Succinctness Meets Transparency A Linking Framework Combining Hyrax and KZG, and Its Applications

Abstract

The thesis begins by introducing the concept of zero knowledge and exploring its various paradigms. It then focuses on a interesting problem: the construction of linking proofs. Chap- ter 2 addresses the challenge of binding two distinct polynomial commitment schemes so that they are consistent at a common evaluation point. The chapter further introduces a method for equating two different polynomials with different domains using an affine line construction. By applying certain optimizations to the bulletproofs protocol, the work achieves a reduction in both the prover’s time and the number of rounds required for proving a large committed inner product. Chapter 3 examines how the proposed linking proof can be used as a foundation for build- ing a zero-knowledge virtual machine (zkVM). This zkVM is designed using the abstraction of the RAM computational model. The main goals are to prove instruction membership and to verify the correct application of the state transition function, ensuring that it uses only instruc- tions whose membership has already been established on that particular step. The differences between this zkVM and existing approaches are discussed in detail. To prove the state transition function, the approach generates a circuit of size O(T ¨ log T), matching the prover’s time complexity. Memory consistency is addressed using De Bruijn graphs, which helps to eliminate the challenges of arithmetizing sorting algorithms within circuits. A compatible SNARK protocol (Hyrax) is used to produce succinct proofs. For the instruction membership problem, the membership lookup is moved outside the circuit and handled as a separate component. The proof is committed using the KZG scheme, and the two components are then efficiently linked using the proposed linking proof construction.