Sensitivity Conjecture and Log-rank Conjecture for functions with small alternating numbers

The Sensitivity Conjecture and the Log-rank Conjecture are among the most important and challenging problems in concrete complexity. Incidentally, the Sensitivity Conjecture is known to hold for monotone functions, and so is the Log-rank Conjecture for $f(x \wedge y)$ and $f(x\oplus y)$ with monotone functions $f$, where $\wedge$ and $\oplus$ are bit-wise AND and XOR, respectively. In this paper, we extend these results to functions $f$ which alternate values for a relatively small number of times on any monotone path from $0^n$ to $1^n$. These deepen our understandings of the two conjectures, and contribute to the recent line of research on functions with small alternating numbers

A Simple FPTAS for Counting Edge Covers

An edge cover of a graph is a set of edges such that every vertex has at least an adjacent edge in it. Previously, approximation algorithm for counting edge covers is only known for 3 regular graphs and it is randomized. We design a very simple deterministic fully polynomial-time approximation scheme (FPTAS) for counting the number of edge covers for any graph. Our main technique is correlation decay, which is a powerful tool to design FPTAS for counting problems. In order to get FPTAS for general graphs without degree bound, we make use of a stronger notion called computationally efficient correlation decay, which is introduced in [Li, Lu, Yin SODA 2012].Comment: To appear in SODA 201

Multi-messenger Study of Galactic Diffuse Emission with LHAASO and IceCube Observations

With the breakthrough in PeV gamma-ray astronomy brought by the LHAASO experiment, the high-energy sky is getting richer than before. Lately, LHAASO Collaboration reported the observation of a gamma-ray diffuse emission with energy up to the PeV level from both the inner and outer Galactic plane. In these spectra, there is one bump that is hard to explain by the conventional cosmic-ray transport scenarios. Therefore, we introduce two extra components corresponding to unresolved sources with exponential-cutoff-power-law (ECPL) spectral shape, one with an index of 2.4, and 20 TeV cutoff energy, and another with index of 2.3 and 2 PeV cutoff energy. With our constructed model, we simulate the Galactic diffuse neutrino flux and find our results are in full agreement with the latest IceCube Galactic plane search. We estimate the Galactic neutrino contributes of $\sim 9\%$ of astrophysical neutrinos at 20 TeV. In the high-energy regime, as expected most of the neutrinos observed by IceCube should be from extragalactic environments.Comment: 10 pages, 8 figures, comments are welcome, accepted by PR

Dynamic Pricing for Air Cargo Revenue Management

We address a dynamic pricing problem for airlines aiming to maximize expected revenue from selling cargo space on a single-leg flight. The cargo shipments' weight and volume are uncertain and their precise values remain unavailable at the booking time. We model this problem as a Markov decision process, and further derive a necessary condition for its optimal pricing strategy. To break the curse of dimensionality, we develop two categories of approximation methods and pricing strategies. One category is based on the quantity of accepted bookings, while the other is founded on the expected weight and volume of accepted bookings. We prove that the pricing strategy of the quantity-based method possesses several inherent structural properties, which are crucial for analytically validating the model and accelerating the computational process. For the weight-volume-based approximation method, we derive a theoretical upper bound for the optimality gap of total expected revenue. For both methods, we further develop augmented strategies to address the extreme pricing issues in scenarios with high product heterogeneity and incorporate the second moment to enhance performance in the scenarios of high uncertainty, respectively. We utilize realistic dataset to conduct extensive numerical tests, and the results show that the average performance gap between the optimal expected revenue and that of each proposed pricing strategy is less than 10%. The quantity-based method requires the least computation, and performs quite well in the scenarios with low product heterogeneity. The augmented quantity-based method and the weight-volume-based method further enhance the resilience to product heterogeneity. The augmented weight-volume-based method significantly improves the revenue when there are high penalties for overbooking and high uncertainty

Ring-LWE: Enhanced Foundations and Applications

Ring Learning With Errors assumption has become an important building block in many modern cryptographic applications, such as (fully) homomorphic encryption and post-quantum cryptosystems like the recently announced NIST CRYSTALS-Kyber public key encryption scheme. In this thesis, we provide an enhanced security foundation for Ring-LWE based cryptosystems and demonstrate their practical potential in real world applications. Enhanced Foundation. We extend the known pseudorandomness of Ring-LWE to be based on ideal lattices of non Dedekind domains. In earlier works of Lyubashevsky, Perkert and Regev (EUROCRYPT 2010), and Peikert, Regev and Stephens-Davidowitz (STOC 2017), the hardness of RLWE was established on ideal lattices of ring of integers of number fields, which are known to be Dedekind domains. These works extended Regev's (STOC 2005) quantum polynomial-time reduction for LWE, thus allowing more efficient and more structured cryptosystems. However, the additional algebraic structure of ideals of Dedekind domains leaves open the possibility that such ideal lattices are not as hard as general lattices. We show that, the Ring-LWE hardness can be based on the polynomial ring, which is potentially be a strict sub-ring of the ring of integers of a number field, and hence not be a Dedekind domain. We present a novel proof technique that builds an algebraic theory for general such rings that also include cyclotomic rings. We also recommend a ``twisted'' cyclotomic field as an alternative for the cyclotomic field used in CRYSTALS-Kyber, as it leads to a more efficient implementation and is based on hardness of ideals in a non Dedekind domain. We leverages the polynomial nature of Ring-LWE, and introduce XSPIR, a new symmetrically private information retrieval (SPIR) protocol, which provides a stronger security guarantee than existing efficient PIR protocols. Like other PIR protocol, XSPIR allows a client to retrieve a specific entry from a server's database without revealing which entry is retrieved. Moreover, the semi-honest client learns no additional information about the database except for the retrieved entry. We demonstrate through analyses and experiments that XSPIR has only a slight overhead compared to state-of-the-art PIR protocols, and provides a stronger security guarantee while enabling the client to perform more complicated queries than simple retrievals

Time delay estimation in the ultrasonic flowmeter in the oil well

AbstractA new prototype of ultrasonic flowmeter used in the oil well is presented. The flowmeter depends on the time delay between the propagating times of the downstream and upstream ultrasonic pulses. The ultrasonic passageway is slanted to prevent the disadvantage introduced by the high viscosity of the oil. Two method of time delay estimation: threshold and cross-correlation are both studied and realized

Ring-LWE Hardness Based on Non-invertible Ideals

We extend the known pseudorandomness of Ring-LWE to be based on lattices that do not correspond to any ideal of any order in the underlying number field. In earlier works of Lyubashevsky et al (EUROCRYPT 2010) and Peikert et al (STOC 2017), the hardness of RLWE was based on ideal lattices of ring of integers of number fields, which are known to be Dedekind domains. While these works extended Regev\u27s (STOC 2005) quantum polynomial-time reduction for LWE, thus allowing more efficient and more structured cryptosystems, the additional algebraic structure of ideals of Dedekind domains leaves open the possibility that such ideal lattices are not as hard as general lattices. In this work we show that hardness of $q$-Ring-LWE can be based on worst-case hardness of ideal lattices in arbitrary orders $O$, as long as the order $O$ satisfies the property that $\frac{1}{m}\cdot O$ contains the ring of integers, for some $m$ co-prime to $q$. The reduction requires that the noise be a factor $m$ more than the original Ring-LWE reduction. We also show that for the power-of-two cyclotomic number fields, there exist orders with $m=4$ such that non-trivial ideals of the order, which are not contained in the conductor, are non-invertible. Since the conductor itself is non-invertible, this gives a non-trivial multiplicative set that lies outside the ideal class group. Another reduction shows that hardness of $q$-Ring-LWE can be based on worst-case hardness of lattices that correspond to sum of ideal-lattices in arbitrary and different orders in the number field, as long as the (set of) orders $\{O_i\}$ satisfy the property that $\frac{1}{m}\cdot O_i$ contains the ring of integers, for some $m$ co-prime to $q$. We also show that for the power-of-two cyclotomic number fields, there exist orders $O_1, O_2$ with $m=8$ such that there are ideals $I_1, I_2$ of $O_1, O_2$ resp. with $I_1+ I_2$ not an ideal of any order in the number field

Enhancing Ring-LWE Hardness using Dedekind Index Theorem

In this work we extend the known pseudorandomness of Ring-LWE (RLWE) to be based on ideal lattices of non Dedekind domains. In earlier works of Lyubashevsky et al (EUROCRYPT 2010) and Peikert et al (STOC 2017), the hardness of RLWE was based on ideal lattices of ring of integers of number fields, which are known to be Dedekind domains. While these works extended Regev\u27s (STOC 2005) quantum polynomial-time reduction for LWE, thus allowing more efficient and more structured cryptosystems, the additional algebraic structure of ideals of Dedekind domains leaves open the possibility that such ideal lattices are not as hard as general lattices. To mitigate this issue, Bolboceanu et al (Asiacrypt 2019) defined q-Order-LWE over any order (modulo q) in a number field and based its hardness on worst-case hard problems of ideal lattices of the same order, but restricted to invertible ideals. Orders generalize the ring of integers to non-Dedekind domains. In a subsequent work in 2021, they proved a non-effective ``ideal-clearing lemma for q-Order-LWE for any q that is co-prime to index of the order in the ring of integers. This work can be shown to give an efficient reduction from any ideal of the same order. However, this requires factorization of arbitrary integers, namely the norm of the given ideal. In this work we give a novel approach to proving the ``ideal-clearing lemma for q-Order-LWE by showing that all ideals I of an order are principal modulo qI, for any q that is co-prime to index of the order in the ring of integers. Further, we give a rather simple (classical) randomized algorithm to find a generator for this principal ideal, which makes our hardness reduction (from all ideals of the order) not require any further quantum steps on top of the quantum Gaussian sampling of the original Regev reduction. This also removes the ``known factorization requirement on q for the original RLWE hardness result of Peikert et al. Finally, we recommend a ``twisted\u27\u27 cyclotomic field as an alternative for the cyclotomic field used in NIST PQC algorithm CRYSTALS-Kyber, as it leads to a more efficient implementation and is based on hardness of ideals in a non-Dedekind domain following Dedekind index theorem