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<p>Commit-and-prove succinct non-interactive arguments of knowledge (CP-SNARKs) are an important class of SNARKs that allow proving the knowledge of a witness committed beforehand. They are crucial in proving composite statements that combine algebraic and non-algebraic statements. However, existing CP-SNARKs only support constraint systems with degree-2 gates, limiting their ability to express non-algebraic statements succinctly. We propose a new family of CP-SNARKs for <i>Customizable Constraint Systems</i>. The new family, named CP-SuperSpartan, supports high-degree gates, eliminates expensive fast Fourier transform operations and supports a general commit-and-prove relation including multiple commitments to different slots of the witness. CP-SuperSpartan has several instantiations based on different multilinear polynomial commitment schemes (PCS). 1) When using pairing-based PCS, CP-SuperSpartan provides the same universally updatable setup as LegoSNARK (CCS’19), but it reduces the proof size and verifier complexity from <i>O</i>(log<sup>2</sup>∣<i>C</i>∣) to <i>O</i>(log∣<i>C</i>∣) while maintaining the same prover complexity, where ∣<i>C</i>∣ is the circuit size. 2) When using discrete-logarithm-based PCS, CP-SuperSpartan provides a transparent setup and achieves <i>O</i>(log ∣<i>C</i>∣) proof size while the smallest proof size of existing transparent CP-SNARKs is <i>O</i>(log<sup><i>O</i>(1)</sup> ∣<i>C</i>∣). 3) When using PCS based on interactive oracle proof of proximity, CP-SuperSpartan provides a transparent setup and firstly achieves <i>O</i>(∣<i>C</i>∣) prover complexity, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(O(\sqrt{|C|})\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>O</mi> <mo stretchy="false">(</mo> <msqrt> <mrow> <mo stretchy="false">|</mo> </mrow> <mi>C</mi> <mrow> <mo stretchy="false">|</mo> </mrow> </msqrt> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> proof size and verifier complexity while the number of public-key operations for both the prover and verifier is independent of ∣<i>C</i>∣.</p>

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CP-SuperSpartan: commit-and-prove SNARKs for customizable constraint systems

  • 期刊论文

  • 发表日期：

    2026年2月20日

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  • Zibo Zhou,
  • Zongyang Zhang,
  • Feng Hao,
  • Jianwei Liu

摘要

Commit-and-prove succinct non-interactive arguments of knowledge (CP-SNARKs) are an important class of SNARKs that allow proving the knowledge of a witness committed beforehand. They are crucial in proving composite statements that combine algebraic and non-algebraic statements. However, existing CP-SNARKs only support constraint systems with degree-2 gates, limiting their ability to express non-algebraic statements succinctly. We propose a new family of CP-SNARKs for Customizable Constraint Systems. The new family, named CP-SuperSpartan, supports high-degree gates, eliminates expensive fast Fourier transform operations and supports a general commit-and-prove relation including multiple commitments to different slots of the witness. CP-SuperSpartan has several instantiations based on different multilinear polynomial commitment schemes (PCS). 1) When using pairing-based PCS, CP-SuperSpartan provides the same universally updatable setup as LegoSNARK (CCS’19), but it reduces the proof size and verifier complexity from O(log2∣C∣) to O(log∣C∣) while maintaining the same prover complexity, where ∣C∣ is the circuit size. 2) When using discrete-logarithm-based PCS, CP-SuperSpartan provides a transparent setup and achieves O(log ∣C∣) proof size while the smallest proof size of existing transparent CP-SNARKs is O(logO(1) ∣C∣). 3) When using PCS based on interactive oracle proof of proximity, CP-SuperSpartan provides a transparent setup and firstly achieves O(∣C∣) prover complexity, \(O(\sqrt{|C|})\) O ( | C | ) proof size and verifier complexity while the number of public-key operations for both the prover and verifier is independent of ∣C∣.