Elliptic curves, cryptographic proofs and signature schemes: a comprehensive analysis of the Pinocchio protocol and implementation frameworks

Abstract

This paper investigates the mathematical and cryptographic principles underlying elliptic curves and their critical role in modern secure communication systems. It begins with an exploration of elliptic curves and their group properties, emphasizing their utility in cryptography through problems like the discrete logarithm. The discussion transitions to the Pinocchio protocol, a zero-knowledge succinct non-interactive argument of knowledge (zk-SNARK), detailing its use of Rank-1 Constraint Systems (R1CS), Quadratic Arithmetic Programs (QAPs), and cryptographic commitments. The final section delves into practical implementations, employing CIRCOM for circuit definition and testing, with specific applications to ECDSA and BLS signature schemes. Through this comprehensive approach, the paper bridges theoretical cryptographic advancements with practical implementation, offering insights into the efficiency, security, and application of elliptic curve-based protocols.