Optimal Bounds for the k -cut Problem

In the k -cut problem, we want to find the lowest-weight set of edges whose deletion breaks a given (multi)graph into k connected components. Algorithms of Karger and Stein can solve this in roughly O ( n 2k ) time. However, lower bounds from conjectures about the k -clique problem imply that Ω ( n (1- o (1)) k ) time is likely needed. Recent results of Gupta, Lee, and Li have given new algorithms for general k -cut in n 1.98k + O(1) time, as well as specialized algorithms with better performance for certain classes of graphs (e.g., for small integer edge weights). In this work, we resolve the problem for general graphs. We show that the Contraction Algorithm of Karger outputs any fixed k -cut of weight α λ k with probability Ω k ( n - α k ), where λ k denotes the minimum k -cut weight. This also gives an extremal bound of O k ( n k ) on the number of minimum k -cuts and an algorithm to compute λ k with roughly n k polylog( n ) runtime. Both are tight up to lower-order factors, with the algorithmic lower bound assuming hardness of max-weight k -clique. The first main ingredient in our result is an extremal bound on the number of cuts of weight less than 2 λ k / k , using the Sunflower lemma. The second ingredient is a fine-grained analysis of how the graph shrinks—and how the average degree evolves—in the Karger process.

Provably Secure Lattice-Based Self-Certified Signature Scheme

Digital signatures are crucial network security technologies. However, in traditional public key signature schemes, the certificate management is complicated and the schemes are vulnerable to public key replacement attacks. In order to solve the problems, in this paper, we propose a self-certified signature scheme over lattice. Using the self-certified public key, our scheme allows a user to certify the public key without an extra certificate. It can reduce the communication overhead and computational cost of the signature scheme. Moreover, the lattice helps prevent quantum computing attacks. Then, based on the small integer solution problem, our scheme is provable secure in the random oracle model. Furthermore, compared with the previous self-certified signature schemes, our scheme is more secure.

The Development of Rhythmic Categories as Revealed Through an Iterative Production Task

Both humans and some non-human animals (e.g., birds and primates) demonstrate bias toward simple integer ratios in auditory rhythms. In humans, biases are found for small integer-ratio rhythms in general. In addition, there are biases for the specific small integer-ratio rhythms common to one’s cultural listening experience. To better understand the developmental trajectory of these biases, we estimated children’s rhythm priors across the entire human rhythm production space of simple rhythms. North American children aged 6-11 years completed an iterative rhythm production task, in which they tapped in synchrony with repeating three-interval rhythms. For each rhythm, the child’s produced rhythm was presented back to them as the stimulus, and over the course of 5 iterations we used their final reproductions to estimate their rhythmic biases or priors. Results suggest that children’s rhythmic priors are (nearly) integer ratios, and the relative weights of the categories observed in children are highly correlated with those of adults. However, we also observed age-related changes especially for the ratio types that vary most across cultures. In an additional rhythm perception task, children were better at detecting rhythmic disruptions to a culturally familiar rhythm (in 4/4 meter with 2:1:1 ratio pattern) than to a culturally unfamiliar rhythm (7/8 meter with 3:2:2 ratios), and performance in this task was correlated with tapping variability in the iterative task. Taken together, our findings provide evidence that children as young as 6 years old exhibit categorical rhythm priors in their rhythm production that closely resemble those of adults in the same culture.

Non-isochronous Metre in Music from Mali

The basic building blocks for rhythmic structure in music are widely believed to be universally confined to small-integer ratios. In particular, basic metric processes such as pulse perception are assumed to depend on the recognition and anticipation of even, categorically equivalent durations or inter-onset intervals, which are related by the ratio of 1:1 (isochrony). Correspondingly, uneven (non-isochronous) beat subdivisions are theorized as instances of expressive microtiming variation, i.e. as performance deviations from some underlying, categorically isochronous temporal structure. By contrast, ethnographic experience suggests that the periodic patterns of uneven beat subdivision timing in various styles of music from Mali themselves constitute rhythmic and metric structures. The present chapter elaborates this hypothesis and surveys a series of empirical research projects that have found evidence for it. These findings have implications for metric theory as well as for our broader understanding of how human perception relates to cultural environments. They suggest that the bias towards isochrony, which according to many accounts of rhythm and metre underlies pulse perception, is culturally specific rather than universal. Claims regarding cultural diversity in the study of music typically concern styles and meanings of performance practices. In this chapter, I will claim that basic structures of perception can vary across cultural groups too.

Identity-Based Linkable Ring Signature on NTRU Lattice

Although most existing linkable ring signature schemes on lattice can effectively resist quantum attacks, they still have the disadvantages of excessive time and storage overhead. This paper constructs an identity-based linkable ring signature (LRS) scheme over NTRU lattice by employing the technologies of trapdoor generation and rejection sampling. The security of this scheme relies on the small integer solution (SIS) problem on NTRU lattice. We prove that this scheme has unconditional anonymity, unforgeability, and linkability under the random oracle model (ROM). Through the performance analysis, this scheme has a shorter size of public/private keys, and when the number of ring members is small (such as N ≤ 8 ), this scheme has a shorter signature size compared with other existing latest lattice-based LRS schemes. The computational efficiency of signature has also been further improved since it only involves multiplication in the polynomial ring and modular operations of small integers. Finally, we implemented our scheme and other similar schemes, and it is shown that the time for the signature generation and verification of this scheme decreases roughly by 44.951% and 33.503%, respectively.

Cross-frequency coupling explains the preference for simple ratios in rhythmic behaviour and the relative stability across non-synchronous patterns

Rhythms are important for understanding coordinated behaviours in ecological systems. The repetitive nature of rhythms affords prediction, planning of movements and coordination of processes within and between individuals. A major challenge is to understand complex forms of coordination when they differ from complete synchronization. By expressing phase as ratio of a cycle, we adapted levels of the Farey tree as a metric of complexity mapped to the range between in-phase and anti-phase synchronization. In a bimanual tapping task, this revealed an increase of variability with ratio complexity, a range of hidden and unstable yet measurable modes, and a rank-frequency scaling law across these modes. We use the phase-attractive circle map to propose an interpretation of these findings in terms of hierarchical cross-frequency coupling (CFC). We also consider the tendency for small-integer attractors in the single-hand repeated tapping of three-interval rhythms reported in the literature. The phase-attractive circle map has wider basins of attractions for such ratios. This work motivates the question whether CFC intrinsic to neural dynamics implements low-level priors for timing and coordination and thus becomes involved in phenomena as diverse as attractor states in bimanual coordination and the cross-cultural tendency for musical rhythms to have simple interval ratios. This article is part of the theme issue ‘Synchrony and rhythm interaction: from the brain to behavioural ecology’.

Privacy Preserving Classification of EEG Data Using Machine Learning and Homomorphic Encryption

Data privacy is a major concern when accessing and processing sensitive medical data. A promising approach among privacy-preserving techniques is homomorphic encryption (HE), which allows for computations to be performed on encrypted data. Currently, HE still faces practical limitations related to high computational complexity, noise accumulation, and sole applicability the at bit or small integer values level. We propose herein an encoding method that enables typical HE schemes to operate on real-valued numbers of arbitrary precision and size. The approach is evaluated on two real-world scenarios relying on EEG signals: seizure detection and prediction of predisposition to alcoholism. A supervised machine learning-based approach is formulated, and training is performed using a direct (non-iterative) fitting method that requires a fixed and deterministic number of steps. Experiments on synthetic data of varying size and complexity are performed to determine the impact on runtime and error accumulation. The computational time for training the models increases but remains manageable, while the inference time remains in the order of milliseconds. The prediction performance of the models operating on encoded and encrypted data is comparable to that of standard models operating on plaintext data.

Lattice-Based Logarithmic-Size Non-Interactive Deniable Ring Signatures

Deniable ring signature can be regarded as group signature without group manager, in which a singer is capable of singing a message anonymously, but, if necessary, each ring member is allowed to confirm or disavowal its involvement in the signature via an interactive mechanism between the ring member and the verifier. This attractive feature makes the deniable ring signature find many applications in the real world. In this work, we propose an efficient scheme with signature size logarithmic to the cardinality of the ring. From a high level, we adapt Libert et al.’s zero-knowledge argument system (Eurocrypt 2016) to allow the prover to convince the verifier that its witness satisfies an additional condition. Then, using the Fait-Shamir transformation, we get a non-interactive deniable ring signature scheme that satisfies the anonymity, traceability, and non-frameability under the small integer solution assumption in the random oracle model.

Universality and cross-cultural variation in mental representations of music revealed by global comparison of rhythm priors

Music is present in every known society, yet varies from place to place. What is universal to the perception of music? We measured a signature of mental representations of rhythm in 923 participants from 39 participant groups in 15 countries across 5 continents, spanning urban societies, indigenous populations, and online participants. Listeners reproduced random ‘‘seed’’ rhythms; their reproductions were fed back as the stimulus (as in the game of “telephone”), such that their biases (the prior) could be estimated from the distribution of reproductions. Every tested group showed a prior with peaks at integer ratio rhythms, suggesting that discrete rhythm “categories” at small integer ratios are universal. The occurrence and relative importance of different integer ratio categories varied across groups, often reflecting local musical systems. However, university students and online participants in non-Western countries tended to resemble Western participants, underrepresenting the variability otherwise evident across cultures. The results suggest the universality of discrete mental representations of music while showing their interaction with culture-specific traditions.

Fuzzy Identity-Based Ring Signature from Lattices

In this paper, a construction of a fuzzy identity-based ring signature scheme (LFIBRS) is proposed. Our LFIBRS combines the characteristics of both the fuzzy identity-based signature (FIBS) and the ring signature. On the one hand, a signature issued under an identity ID can be verified by any identity ID ′ that is “close enough” to the identity ID . Since biometric identification is the well-known most popular and reliable identification method, our LFIBRS can be applied in such a situation whenever it is required for official audit or supervision that the signer’s real identity is needed to be authenticated. On the other hand, LFIBRS provides anonymity under the random oracle model. In addition, LFIBRS provides unforgeability under the small integer solution (SIS) lattice hardness assumption which can resist large-scale quantum computer attacks in the future.