Binary Huff Curves over Non-Local Rings: The Yak Protocol and Blockchain

Abstract

Motivated by the growing need for secure and efficient cryptographic solutions in blockchain technology, this study explores the cryptographic potential of binary Huff curves defined over the non-local ring <img src=image/13442212_01.gif>, introducing novel group structures for advanced blockchain applications. The research aims to enhance the security and efficiency of cryptographic primitives by leveraging the algebraic properties of these curves, particularly for resource-constrained devices in blockchain ecosystems. We establish a bijection between the Huff curve <img src=image/13442212_02.gif> and the product <img src=image/13442212_03.gif>, enabling an efficient group law that increases the complexity of the discrete logarithm problem (DLP). Methodologically, we define arithmetic operations in <img src=image/13442212_04.gif>, prove the bijection, and derive addition formulas for the curve. These results are applied to adapt the Yak key exchange protocol, enhancing its resistance to DLP-based attacks through the non-local ring's structure. Principal findings demonstrate that <img src=image/13442212_02.gif> achieves approximately <img src=image/13442212_05.gif> group order, doubling the DLP security to <img src=image/13442212_06.gif>-bit compared to <img src=image/13442212_06.gif>/2-bit for standard curves over <img src=image/13442212_07.gif>, with computational efficiency suitable for Internet of Things (IoT) devices. The study contributes to cryptography by proposing a robust framework for blockchain transaction security and secure data management, notably in multi-party computation and zero-knowledge proofs. Key conclusions highlight the curves' potential to secure blockchain validators and IoT nodes, as exemplified in supply chain applications. Novel aspects include the non-local ring's algebraic constraints and the Yak protocol's adaptation for blockchain. Limitations include the need for practical implementation and benchmarking against curves like secp256k1. Practical implications involve improved transaction security and data privacy in blockchain, while social implications include enabling secure, decentralized systems for healthcare and supply chain tracking. Future research should validate performance in real-world blockchain environments and assess resistance to side-channel attacks.