EIP-1895: Support for an Elliptic Curve Cycle #1895

AlexandreBelling · 2019-04-01T12:30:13Z

The EVM currently supports elliptic curves operations for curve alt-bn128 thanks to precompiles ecadd and ecmul and ecpairing. The classes MNT4 and 6 contains cycles of curves and this enable doing operation on one curve inside a SNARK on the other (end reversely). This EIP suggests adding support for those curves.

nicksavers

Thanks for the Draft. Added some minor suggestions. Please go over the text and correct the grammar and spelling mistakes. These make the proposal a bit more difficult to read.

nicksavers · 2019-04-01T20:48:45Z

EIPS/eip-1895.md

+
+More concretely, today if the EVM has to deal with 1000s of SNARK verification it would take around 1.5 Billion gas and would be impractical for Ethereum. Recursive snarks for instance make it possible to aggregate multiple proofs into a single one that can be verified like any other SNARKs. It massively reduces the cost of verification then.
+
+However, this is impossible using *alt-bn128* and in my knowledge, the only family of pairing-friendly curves known to produce cycles are MNT4 and MNT6. A complete characterization of the cycles existing between thoses two families is proposed [here](https://arxiv.org/pdf/1803.02067.pdf)


For a Draft it's ok, but please refrain from referencing yourself in the motivation ('my knowledge').

For clarity it's perhaps better to link to the paper as a Reference or at least describe the title of the paper that is linked to.

nicksavers · 2019-04-01T20:49:52Z

EIPS/eip-1895.md

+
+### The curve
+
+The proposed curves are described in this [paper](https://eprint.iacr.org/2014/595.pdf).


Similar to the link above, maybe use a reference and better description of the link.
Also, the same link is referenced twice in the proposal

EIPS/eip-1895.md

nicksavers · 2019-04-01T21:02:56Z

EIPS/eip-1895.md

+
+It has 80 bits (the curves respectively have 80 and 120 bits of security) of security which might be considered not enough for some application, but enough for others.
+
+If not, another curve is being used by Coda but is defined over a 727 bits which might be considered prohibitively slow. *Note, that the curves and field elements are represented on field larger than 256 bits (even for the 80 bits of security), therefore it might be necessary to add support for larger field size operations.


If not what?

nicksavers · 2019-04-01T21:13:42Z

EIPS/eip-1895.md

+
+## Simple Summary
+
+The EVM currently supports elliptic curves operations for curve *alt-bn128* thanks to precompiles `ecadd` and `ecmul` and `ecpairing`. The classes MNT4 and 6 contains cycles of curves and this enable doing operation on one curve inside a SNARK on the other (end reversely). This EIP suggests adding support for those curves.


I'm not sure how to read this sentence:

"The classes MNT4 and 6 contains cycles of curves and this enable doing operation on one curve inside a SNARK on the other (end reversely)."

Could you perhaps clarify it?

@AlexandreBelling Could you still clarify the sentence?

Does it make more sense if I rephrase like that ?

"An elliptic curve cycle is a set of elliptic curves with the property that the inner-field of one curve is the outer-field of the other one. Cycles make it possible to do elliptic curve operations for one of the curves inside zkSNARK circuit that is proven and verified using the other curve of the cycle."

That's an explanation, but the sentence structure is still unclear to me. It looks like the grammar is not correct. So you could leave the sentence in there, just fix the grammar. Perhaps ask an English native speaker to correct it.

Hum ok, I have asked someone to help me fix it. I now believe it means what it's supposed to mean :)

This might be "Two elliptic curves were generated such that group order of one curve is coordinates field characteristic of the other curve, and reverse."

We have a good arXiv paper referenced describing this property already, making it unreasonable explaining cycle here on github. I would also refer to my favorite IACR 2014/595 preprint.

nicksavers · 2019-04-01T21:21:25Z

EIPS/eip-1895.md

+
+Elliptic curve is the basic block of recursive SNARKSs (ie: verifying a SNARK inside a SNARK) and this addresses the issue of scalable zero-knowledge. More generally this addresses partly the scalability issue as SNARKs verification are constant time in the size of the circuit being verified.
+
+More concretely, today if the EVM has to deal with 1000s of SNARK verification it would take around 1.5 Billion gas and would be impractical for Ethereum. Recursive snarks for instance make it possible to aggregate multiple proofs into a single one that can be verified like any other SNARKs. It massively reduces the cost of verification then.


'verification' -> 'verifications'
'Billion' -> 'billion'
'SNARKs' -> 'SNARK'

nicksavers · 2019-04-01T21:25:12Z

EIPS/eip-1895.md

+
+## Rationale
+
+The only usable cycle in my knowledge are introduced in this [paper](https://eprint.iacr.org/2014/595.pdf).


Again, don't reference yourself in the Rationale

nicksavers · 2019-04-01T21:29:27Z

EIPS/eip-1895.md

+
+The only usable cycle in my knowledge are introduced in this [paper](https://eprint.iacr.org/2014/595.pdf).
+
+It has 80 bits (the curves respectively have 80 and 120 bits of security) of security which might be considered not enough for some application, but enough for others.


Perhaps better to rephrase as:

"The curve has 80 bits of security (whereas MNT6 has 120 bits) which might not be considered enough for some applications, .... ."

And clarify which type of applications you mean with 'some applications' and 'others'.

nicksavers · 2019-04-01T21:32:18Z

EIPS/eip-1895.md

+
+It has 80 bits (the curves respectively have 80 and 120 bits of security) of security which might be considered not enough for some application, but enough for others.
+
+If not, another curve is being used by Coda but is defined over a 727 bits which might be considered prohibitively slow. *Note, that the curves and field elements are represented on field larger than 256 bits (even for the 80 bits of security), therefore it might be necessary to add support for larger field size operations.


There is no reference to Coda for the reader.

nicksavers · 2019-04-01T21:34:53Z

EIPS/eip-1895.md

+
+It has 80 bits (the curves respectively have 80 and 120 bits of security) of security which might be considered not enough for some application, but enough for others.
+
+If not, another curve is being used by Coda but is defined over a 727 bits which might be considered prohibitively slow. *Note, that the curves and field elements are represented on field larger than 256 bits (even for the 80 bits of security), therefore it might be necessary to add support for larger field size operations.


Why add your rationale as a note? It seems like a perfectly valid argument that could just be a normal sentence as the others.

AlexandreBelling · 2019-04-02T11:34:12Z

Hi nicksavers, thanks for the review.

Updated the draft as requested. Please, let me know if you have any comment.

nicksavers · 2019-04-02T13:45:26Z

EIPS/eip-1895.md

+eip: 1895
+title: Support for an Elliptic Curve Cycles
+author: Alexandre Belling <[email protected]>
+discussions-to: https://ethresear.ch/t/reducing-the-verification-cost-of-a-snark-through-hierarchical-aggregation/5128/7


This link doesn't go to the top of start of the discussion, but to one particular comment by you.

nicksavers · 2019-04-02T13:54:15Z

EIPS/eip-1895.md

+
+## References
+
+* [Scalable Zero Knowledge via Cycles of Elliptic Curves](https://eprint.iacr.org/2014/595.pdf)


For the references could you display the full URL to the reader? Normally with references to articles, the authors and date are also mentioned.

Have a look at Harvard Referencing to make good references.
If you want to really go all out, have a look at this list. https://libguides.ioe.ac.uk/harvard/AZlisting

AlexandreBelling · 2019-04-02T21:15:12Z

Done. Thanks again.

nicksavers · 2019-04-02T22:00:55Z

I'd say this is ready to be merged as Draft.

rex4539 · 2019-04-03T15:56:16Z

EIPS/eip-1895.md

+
+More concretely, today if the EVM has to deal with 1000s of SNARK verification it would take around 1.5 billion gas and would be impractical for Ethereum. Recursive SNARKs for instance make it possible to aggregate multiple proofs into a single one that can be verified like any other SNARK. It massively reduces the cost of verifications then.
+
+However, this is impossible using *alt-bn128* and in my knowledge, the only family of pairing-friendly curves known to produce cycles are MNT4 and MNT6. A complete characterization of the cycles existing between thoses two families is proposed in [On cycles of pairing-friendly elliptic curves


Typo: "thoses"

rex4539 · 2019-04-03T15:59:22Z

EIPS/eip-1895.md

+
+Independently of the cycle chosen, the groups and field elements are represented with integers larger than 256 bits (even for the 80 bits of security), therefore it might be necessary to also add support for larger field size operations.
+
+We currently don't know more efficients pairing-friendly cycles and don't know if there are. It might be possible to circumvent this problem though by relaxing the constraint that all the curves of the cycle must be pairing friendly). If we had a cycle with only one pairing friendly curve we would still be able to compose proofs by alternating between SNARKs and any other general purpose zero-knowledge cryptosystems.


efficients -> efficient

nicksavers · 2019-04-03T18:44:06Z

EIPS/eip-1895.md

+created: 2018-31-03
+---
+
+# Support for an Elliptic Curve Cycles


The title doesn't need to be here. During parsing the title from the header is used.

nicksavers · 2019-04-03T18:45:42Z

EIPS/eip-1895.md

@@ -0,0 +1,166 @@
+---
+eip: 1895
+title: Support for an Elliptic Curve Cycles


I'm thinking you'd want the title to either say

Support for Elliptic Curve Cycles
or

Support for an Elliptic Curve Cycle

not

...an Elliptic Curve Cycles

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AlexandreBelling added 7 commits April 1, 2019 13:05

new(EIP): draft for elliptic curve cycles

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fix(typo): remove trailing copy-pasta

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typo(clean): encoding table for y

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doc(email): Add contact by mail

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fix(typo): Correct format for email

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fix(typo): Remove duplicate header

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feature(eipnum): Adds EIPnum

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AlexandreBelling changed the title ~~EIP: Support for Elliptic Curve Cycles~~ EIP-1895 : Support for Elliptic Curve Cycles Apr 1, 2019

AlexandreBelling changed the title ~~EIP-1895 : Support for Elliptic Curve Cycles~~ EIP-1895: Support for Elliptic Curve Cycles Apr 1, 2019

AlexandreBelling added 4 commits April 1, 2019 14:51

doc(summary): Update summary

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rename(1895): Rename the eip in order to pass CI

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feature(1895): Rename eip file

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fix(typo): Adds missing s to the type

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fix(typo): Fix spelling mistakes and rephrases

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AlexandreBelling added 2 commits April 2, 2019 22:46

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AlexandreBelling changed the title ~~EIP-1895: Support for Elliptic Curve Cycles~~ EIP-1895: Support for an Elliptic Curve Cycle Apr 4, 2019

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nicksavers merged commit d4efb6c into ethereum:master Apr 24, 2019

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EIP-1895: Support for an Elliptic Curve Cycle (ethereum#1895)

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EIP-1895: Support for an Elliptic Curve Cycle #1895

EIP-1895: Support for an Elliptic Curve Cycle #1895

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		More concretely, today if the EVM has to deal with 1000s of SNARK verification it would take around 1.5 Billion gas and would be impractical for Ethereum. Recursive snarks for instance make it possible to aggregate multiple proofs into a single one that can be verified like any other SNARKs. It massively reduces the cost of verification then.

		However, this is impossible using alt-bn128 and in my knowledge, the only family of pairing-friendly curves known to produce cycles are MNT4 and MNT6. A complete characterization of the cycles existing between thoses two families is proposed [here](https://arxiv.org/pdf/1803.02067.pdf)


		### The curve

		The proposed curves are described in this [paper](https://eprint.iacr.org/2014/595.pdf).


		It has 80 bits (the curves respectively have 80 and 120 bits of security) of security which might be considered not enough for some application, but enough for others.

		If not, another curve is being used by Coda but is defined over a 727 bits which might be considered prohibitively slow. *Note, that the curves and field elements are represented on field larger than 256 bits (even for the 80 bits of security), therefore it might be necessary to add support for larger field size operations.


		## Simple Summary

		The EVM currently supports elliptic curves operations for curve alt-bn128 thanks to precompiles `ecadd` and `ecmul` and `ecpairing`. The classes MNT4 and 6 contains cycles of curves and this enable doing operation on one curve inside a SNARK on the other (end reversely). This EIP suggests adding support for those curves.


		Elliptic curve is the basic block of recursive SNARKSs (ie: verifying a SNARK inside a SNARK) and this addresses the issue of scalable zero-knowledge. More generally this addresses partly the scalability issue as SNARKs verification are constant time in the size of the circuit being verified.

		More concretely, today if the EVM has to deal with 1000s of SNARK verification it would take around 1.5 Billion gas and would be impractical for Ethereum. Recursive snarks for instance make it possible to aggregate multiple proofs into a single one that can be verified like any other SNARKs. It massively reduces the cost of verification then.


		## Rationale

		The only usable cycle in my knowledge are introduced in this [paper](https://eprint.iacr.org/2014/595.pdf).


		## References

		* [Scalable Zero Knowledge via Cycles of Elliptic Curves](https://eprint.iacr.org/2014/595.pdf)


		Independently of the cycle chosen, the groups and field elements are represented with integers larger than 256 bits (even for the 80 bits of security), therefore it might be necessary to also add support for larger field size operations.

		We currently don't know more efficients pairing-friendly cycles and don't know if there are. It might be possible to circumvent this problem though by relaxing the constraint that all the curves of the cycle must be pairing friendly). If we had a cycle with only one pairing friendly curve we would still be able to compose proofs by alternating between SNARKs and any other general purpose zero-knowledge cryptosystems.

EIP-1895: Support for an Elliptic Curve Cycle #1895

EIP-1895: Support for an Elliptic Curve Cycle #1895

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