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Image encryption is an effective way to protect image data. However, existing image encryption algorithms are still unable to strike a good balance between security and efficiency. To overcome the shortcomings of these algorithms, an image encryption algorithm based on plane-level image filtering and discrete logarithmic transformation (IEA-IF-DLT) is proposed. By utilizing the hash value more rationally, our proposed IEA-IF-DLT avoids the overhead caused by repeated generations of chaotic sequences and further improves the encryption efficiency through plane-level and three-dimensional (3D) encryption operations. Aiming at the problem that common modular addition and XOR operations are subject to differential attacks, IEA-IF-DLT additionally includes discrete logarithmic transformation to boost security. In IEA-IF-DLT, the plain image is first transformed into a 3D image, and then three rounds of plane-level permutation, plane-level pixel filtering, and 3D chaotic image superposition are performed. Next, after a discrete logarithmic transformation, a random pixel swapping is conducted to obtain the cipher image. To demonstrate the superiority of IEA-IF-DLT, we compared it with some state-of-the-art algorithms. The test and analysis results show that IEA-IF-DLT not only has better security performance, but also exhibits significant efficiency advantages.

We describe a new Schnorr-based multi-signature scheme (i.e., a protocol which allows a group of signers to produce a short, joint signature on a common message) called MuSig , provably secure under the Discrete Logarithm assumption and in the plain public-key model (meaning that signers are only required to have a public key, but do not have to prove knowledge of the private key corresponding to their public key to some certification authority or to other signers before engaging the protocol). MuSig improves over the state-of-art scheme of Bellare and Neven (ACM Conference on Computer and Communications Security-CCS 2006) and its variants by Bagherzandi et al. (ACM Conference on Computer and Communications Security-CCS 2008) and Ma et al. (Des Codes Cryptogr 54(2):121–133, 2010) in two respects: (i) it is simple and efficient, having the same key and signature size as standard Schnorr signatures; (ii) it allows key aggregation, which informally means that the joint signature can be verified exactly as a standard Schnorr signature with respect to a single “aggregated” public key which can be computed from the individual public keys of the signers. To the best of our knowledge, this is the first multi-signature scheme provably secure under the Discrete Logarithm assumption in the plain public-key model which allows key aggregation. As an application, we explain how our new multi-signature scheme could improve both performance and user privacy in Bitcoin.

A number of important observables exhibit logarithms in their perturbative description that are induced by emissions at widely separated rapidities. These include transverse-momentum (qT) logarithms, logarithms involving heavy-quark or electroweak gauge boson masses, and small-x logarithms. In this paper, we initiate the study of rapidity logarithms, and the associated rapidity divergences, at subleading order in the power expansion. This is accomplished using the soft collinear effective theory (SCET). We discuss the structure of subleading-power rapidity divergences and how to consistently regulate them. We introduce a new pure rapidity regulator and a corresponding MS¯\\[ MS \\]-like scheme, which handles rapidity divergences while maintaining the homogeneity of the power expansion. We find that power-law rapidity divergences appear at subleading power, which give rise to derivatives of parton distribution functions. As a concrete example, we consider the qT spectrum for color-singlet production, for which we compute the complete qT2/Q2 suppressed power corrections at Oαs\\[ O(_s) \\], including both logarithmic and nonlogarithmic terms. Our results also represent an important first step towards carrying out a resummation of subleading-power rapidity logarithms.

In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin). We establish the following results: The key tools behind our results are limiting interpolation techniques and new characterizations of Besov and Sobolev norms in terms of the behavior of the Fourier transforms for functions such that their Fourier transforms are of monotone type or lacunary series.

Given any line in the plane, we strengthen the Littlewood conjecture by two logarithms for almost every point on the line, thereby generalising the fibre result of Beresnevich, Haynes, and Velani. To achieve this, we prove an effective asymptotic equidistribution result for one-parameter unipotent orbits in SL(3,R)/SL(3,Z). We also provide a complementary convergence statement, by developing the structural theory of dual Bohr sets: at the cost of a slightly stronger Diophantine assumption, this sharpens a result of Kleinbock’s from 2003. Finally, we refine the theory of logarithm laws in homogeneous spaces.

We study the detection of a sparse change in a high-dimensional mean vector as a minimax testing problem. Our first main contribution is to derive the exact minimax testing rate across all parameter regimes for n independent, p-variate Gaussian observations. This rate exhibits a phase transition when the sparsity level is of order √plog log(8n) and has a very delicate dependence on the sample size: in a certain sparsity regime, it involves a triple iterated logarithmic factor in n. Further, in a dense asymptotic regime, we identify the sharp leading constant, while in the corresponding sparse asymptotic regime, this constant is determined to within a factor of √2. Extensions that cover spatial and temporal dependence, primarily in the dense case, are also provided.