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Gemini: Elastic SNARKs for Diverse Environments

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Advances in Cryptology – EUROCRYPT 2022 (EUROCRYPT 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13276))

Abstract

We introduce a new class of succinct arguments, that we call elastic. Elastic SNARKs allow the prover to allocate different resources (such as memory and time) depending on the execution environment and the statement to prove. The resulting output is independent of the prover’s configuration. To study elastic SNARKs, we extend the streaming paradigm of [Block et al., TCC’20]. We provide a definitional framework for elastic polynomial interactive oracle proofs for R1CS instances and design a compiler which transforms an elastic PIOP into a preprocessing argument system that supports streaming or random access to its inputs. Depending on the configuration, the prover will choose different trade-offs for time (either linear, or quasilinear) and memory (either linear, or logarithmic). We prove the existence of elastic SNARKS by presenting Gemini, a novel FFT-free preprocessing argument. We prove its security and develop a proof-of-concept implementation in Rust based on the arkworks framework. We provide benchmarks for large R1CS instances of tens of billions of gates on a single machine.

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Notes

  1. 1.

    https://ethereum.org/en/developers/docs/scaling/layer-2-rollups/.

  2. 2.

    https://research.protocol.ai/sites/snarks/.

  3. 3.

    Note that \({M}= \varOmega ({N})\) without loss of generality because if \({M}< {N}/3\) then there are variables of \(\mathbf {z}\) that do not participate in any constraint, which can be dropped. Thus the main size measure for R1CS is the sparsity parameter \({M}\).

  4. 4.

    The canonical stream of a vector consists of the sequence of its entries, from last to first.

  5. 5.

    The argument prover and argument verifier emulate the underlying probabilistic proof, with the argument prover sending commitments to proof messages and sending answers to queries together with commitment openings to authenticate those answers.

  6. 6.

    For example, if one polynomial consists of all of the even coefficients of another, one can produce streams of the coefficients of both polynomials simultaneously, in half the number of passes required to compute streams of each polynomial one at a time.

  7. 7.

    Here \(|\mathbb {G}_1|=|\mathbb {G}_2|=|\mathbb {G}_T|=q\), \(G\) generates \(\mathbb {G}_1\), \(H\) generates \(\mathbb {G}_2\), and \(e:\mathbb {G}_1\times \mathbb {G}_2\rightarrow \mathbb {G}_T\) is a non-degenerate bilinear map.

  8. 8.

    This restriction is merely didactical. Given any \(\mathbf {f}\in \mathbb {F}^{N}\), representing the coefficients of a degree \(N-1\) polynomial, it is easy to simulate polynomial-evaluation query access to \((\mathbf {f},1)\) using the polynomial \(\mathbf {f}({X}) + {X}^{N+1}\). For any evaluation query in \(x \in \mathbb {F}\), forward evaluation queries to \(\mathbf {f}\) and add \(x^{N+1}\) before returning. This costs \(O(\log N)\) \(\mathbb {F}\)-ops.

  9. 9.

    See https://github.com/arkworks-rs/gemini.

  10. 10.

    cf. https://github.com/arkworks-rs/marlin.

  11. 11.

    This is vm.overcommit=2. See https://www.kernel.org/doc/Documentation/vm/overcommit-accounting.

  12. 12.

    source: https://calculator.aws.

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Authors and Affiliations

  1. IBM Research, Zurich, Switzerland

    Jonathan Bootle

  2. École polytechnique fédérale de Lausanne, Lausanne, Switzerland

    Alessandro Chiesa

  3. University of California, Berkeley, Berkeley, USA

    Alessandro Chiesa, Yuncong Hu & Michele Orrú

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  1. Jonathan Bootle
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  2. Alessandro Chiesa
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  3. Yuncong Hu
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  4. Michele Orrú
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Correspondence to Jonathan Bootle or Michele Orrú .

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Editors and Affiliations

  1. University of Haifa, Haifa, Haifa, Israel

    Orr Dunkelman

  2. University of Warsaw, Warsaw, Poland

    Stefan Dziembowski

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Bootle, J., Chiesa, A., Hu, Y., Orrú, M. (2022). Gemini: Elastic SNARKs for Diverse Environments. In: Dunkelman, O., Dziembowski, S. (eds) Advances in Cryptology – EUROCRYPT 2022. EUROCRYPT 2022. Lecture Notes in Computer Science, vol 13276. Springer, Cham. https://doi.org/10.1007/978-3-031-07085-3_15

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