Quantum Algorithms for Boolean Equation Solving and Quantum Algebraic Attack on Cryptosystems

Decision of whether a Boolean equation system has a solution is an NPC problem and finding a solution is NP hard. In this paper, we present a quantum algorithm to decide whether a Boolean equation system FS has a solution and compute one if FS does have solutions with any given success probability. The runtime complexity of the algorithm is polynomial in the size of FS and the condition number of FS. As a consequence, we give a polynomial-time quantum algorithm for solving Boolean equation systems if their condition numbers are small, say polynomial in the size of FS. We apply our quantum algorithm for solving Boolean equations to the cryptanalysis of several important cryptosystems: the stream cipher Trivum, the block cipher AES, the hash function SHA-3/Keccak, and the multivariate public key cryptosystems, and show that they are secure under quantum algebraic attack only if the condition numbers of the corresponding equation systems are large. This leads to a new criterion for designing cryptosystems that can against the attack of quantum computers: their corresponding equation systems must have large condition numbers

Is admiration a source of indeterminacy when the speed of habit formation is finite?

In an economy where the time preference rate is sufficiently decreasing in individual consumption, Chen and Hsu (2007) find that a consumption admiration effect can be a source of local indeterminacy, whereby average consumption flows exert a positive external effect on an individual's utility. In our paper, average consumption habits externally increase an individual's utility. The increase in average consumption habits is the difference between average consumption flows and existing average consumption habits adjusted for by the speed of the consumption habit formation. The model in Chen and Hsu (2007) is a special case that emerges only when the speed of habit formation is infinite. In our general model, an admiration effect is no longer a source of equilibrium indeterminacy unless the speed of consumption habit formation is infinite.Neoclassical growth model, consumption habit externalities, indeterminacy

From patterned response dependency to structured covariate dependency: categorical-pattern-matching

Data generated from a system of interest typically consists of measurements from an ensemble of subjects across multiple response and covariate features, and is naturally represented by one response-matrix against one covariate-matrix. Likely each of these two matrices simultaneously embraces heterogeneous data types: continuous, discrete and categorical. Here a matrix is used as a practical platform to ideally keep hidden dependency among/between subjects and features intact on its lattice. Response and covariate dependency is individually computed and expressed through mutliscale blocks via a newly developed computing paradigm named Data Mechanics. We propose a categorical pattern matching approach to establish causal linkages in a form of information flows from patterned response dependency to structured covariate dependency. The strength of an information flow is evaluated by applying the combinatorial information theory. This unified platform for system knowledge discovery is illustrated through five data sets. In each illustrative case, an information flow is demonstrated as an organization of discovered knowledge loci via emergent visible and readable heterogeneity. This unified approach fundamentally resolves many long standing issues, including statistical modeling, multiple response, renormalization and feature selections, in data analysis, but without involving man-made structures and distribution assumptions. The results reported here enhance the idea that linking patterns of response dependency to structures of covariate dependency is the true philosophical foundation underlying data-driven computing and learning in sciences.Comment: 32 pages, 10 figures, 3 box picture

The Minimal GUT with Inflaton and Dark Matter Unification

Giving up the solutions to the fine-tuning problems, we propose the non-supersymmetric flipped $SU(5)\times U(1)_X$ model based on the minimal particle content principle, which can be constructed from the four-dimensional $SO(10)$ models, five-dimensional orbifold $SO(10)$ models, and local F-theory $SO(10)$ models. To achieve gauge coupling unification, we introduce one pair of vector-like fermions, which form complete $SU(5)\times U(1)_X$ representation. Proton lifetime is around $5\times 10^{35}$ years, neutrino masses and mixing can be explained via seesaw mechanism, baryon asymmetry can be generated via leptogenesis, and vacuum stability problem can be solved as well. In particular, we propose that inflaton and dark matter particle can be unified to a real scalar field with $Z_2$ symmetry, which is not an axion and does not have the non-minimal coupling to gravity. Such kind of scenarios can be applied to the generic scalar dark matter models. Also, we find that the vector-like particle corrections to the $B_s^0$ masses can be about 6.6%, while their corrections to the $K^0$ and $B_d^0$ masses are negligible.Comment: 5 pages, 4 figures;V2: published versio

Out-of-distribution forgetting: vulnerability of continual learning to intra-class distribution shift

Continual learning (CL) is an important technique to allow artificial neural networks to work in open environments. CL enables a system to learn new tasks without severe interference to its performance on old tasks, i.e., overcome the problems of catastrophic forgetting. In joint learning, it is well known that the out-of-distribution (OOD) problem caused by intentional attacks or environmental perturbations will severely impair the ability of networks to generalize. In this work, we reported a special form of catastrophic forgetting raised by the OOD problem in continual learning settings, and we named it out-of-distribution forgetting (OODF). In continual image classification tasks, we found that for a given category, introducing an intra-class distribution shift significantly impaired the recognition accuracy of CL methods for that category during subsequent learning. Interestingly, this phenomenon is special for CL as the same level of distribution shift had only negligible effects in the joint learning scenario. We verified that CL methods without dedicating subnetworks for individual tasks are all vulnerable to OODF. Moreover, OODF does not depend on any specific way of shifting the distribution, suggesting it is a risk for CL in a wide range of circumstances. Taken together, our work identified an under-attended risk during CL, highlighting the importance of developing approaches that can overcome OODF