This paper examines the application of the Verkle tree—an efficient data structure that leverages commitments and a novel proof technique in cryptographic solutions.Unlike traditional Merkle trees,the Verkle tree sig...This paper examines the application of the Verkle tree—an efficient data structure that leverages commitments and a novel proof technique in cryptographic solutions.Unlike traditional Merkle trees,the Verkle tree significantly reduces signature size by utilizing polynomial and vector commitments.Compact proofs also accelerate the verification process,reducing computational overhead,which makes Verkle trees particularly useful.The study proposes a new approach based on a non-positional polynomial notation(NPN)employing the Chinese Remainder Theorem(CRT).CRT enables efficient data representation and verification by decomposing data into smaller,indepen-dent components,simplifying computations,reducing overhead,and enhancing scalability.This technique facilitates parallel data processing,which is especially advantageous in cryptographic applications such as commitment and proof construction in Verkle trees,as well as in systems with constrained computational resources.Theoretical foundations of the approach,its advantages,and practical implementation aspects are explored,including resistance to potential attacks,application domains,and a comparative analysis with existing methods based on well-known parameters and characteristics.An analysis of potential attacks and vulnerabilities,including greatest common divisor(GCD)attacks,approximate multiple attacks(LLL lattice-based),brute-force search for irreducible polynomials,and the estimation of their total number,indicates that no vulnerabilities have been identified in the proposed method thus far.Furthermore,the study demonstrates that integrating CRT with Verkle trees ensures high scalability,making this approach promising for blockchain systems and other distributed systems requiring compact and efficient proofs.展开更多
Let M be a C^(2)-smooth Riemannian manifold with boundary and X be a metric space with non-positive curvature in the sense of Alexandrov.Let u:M→X be a Sobolev mapping in the sense of Korevaar and Schoen.In this shor...Let M be a C^(2)-smooth Riemannian manifold with boundary and X be a metric space with non-positive curvature in the sense of Alexandrov.Let u:M→X be a Sobolev mapping in the sense of Korevaar and Schoen.In this short note,we introduce a notion of p-energy for u which is slightly different from the original definition of Korevaar and Schoen.We show that each minimizing p-harmonic mapping(p≥2)associated to our notion of p-energy is locally Holder continuous whenever its image lies in a compact subset of X.展开更多
We give the sharp estimates for the degree of symmetry and the semi-simple degree of symmetry of certain compact fiber bundles with non-trivial four dimensional fibers in the sense of cobordism, by virtue of the rigid...We give the sharp estimates for the degree of symmetry and the semi-simple degree of symmetry of certain compact fiber bundles with non-trivial four dimensional fibers in the sense of cobordism, by virtue of the rigidity theorem of harmonic maps due to Schoen and Yau (Topology, 18, 1979, 361-380). As a corollary of this estimate, we compute the degree of symmetry and the semi-simple degree of symmetry of CP2×V, where V is a closed smooth manifold admitting a real analytic Riemannian metric of non-positive curvature. In addition, by the Albanese map, we obtain the sharp estimate of the degree of symmetry of a compact smooth manifold with some restrictions on its one dimensional cohomology.展开更多
基金funded by the Ministry of Science and Higher Education of Kazakhstan and carried out within the framework of the project AP23488112“Development and study of a quantum-resistant digital signature scheme based on a Verkle tree”at the Institute of Information and Computational Technologies.
文摘This paper examines the application of the Verkle tree—an efficient data structure that leverages commitments and a novel proof technique in cryptographic solutions.Unlike traditional Merkle trees,the Verkle tree significantly reduces signature size by utilizing polynomial and vector commitments.Compact proofs also accelerate the verification process,reducing computational overhead,which makes Verkle trees particularly useful.The study proposes a new approach based on a non-positional polynomial notation(NPN)employing the Chinese Remainder Theorem(CRT).CRT enables efficient data representation and verification by decomposing data into smaller,indepen-dent components,simplifying computations,reducing overhead,and enhancing scalability.This technique facilitates parallel data processing,which is especially advantageous in cryptographic applications such as commitment and proof construction in Verkle trees,as well as in systems with constrained computational resources.Theoretical foundations of the approach,its advantages,and practical implementation aspects are explored,including resistance to potential attacks,application domains,and a comparative analysis with existing methods based on well-known parameters and characteristics.An analysis of potential attacks and vulnerabilities,including greatest common divisor(GCD)attacks,approximate multiple attacks(LLL lattice-based),brute-force search for irreducible polynomials,and the estimation of their total number,indicates that no vulnerabilities have been identified in the proposed method thus far.Furthermore,the study demonstrates that integrating CRT with Verkle trees ensures high scalability,making this approach promising for blockchain systems and other distributed systems requiring compact and efficient proofs.
基金supported by the Qilu funding of Shandong University (62550089963197)financially supported by the National Natural Science Foundation of China (11701045)the Yangtze Youth Fund (2016cqn56)
文摘Let M be a C^(2)-smooth Riemannian manifold with boundary and X be a metric space with non-positive curvature in the sense of Alexandrov.Let u:M→X be a Sobolev mapping in the sense of Korevaar and Schoen.In this short note,we introduce a notion of p-energy for u which is slightly different from the original definition of Korevaar and Schoen.We show that each minimizing p-harmonic mapping(p≥2)associated to our notion of p-energy is locally Holder continuous whenever its image lies in a compact subset of X.
基金Project supported by the Japanese Government Scholarshipthe Japan Society for the Promotion of Science Postdoctoral Fellowship for Foreign Researchers+2 种基金the Focused Research Group Postdoctoral Fellowshipthe Program of Visiting Scholars at Chern Institute of Mathematicsthe National Natural Science Foundation of China (No. 10601053).
文摘We give the sharp estimates for the degree of symmetry and the semi-simple degree of symmetry of certain compact fiber bundles with non-trivial four dimensional fibers in the sense of cobordism, by virtue of the rigidity theorem of harmonic maps due to Schoen and Yau (Topology, 18, 1979, 361-380). As a corollary of this estimate, we compute the degree of symmetry and the semi-simple degree of symmetry of CP2×V, where V is a closed smooth manifold admitting a real analytic Riemannian metric of non-positive curvature. In addition, by the Albanese map, we obtain the sharp estimate of the degree of symmetry of a compact smooth manifold with some restrictions on its one dimensional cohomology.
