Privacy attacks and poisoning attacks are two of the thorniest problems in federation learning (FL). Homomorphic encryption (HE), which allows certain mathematical operations to be done in the ciphertext state, provides a way to solve these two problems simultaneously. However, existing Paillier-based and CKKS-based privacy-preserving byzantine-robust FL (PBFL) solutions not only suffer from low efficiency but also expose the final model to the server. Additionally, these methods are limited to one robust aggregation algorithm (AGR) and are therefore vulnerable to AGR-tailored poisoning attacks. In this paper, we present AegisFL, an efficient PBLF system that provides the flexibility to change the AGR. We first observe that the core of the existing advanced AGRs is to calculate the inner products, $L_2$ norms and mean values for vectors. Based on this observation, we tailor a packing scheme for PBFL, which fits perfectly with RLWE-based fully homomorphic encryption. Under this packing scheme, the server only needs to perform one ciphertext multiplication to construct any required AGR, while the global model only belongs to honest clients. Finally, we conduct extensive experiments on different datasets and adversary settings, which also confirm the effectiveness and efficiency of our scheme.

This letter addresses some security issues of an identity-based provable data possession protocol with compressed cloud storage (published in IEEE TIFS, doi:10.1109/TIFS.2022. 3159152). Some serious flaws are identified and an attack to the protocol is designed. This attack is able to recover the ephemeral secret keys from two encrypted blocks with high probability to reveal the original plaintext file completely. Moreover, an adversary can impersonate a data owner to outsource any file to the cloud in a malicious way. The main ingredients of the attack is some classical number theoretic results.

Given a binary sequence with length$n$, determining its minimal rational fraction representation (MRFR) has important applications in the design and cryptanalysis of stream ciphers. There are many studies of this problem since Klapper and Goresky first introduced an adaptive rational approximation algorithm with a time complexity of$O(n^{2}\log n\log \log n)$. In this paper, we revisit this problem by considering both adaptive and non-adaptive efficient algorithms. Compared with the state-of-art methods, we make several contributions to the problem of finding MRFR. Firstly, we find a general and precise recursive relationship between the minimal bases for two adjacent lattices generated by successive truncation sequences. This enables us to improve the currently fastest adaptive algorithm proposed by Liet al.. Secondly, by optimizing a time-consuming step of the well-known Lagrange reduction algorithm for 2-dimensional lattices, we obtain a non-adaptive, and yet practically faster MRFR-solving algorithm namedglobalEuclidean algorithm. Thirdly, we identify theoretical flaws on some non-adaptive methods in the literature by counter-examples and correct the problems by designing modified Euclidean algorithm namedpartialEuclidean algorithm. Meanwhile, we further reduce the time complexity of existing algorithm from$O(n^{2})$to$O(n\log ^{2}n\log \log n)$by invoking the half-gcd algorithm. We also conduct a comprehensive experimental comparative analysis on the above algorithms to validate our theoretical analysis.

Koblitz curves have been an important subject of consideration for both theoretical andpractical interests. The window$\tau$-adic algorithm of Solinas (window$\tau$NAF) is the most powerful method for computing point multiplication for Koblitz curves.Pre-computation plays an important role in improving the performance of point multiplication. In this paper, the concept of optimal pre-computation for window$\tau$NAF is formulated. In this setting, an optimal pre-computation has some mathematically natural and clean forms, and requires$2^{w-2}-1$point additions and two evaluations of the Frobenius map$\tau$, where$w$is the window width. One of the main results of this paper is to construct an optimal pre-computation scheme for each window width$w$from$4$to$15$(more than practical needs).These pre-computations can be easily incorporated into implementations of window$\tau$NAF. The ideas in the paper can also be used to construct other suitable pre-computations. This paper includes a discussion of coefficient sets for window$\tau$NAF and the divisibility by powers of$\tau$through different approaches. Some issues of implementation are also discussed.

The class of Fourier matrices is of special importance in compressed sensing (CS). This paper concerns deterministic construction of CS matrices from Fourier matrices. Using Katz' character sum estimation, we are able to design a deterministic procedure to select rows from a Fourier matrix to form a good CS matrix for sparse recovery. The sparsity bound in our construction is similar to that of binary CS matrices constructed by DeVore, which greatly improves previous results for CS matrices from Fourier matrices. Our approach also provides more flexibility in terms of the dimension of CS matrices. This paper also contains a useful improvement to Katz' character sum estimation for quadratic extensions, with an elementary and transparent proof. Based on this improvement, we construct a class of special CS matrices consisting of partial Fourier matrices whose columns are a union of orthonormal bases. As a consequence, our construction yields an approximately mutually unbiased bases from Fourier matrices which is of particular interest to quantum information theory. Some numerical examples are also included.

This article considers sparse signal recovery in the presence of noise. A mutual incoherence condition which was previously used for exact recovery in the noiseless case is shown to be sufficient for stable recovery in the noisy case. Furthermore, the condition is proved to be sharp. A specific counterexample is given. In addition, an oracle inequality is derived under the mutual incoherence condition in the case of Gaussian noise.

This paper discusses new bounds for restricted isometry constants in compressed sensing. Let Φ be an n × p real matrix and A; be a positive integer with k ≤ n. One of the main results of this paper shows that if the restricted isometry constant δkof Φ satisfies δk1minimization when no noise is present and k-sparse signals can be estimated stably in the noisy case. It is also shown that the bound cannot be substantially improved. An explicit example is constructed in which δk= k-1/2k-1 <; 0.5, but it is impossible to recover certain k-sparse signals.

A Steiner star for a set P of n points in ℝd connects an arbitrary center point to all points of P, while a star connects a point p ∊ P to the remaining n − 1 points of P. All connections are realized by straight line segments. Fekete and Meijer showed that the minimum star is at most √2 times longer than the minimum Steiner star for any finite point configuration in ℝd. The maximum ratio between them, over all finite point configurations in ℝd, is called the star Steiner ratio in ℝd. It is conjectured that this ratio is 4/π = 1.2732… in the plane and 4/3 = 1.3333… in three dimensions. Here we give upper bounds of 1.3631 in the plane, and 1.3833 in 3-space, thereby substantially improving recent upper bounds of 1.3999, and √2 — 10−-4, respectively. Our results also imply improved bounds on the maximum ratios between the minimum star and the maximum matching in two and three dimensions. Our method exploits the connection with the classical problem of estimating the maximum sum of pairwise distances among n points on the unit sphere, first studied by László Fejes Tóth. It is quite general and yields the first non-trivial estimates below √2 on the star Steiner ratios in arbitrary dimensions. We show, however, that the star Steiner ratio in ℝd tends to √2, the upper bound given by Fekete and Meijer, as d goes to infinity. Our estimates on the star Steiner ratios are therefore much closer to the conjectured values in higher dimensions! As it turns out, our estimates as well as the conjectured values of the Steiner ratios (in the limit, for n going to infinity) are related to the classical infinite Wallis product: .

This paper considers constrained lscr1minimization methods in a unified framework for the recovery of high-dimensional sparse signals in three settings: noiseless, bounded error, and Gaussian noise. Both lscr1minimization with an lscrinfinconstraint (Dantzig selector) and lscr1minimization under anllscr2constraint are considered. The results of this paper improve the existing results in the literature by weakening the conditions and tightening the error bounds. The improvement on the conditions shows that signals with larger support can be recovered accurately. In particular, our results illustrate the relationship between lscr1minimization with anllscr2constraint and lscr1minimization with an lscrinfinconstraint. This paper also establishes connections between restricted isometry property and the mutual incoherence property. Some results of Candes, Romberg, and Tao (2006), Candes and Tao (2007), and Donoho, Elad, and Temlyakov (2006) are extended.

The main topic of the paper is virtual spectrophotometers, virtual spectrophotometry and strategies for reducing the number of samples required to obtain asymptotically convergent measurements. A brief section on applications of virtual spectrophotometry is presented. This is followed by a careful formulation of virtual spectrophotometers. A virtual spectrophotometer, being a virtual device, is only described by its geometry.