resistant hash functions; 4) we instantiate our privacy amplification protocol with the above mentioned non-malleable two-source extractors in the CRS model, leading to explicit, computationally-secure protocols. This is not immediate from (1) because in the computational setting we need to make sure that, in particular, all randomness sources remain samplable throughout the proof. This requires upgrading the assumption of quasi-polynomial hardness of DDH to sub-exponential hardness of DDH.We emphasize that each of the first three results can be read independently.

We show that the most common flavors of noisy leakage can be simulated in the information-theoretic setting using a single query of bounded leakage, up to a small statistical simulation error and a slight loss in the leakage parameter. The latter holds true in particular for one of the most used noisy-leakage models, where the noisiness is measured using the conditional average min-entropy (Naor and Segev, CRYPTO’09 and SICOMP’12). Our reductions between noisy and bounded leakage are achieved in two steps. First, we put forward a new leakage model (dubbed the dense leakage model) and prove that dense leakage can be simulated in the information-theoretic setting using a single query of bounded leakage, up to small statistical distance. Second, we show that the most common noisy-leakage models fall within the class of dense leakage, with good parameters. Third, we prove lower bounds on the amount of bounded leakage required for simulation with sub-constant error, showing that our reductions are nearly optimal. In particular, our results imply that useful general simulation of noisy leakage based on statistical distance and mutual information is impossible. We also provide a complete picture of the relationships between different noisy-leakage models. Our result finds applications to leakage-resilient cryptography, where we are often able to lift security in the presence of bounded leakage to security in the presence of noisy leakage, both in the information-theoretic and in the computational setting. Remarkably, this lifting procedure makes only black-box use of the underlying schemes. Additionally, we show how to use lower bounds in communication complexity to prove that bounded-collusion protocols (Kumar, Meka, and Sahai, FOCS’19) for certain functions do not only require long transcripts, but also necessarily need to reveal enough information about the inputs.

Mean-based reconstruction is a fundamental, natural approach to worst-case trace reconstruction over channels with synchronization errors. It is known that$\exp (\Theta (n^{1/3}))$traces are necessary and sufficient for mean-based worst-case trace reconstruction over the deletion channel, and this result was also extended to certain channels combining deletions and geometric insertions of uniformly random bits. In this work, we use a simple extension of the original complex-analytic approach to show that these results are examples of a much more general phenomenon. We introduceoblivious synchronization channels, which map each input bit to an arbitrarily distributed sequence of replications and insertions of random bits. This general class captures all previously considered synchronization channels. We show that for any oblivious synchronization channel whose output length follows a sub-exponential distribution either mean-based trace reconstruction is impossible or$\exp (O(n^{1/3}))$traces suffice for this task.

Mean-based reconstruction is a fundamental, natural approach to worst-case trace reconstruction over channels with synchronization errors. It is known that$\exp(O(n^{1/3}))$traces are necessary and sufficient for mean-based worst-case trace reconstruction over the deletion channel, and this result was also extended to certain channels combining deletions and geometric insertions of uniformly random bits. In this work, we use a simple extension of the original complex-analytic approach to show that these results are examples of a much more general phenomenon:$\exp(O(n^{1/3}))$traces suffice for mean-based worst-case trace reconstruction over any memoryless channel that maps each input bit to an arbitrarily distributed sequence of replications and insertions of random bits, provided the length of this sequence follows a sub-exponential distribution.

Synchronization channels, such as the well-known deletion channel, are surprisingly harder to analyze than memoryless channels, and they are a source of many fundamental problems in information theory and theoretical computer science. One of the most basic open problems regarding synchronization channels is the derivation of an exact expression for their capacity. Unfortunately, most of the classic information-theoretic techniques at our disposal fail spectacularly when applied to synchronization channels. Therefore, new approaches must be considered to tackle this problem. This survey gives an account of the great effort made over the past few decades to better understand the (broadly defined) capacity of synchronization channels, including both the main results and the novel techniques underlying them. Besides the usual notion of channel capacity, we also discuss the zero-error capacity of adversarial synchronization channels.

We revisit the fundamental problem of determining seed length lower bounds for strong extractors and natural variants thereof. These variants stem from a "change in quantifiers" over the seeds of the extractor: While a strong extractor requires that the average output bias (over all seeds) is small for all input sources with sufficient min-entropy, a somewhere extractor only requires that there exists a seed whose output bias is small. More generally, we study what we call probable extractors, which on input a source with sufficient min-entropy guarantee that a large enough fraction of seeds have small enough associated output bias. Such extractors have played a key role in many constructions of pseudorandom objects, though they are often defined implicitly and have not been studied extensively. Prior known techniques fail to yield good seed length lower bounds when applied to the variants above. Our novel approach yields significantly improved lower bounds for somewhere and probable extractors. To complement this, we construct a somewhere extractor that implies our lower bound for such functions is tight in the high min-entropy regime. Surprisingly, this means that a random function is far from an optimal somewhere extractor in this regime. The techniques that we develop also yield an alternative, simpler proof of the celebrated optimal lower bound for strong extractors originally due to Radhakrishnan and Ta-Shma (SIAM J. Discrete Math., 2000).

Motivated by applications in topological DNA-based data storage, we introduce and study a novel setting of Non-Adaptive Group Testing (NAGT) with runlength constraints on the columns of the test matrix, in the sense that any two 1's must be separated by a run of at least d 0's. We describe and analyze a probabilistic construction of a runlength-constrained scheme in the zero-error and vanishing error settings, and show that the number of tests required by this construction is optimal up to logarithmic factors in the runlength constraint d and the number of defectives k in both cases. Our results reveal that runlength-constrained NAGT is not more restrictive than unconstrained NAGT when d = O(k), and that for almost all choices of d and k it is not more restrictive than NAGT with a column Hamming weight constraint only.

Construction of error-correcting codes achieving a designated minimum distance parameter is a central problem in coding theory. A classical and algebraic family of error-correcting codes studied for this purpose are the BCH codes. In this work, we study a very simple construction of linear codes that achieve a given distance parameter K. Moreover, we design a simple, nearly optimal syndrome decoder for the code as well. The running time of the decoder is only logarithmic in the block length of the code, and nearly linear in the distance parameter K. This decoder can be applied to exact for-all sparse recovery over any field, improving upon previous results with the same number of measurements. Furthermore, computation of the syndrome from a received word can be done in nearly linear time in the block length. We also demonstrate an application of these techniques in non-adaptive group testing, and construct simple explicit measurement schemes with O(K2log2N) tests and O(K3log2N) recovery time for identifying up to K defectives in a population of size N.

Motivated by average-case trace reconstruction and coding for portable DNA-based storage systems, we initiate the study of coded trace reconstruction, the design and analysis of high-rate efficiently encodable codes that can be efficiently decoded with high probability from few reads (also called traces) corrupted by edit errors. Codes used in current portable DNA-based storage systems with nanopore sequencers are largely based on heuristics, and have no provable robustness or performance guarantees even for an error model with i.i. d. deletions and constant deletion probability. Our work is a first step towards the design of efficient codes with provable guarantees for such systems. We consider a constant rate of i.i. d. deletions, and begin by analyzing marker-based code-constructions coupled with worst-case trace reconstruction algorithms. Then, we show how a more careful design of the code allows us to exploit ideas from average-case trace reconstruction to reduce the number of traces required with the same redundancy.

New capacity upper bounds are presented for the discrete-time Poisson channel with no dark current and an average-power constraint. These bounds are a consequence of techniques developed for the seemingly unrelated problem of upper bounding the capacity of binary deletion and repetition channels. Previously, the best known capacity upper bound in the regime where the average-power constraint does not approach zero was due to Martinez (JOSA B, 2007), which is re-derived as a special case of the framework developed in this paper. Furthermore, this framework is carefully instantiated in order to obtain a closed-form bound that improves the result of Martinez everywhere. Finally, capacity-achieving distributions for the discrete-time Poisson channel are studied under an average-power constraint and/or a peak-power constraint and arbitrary dark current. In particular, it is shown that the support of the capacity-achieving distribution under an average-power constraint must only be countably infinite. This settles a conjecture of Shamai (IEE Proceedings I, 1990) in the affirmative. Previously, it was only known that the support must be an unbounded set.

We present new capacity upper bounds for the discrete-time Poisson channel with no dark current and an average-power constraint. These bounds are a simple consequence of techniques developed by one of the authors for the seemingly unrelated problem of upper bounding the capacity of binary deletion and repetition channels. Previously, the best known capacity upper bound in the regime where the average-power constraint does not approach zero was due to Martinez (JOSA B, 2007), which we re-derive as a special case of our framework. Furthermore, we instantiate our framework to obtain a closed-form bound that noticeably improves the result of Martinez everywhere.

In this paper we provide a generalization of weak oblivious transfer through the constructive cryptography framework. This generalization requires the global order of the inputs and outputs from and to two parties called Alice and Bob to be completely defined, a subtlety which has been overlooked by previous work on the subject. We provide evidence that the order of inputs and outputs in weak oblivious transfer matters. In particular, it may influence the kind and strength of symmetry results which can be obtained about such resources.