Title Page
Contents
ABSTRACT 8
Ⅰ. Introduction 9
1. Contributions 11
2. Related works 13
Ⅱ. Preliminaries 15
1. Notations 16
2. Security assumptions 17
3. Cryptographic primitives 17
4. Commitment schemes 20
Ⅲ. Mathematical Foundations 22
1. Divisors 24
2. Weil pairing 35
3. Tate pairing 46
Ⅳ. Hadamard Product Argument 48
1. Generic Hadamard Product Argument 49
2. Verifier-friendly Hadamard product argument 61
3. Batching Technique 76
Ⅴ. Applications 78
1. Commit-and-Prove SNARK 78
2. SNARK aggregation 80
3. zk-SNARK system for R1CS 83
4. Relaxed R1CS 89
Ⅵ. Conclusion 93
References 95
Abstract (in Korean) 103
Table 1. This table provides a comparison between our work and other transparent zk-SNARKs with respect to the size of the constraint system, denoted... 12
Table 2. This table shows the comparison of asymptotic performance of CP-SNARKs. In the context of CP-SNARK, ℓ denotes the number of input com-... 80
Table 3. This table provides an asymptotic performance comparison between our proposed method and SnarkPack [22], where n represents the number of... 83
Table 4. This table provides a comparison between our work and other transparent zk-SNARKs in terms of P's time, V's time, and communication size.... 89
Table 5. This table shows the analysis of transparent recursive zk-SNARKs. In the table, n denotes the circuit size of the computation at each incremental... 92