Search

In this paper, we propose an algorithm for computing mod $ \ell $ Galois representations associated to modular forms of weight $ k $ when $ \ell < k-1 $. We also present the corresponding results for the projective Galois representations. Moreover, we apply our algorithms to explicitly compute the mod $ \ell $ projective Galois representations associated to $ \Delta_{k} $ for $ k = 16, 20, 22, 26 $ and all the unexceptional primes $ \ell $, with $ \ell < k-1 $. As an application, for $ k = 16, 20, 22, 26 $, we obtain the new bounds $ B_k $ of $ n $ such that $ a_n(\Delta_k)\ne0 $ for all $ n < B_k $.

Homomorphic encryption (HE) has emerged as a pivotal technology for secure neural network inference (SNNI), offering privacy-preserving computations on encrypted data. Despite active developments in this field, HE-based SNNI frameworks are impeded by three inherent limitations. Firstly, they cannot evaluate non-linear functions such as ReLU, the most widely adopted activation function in neural networks. Secondly, the permitted number of homomorphic operations on ciphertexts is bounded, consequently limiting the depth of neural networks that can be evaluated. Thirdly, the computational overhead associated with HE is prohibitively high, particularly for deep neural networks. In this paper, we introduce a novel paradigm designed to address the three limitations of HE-based SNNI. Our approach is an interactive approach that is solely based on HE, called iLHE. Utilizing the idea of iLHE, we present two protocols: ReLU, which facilitates the direct evaluation of the ReLU function on encrypted data, tackling the first limitation, and HeRefresh, which extends the feasible depth of neural network computations and mitigates the computational overhead, thereby addressing the second and third limitations. Based on HeReLU and HeRefresh protocols, we build a new framework for SNNI, named HeFUN. We prove that our protocols and the HeFUN framework are secure in the semi-honest security model. Empirical evaluations demonstrate that HeFUN surpasses current HE-based SNNI frameworks in multiple aspects, including security, accuracy, the number of communication rounds, and inference latency. Specifically, for a convolutional neural network with four layers on the MNIST dataset, HeFUN achieves 99.16% accuracy with an inference latency of 1.501 s, surpassing the popular HE-based framework CryptoNets proposed by Gilad-Bachrach, which achieves 98.52% accuracy with an inference latency of 3.479 s.

Linkable ring signatures is a useful cryptographic tool for constructing applications such as ones relative to electronic voting (e-voting), digital cashes (e-cashes) as well as cloud computing. Equipped with linkable ring signatures, e-voting, e-cash systems can simultaneously enjoy the privacy and the unreusability properties thanks to the anonymity and the linkability of linkable ring signatures. Likewise, cloud servers can enjoy a privacy-preserving ability, a flexible access control and an efficient security management with linkable ring signatures. Moreover, linkable ring signatures built in the identity-based setting would help to remove the expense of using the conventional public key infrastructure and also could be applied to the user management. This primitive hence would be suitable for huge-scale applications. In this paper, we present the first identity-based linkable ring signatures (IdLRS) in both integer lattice and ideal lattice setting. The proposed IdLRS is proved secure in the random oracle model and based on the hardness of the short integer solution and ring short integer solution assumption. We also implement the proposed idLRS as a proof of concept and then do some experiments to evaluate the running times and the sizes.

In modern applications, such as Electronic Voting, e-Health, e-Cash, there is a need that the validity of a signature should be verified by only one responsible person. This is opposite to the traditional digital signature scheme where anybody can verify a signature. There have been several solutions for this problem, the first one is we combine a signature scheme with an encryption scheme; the second one is to use the group signature; and the last one is to use the strong designated verifier signature scheme with the undeniable property. In this paper, we extend the traditional digital signature scheme to propose a new solution for the aforementioned problem. Our extension is in the sense that only a designated verifier (responsible person) can verify a signer’s signature, and if necessary (in case the signer refuses to admit his/her signature) the designated verifier without revealing his/her secret key is able to prove to anybody that the signer has actually generated the signature. The comparison between our proposed solution and the three existing solutions shows that our proposed solution is the best one in terms of both security and efficiency.