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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, independent 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.

Based on extensive research in Sanskrit sources, Mathematics in India chronicles the development of mathematical techniques and texts in South Asia from antiquity to the early modern period. Kim Plofker reexamines the few facts about Indian mathematics that have become common knowledge--such as the Indian origin of Arabic numerals--and she sets them in a larger textual and cultural framework. The book details aspects of the subject that have been largely passed over in the past, including the relationships between Indian mathematics and astronomy, and their cross-fertilizations with Islamic scientific traditions. Plofker shows that Indian mathematics appears not as a disconnected set of discoveries, but as a lively, diverse, yet strongly unified discipline, intimately linked to other Indian forms of learning.

This chapter contains sections titled: Rotation of Earth: Latitude and Longitude Celestial Sphere Treatment in Solar Time Treatment in Standard Time Problems

This chapter contains sections titled: Introduction Thermodynamics Formal Background mfold and UNAFold Troubleshooting References