Abstract In the case of standard LWE samples $$({\textbf {A}},{\textbf {b = sA + e}})$$ ( A , b = sA + e ) , $${\textbf {A}}$$ A is typically uniformly over $$\mathbb {Z}_q^{n \times m}$$ Z q n × m . Under the $$\textsf {DLWE}$$ DLWE assumption, the conditional distribution of $${\textbf {s}}|({\textbf {A}}, {\textbf {b}})$$ s | ( A , b ) and $${\textbf {s}}$$ s is expected to be consistent. However, in the case where an adversary chooses $${\textbf {A}}$$ A adaptively, the disparity between the two entities may be larger. In this work, our primary focus is on the quantification of the Average Conditional Min-Entropy $$\tilde{H}_\infty ({\textbf {s}}|{\textbf {sA + e}})$$ H ~ ∞ ( s | sA + e ) of $${\textbf {s}}$$ s , where $${\textbf {A}}$$ A is chosen by the adversary. Brakerski and Döttling answered the question in one case: they proved that when $${\textbf {s}}$$ s is uniformly chosen from $$\mathbb {Z}_q^n$$ Z q n , it holds that $$\tilde{H}_\infty ({\textbf {s}}|{\textbf {sA + e}}) \varpropto \rho _\sigma (\varLambda _q({\textbf {A}}))$$ H ~ ∞ ( s | sA + e ) ∝ ρ σ ( Λ q ( A

The current multikey fully homomorphic encryption (MKFHE) needs to add exponential noise in the distributed decryption phase to ensure the simulatability of partial decryption. Such a large noise causes the ciphertext modulus of the scheme to increase exponentially compared to the single‐key fully homomorphic encryption (FHE), further reducing the efficiency of the scheme and making the hardness problem on the lattice on which the scheme relies have a subexponential approximation factor (which means that the security of the scheme is reduced). To address this problem, this paper analyzes in detail the noise in partial decryption of the MKFHE based on the learning with error (LWE) problem. It points out that this part of the noise is composed of private key and the noise in initial ciphertext. Therefore, as long as the encryption scheme is leak‐resistant and the noise in partial decryption is independent of the noise in the initial ciphertext, the semantic security of the ciphertext can be guaranteed. In order to make the noise in the initial ciphertext independent of the noise in the partial decryption, this paper proves the smudging lemma on discrete Gaussian distribution and achieves this goal by multiplying the initial ciphertext by a “dummy” ciphertext with a plaintext of 1. Based on the above method, this paper removes the exponential noise in the distributed decryption phase for the first time and reduces the ciphertext modulus of MKFHE from 2 ω ( λ L log λ ) to 2 O ( λ + L ) as the same level as the FHE.

The hardness of the NTRU problem has not been well understood until 2021, when Pellet-Mary and Stehlé (2021) gave a reduction from the Gap-SVP problem on the ideal lattice to the NTRU-Search problem. Assuming the equivalence of the NTRU-Decision and the NTRU-Search problem, with this reduction together, we construct a leveled homomorphic encryption scheme. Compared to homomorphic schemes based on RLWE such as CKKS and BGV, the ciphertext of our scheme is a single polynomial. As a result, ciphertext multiplication involves only one multiplication of two polynomials, rather than the tensor multiplication of polynomial vectors as in BGV, CKKS schemes. In particular, by introducing a label, the ciphertext of our scheme does not need to be linearized after multiplication. This significantly accelerates the speed of homomorphic evaluation by reducing the number of polynomial multiplications from 6 to 1. Complexity analysis and experimental results indicate that the ciphertext multiplication in our scheme is approximately 4~5 times faster than CKKS and BFV schemes

To solve the problem of dense vehicle target detection in polarimetric synthetic aperture radar (PolSAR) images from urban areas under complex scenarios, this paper proposes a target detection method that combines the superpixel segmentation and the Wishart classifier. Firstly, the buildings are detected based on the different polarimetric scattering characteristics of ground objects. Then, the morphological information of the target is obtained by the local Wishart classifier and the superpixel segmentation. After that, the center points of the target are obtained by the global Wishart classifier. Finally, the region growing procedure is used to fuse the information obtained by above-mentioned classifiers to complete the target detection task.