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

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merged 18 commits into from
Apr 24, 2019

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@AlexandreBelling AlexandreBelling commented Apr 1, 2019

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.

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

EIPS/eip-1895.md Outdated

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)
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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.

EIPS/eip-1895.md Outdated

### The curve

The proposed curves are described in this [paper](https://eprint.iacr.org/2014/595.pdf).
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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 Show resolved Hide resolved
EIPS/eip-1895.md Outdated

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.
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If not what?

EIPS/eip-1895.md Outdated

## 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.
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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?

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@AlexandreBelling Could you still clarify the sentence?

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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."

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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.

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

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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.

EIPS/eip-1895.md Outdated

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

EIPS/eip-1895.md Outdated

## Rationale

The only usable cycle in my knowledge are introduced in this [paper](https://eprint.iacr.org/2014/595.pdf).
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Again, don't reference yourself in the Rationale

EIPS/eip-1895.md Outdated

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.
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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'.

EIPS/eip-1895.md Outdated

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.
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There is no reference to Coda for the reader.

EIPS/eip-1895.md Outdated

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

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Hi nicksavers, thanks for the review.

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

EIPS/eip-1895.md Outdated
eip: 1895
title: Support for an Elliptic Curve Cycles
author: Alexandre Belling <alexandrebelling8@gmail.com>
discussions-to: https://ethresear.ch/t/reducing-the-verification-cost-of-a-snark-through-hierarchical-aggregation/5128/7
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This link doesn't go to the top of start of the discussion, but to one particular comment by you.

EIPS/eip-1895.md Outdated

## References

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

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Done. Thanks again.

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

EIPS/eip-1895.md Outdated

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
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Typo: "thoses"

EIPS/eip-1895.md Outdated

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.
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efficients -> efficient

EIPS/eip-1895.md Outdated
created: 2018-31-03
---

# Support for an Elliptic Curve Cycles
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The title doesn't need to be here. During parsing the title from the header is used.

EIPS/eip-1895.md Outdated
@@ -0,0 +1,166 @@
---
eip: 1895
title: Support for an Elliptic Curve Cycles
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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

@AlexandreBelling 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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* resolved an TODO item as Rationale entry

* Reduced gas cost for static calls made to precompiles (ethereum#2046)

* Reduced gas cost for static calls made to precompiles

* Rename to EIP-2046

* Add discussion URL

* Add 2046 to Istanbul

* Fix header

* Add draft for ESO (extended state oracle) (ethereum#2014)

* Add draft for ESO (extended state oracle)

* Add mention of revert to ESO

* Add EIP-2014 number and rename file

* Add reference to EIP-1959 and EIP-1965

* Add discussions-to URL

* Update eip-181.md

* Update eip-162.md

* State Rent Change A EIP placeholder - State counters contract (ethereum#2029)

* Create eip-StateRentACountersContract.md

* Update eip-StateRentACountersContract.md

* Rename eip-StateRentACountersContract.md to eip-2029.md

* Update eip-2029.md

* Update eip-2029.md

* Fix typos

* Make EIP-2014 into a proper link

* New Opcode to check if a chainID is part of the history of chainIDs (ethereum#1959)

* Automatically merged updates to draft EIP(s) 1679 (ethereum#2055)


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* Copyright 107 (ethereum#2068)

* add copyrights for 107

* use github username

* Automatically merged updates to draft EIP(s) 1155 (ethereum#2074)


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* State Rent change C draft EIP - Net contract storage size accounting (ethereum#2027)

* State Rent change C draft EIP

* Update eip-draft_StateRentCnetContractSizes.md

* Update eip-draft_StateRentCnetContractSizes.md

* Rename eip-draft_StateRentCnetContractSizes.md to eip-2027.md

* Update eip-2027.md

* Update eip-2027.md

* Update eip-2027.md

* Quote "block C" properly

* Update eip-2027.md

* State Rent change B placeholder EIP - net transaction counter (ethereum#2031)

* Create eip-draft_StateRentBNetTransactionCounter.md

* Update and rename eip-draft_StateRentBNetTransactionCounter.md to eip-2031.md

* Update eip-2031.md

* Update eip-2031.md

* Update eip-2031.md

* Fix some typos

* Use canonical link to EIP-2029

* Update eip-2031.md

* Update eip-2031.md

* Stateless Clients: Repricing SLOAD and SSTORE to pay for block proofs (ethereum#2035)

* Create eip-draft_StatelessClientGasRepricing.md

* Update and rename eip-draft_StatelessClientGasRepricing.md to eip-2035.md

* Update eip-2035.md

* Update eip-2035.md

* Update eip-2035.md

* Update eip-2035.md

* Update eip-2035.md

* Update eip-2035.md

* Fix typos

* Use canonical links to other EIPs

* Update eip-2035.md

* Fix link to EIP-2028

* Automatically merged updates to draft EIP(s) 1679 (ethereum#2036)


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* Automatically merged updates to draft EIP(s) 1930 (ethereum#2076)


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* Automatically merged updates to draft EIP(s) 1679 (ethereum#2042)


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* Automatically merged updates to draft EIP(s) 1418 (ethereum#2078)


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* Automatically merged updates to draft EIP(s) 1108 (ethereum#2067)


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* Automatically merged updates to draft EIP(s) 1155 (ethereum#2083)


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* Automatically merged updates to draft EIP(s) 1155 (ethereum#2084)


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* Automatically merged updates to draft EIP(s) 1155 (ethereum#2085)


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* Automatically merged updates to draft EIP(s) 1930 (ethereum#2086)


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* Automatically merged updates to draft EIP(s) 778 (ethereum#2087)


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* Automatically merged updates to draft EIP(s) 1155 (ethereum#2088)


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* Automatically merged updates to draft EIP(s) 1155 (ethereum#2089)


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* Last call for ERC-1155 (ethereum#2091)

* add EIP for particle gas costs (ethereum#2045)

* add EIP for fractional gas costs

* use EIP number 2045, add references, discussion-to

* add EIP-2045 to Istanbul

* Formatting

* Add ewasm to rationale

* Clarify cost reduction examples

* Clarify the increase of particles

* link to eip-2035, edit for clarity

* add reference to eip-2035

* add axic as author

* rename title to "particle gas costs", edit ewasm section

* fix header

* Automatically merged updates to draft EIP(s) 1803 (ethereum#2093)


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* Automatically merged updates to draft EIP(s) 1803, 663 (ethereum#2094)


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* Automatically merged updates to draft EIP(s) 1193 (ethereum#2092)


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* Automatically merged updates to draft EIP(s) 1155 (ethereum#2096)


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* Automatically merged updates to draft EIP(s) 778 (ethereum#2097)


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* Automatically merged updates to draft EIP(s) 1155 (ethereum#2101)


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* Automatically merged updates to draft EIP(s) 1261 (ethereum#2102)


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* fix typo: "as follows" (ethereum#2099)

see: https://ell.stackexchange.com/a/31673

* EIP-2003 - EVMC modules for implementations of precompiled contracts (ethereum#2003)

* Automatically merged updates to draft EIP(s) 1261 (ethereum#2107)


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* Automatically merged updates to draft EIP(s) 1155 (ethereum#2108)


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* Automatically merged updates to draft EIP(s) 1155 (ethereum#2109)


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* Automatically merged updates to draft EIP(s) 1155 (ethereum#2110)


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* Automatically merged updates to draft EIP(s) 2003 (ethereum#2112)


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* Automatically merged updates to draft EIP(s) 1155 (ethereum#2113)


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* Automatically merged updates to draft EIP(s) 1155 (ethereum#2114)


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* Automatically merged updates to draft EIP(s) 1155 (ethereum#2116)


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* Automatically merged updates to draft EIP(s) 1155 (ethereum#2117)


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* Automatically merged updates to draft EIP(s) 1155 (ethereum#2118)


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* Automatically merged updates to draft EIP(s) 1155 (ethereum#2120)


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* ERC-1155 Final Status (ethereum#2122)

* Automatically merged updates to draft EIP(s) 1344 (ethereum#2079)


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* Propose to move EIP-695 to last call status (ethereum#1566)

* Propose to move EIP-695 to last call status

* Add a discussion link

* Formatting fix

* Fix @sorpaas handle in author field

* Use EIP155 formatting for `CHAIN_ID`

* Formatting fix for "chain ID"

The document had places where we use lowercase "id" and uppercase "ID". This unify them to use uppercase "ID".

* Update eip-695.md

* Move EIP-695 to Last Call status and add 155 requirement (ethereum#2128)

* Update eip-1577.md (ethereum#1670)

* Automatically merged updates to draft EIP(s) 1702 (ethereum#2130)


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* Automatically merged updates to draft EIP(s) 1702 (ethereum#2131)


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* Automatically merged updates to draft EIP(s) 695 (ethereum#2133)


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* Automatically merged updates to draft EIP(s) 1202 (ethereum#2134)


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* Automatically merged updates to draft EIP(s) 1702 (ethereum#2140)


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* Automatically merged updates to draft EIP(s) 1702 (ethereum#2141)


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* Funding ETH1.X through a Developer Block Reward for 18 Months (ethereum#2025)

* Automatically merged updates to draft EIP(s) 1702 (ethereum#2149)


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* EIP 747: wallet_watchAsset (ethereum#1426)

* Initial draft

* Add eip-747: watchToken

* Update discussion link

* Update eip-747 to watchAsset

* Add image specification and examples EIP747

* Add asset type error

* Respond to feedback

* Depends on 1474

* Correct type to standards track

* Correct spelling

* Add eip images locally (ethereum#2150)

* Automatically merged updates to draft EIP(s) 823 (ethereum#2151)


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* Automatically merged updates to draft EIP(s) 1193 (ethereum#2057)


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* ERC-1948: Non-fungible Data Token (ethereum#1948)

* first draft

* remove boilerplate

* fix indentations

* typos

* ethereum code highlights

* moved file to id

* syntax

* naming

* link to erc721

* make interface an interface

* typo

* better comments

* Automatically merged updates to draft EIP(s) 1679, 233 (ethereum#2081)


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* Automatically merged updates to draft EIP(s) 1702 (ethereum#2152)


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* Introduced Abandoned status in EIP-1 (ethereum#2051)

* Introduced Abandoned status in EIP-1

* Fix typo

* EIP-1967: Standard Proxy Storage Slots (ethereum#1967)

* Proxy storage slots EIP

* Apply suggestions from code review

Co-Authored-By: spalladino <spalladino@gmail.com>

* Add reference to vyper storage slots

* Fix links to EIPs

Co-Authored-By: spalladino <spalladino@gmail.com>

* Apply suggestions from code review

Co-Authored-By: spalladino <spalladino@gmail.com>

* Applied suggestions by elopio and nventuro

* Add discussions-to

* Add EIP number

* Rename eip file

* Fix type attribute

* Use zOS independent hash preimage

* Implement suggestion by axic

* Automatically merged updates to draft EIP(s) 1474 (ethereum#2155)


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* Automatically merged updates to draft EIP(s) 1613 (ethereum#2163)


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* Automatically merged updates to draft EIP(s) 1066 (ethereum#2164)


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* Mark EIP-158 proper superseded and not replaced (ethereum#2160)

* Update eip_validator to 0.8.2 (ethereum#2161)

* Fix Markdown formatting in EIP-1186

* Update eip_validator to 0.8.2

* Abandone EIP-1355 "Ethash 1a" (ethereum#1785)

* Add information EIP:  Common Prometheus metrics (ethereum#2159)

* Add common metrics EIP.

* Fix spelling error.

* Assign EIP number, added more information on what Prometheus does.

* Add link to prometheus website for further info.

* Fix link.

* Fix discussion link.

* Switch to standards track - interface instead of informational.

* Add motivation.

* superseded-by EIP 1 (ethereum#2165)

* Automatically merged updates to draft EIP(s) 1702 (ethereum#2167)


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* EIP-1753 - Define a smart contract interface for permits and licences (ethereum#1753)

* First draft

* Update eip-x.md

* Update eip-x.md

* Update eip-x.md

* Update eip-x.md

* Update eip-x.md

* Update eip-x.md

* Update eip-x.md

* Update eip-x.md

* Update authors

* Update eip-x.md

* Rename eip md

* Rollback

* Rename md

* Add copyright

* WIP: Re adding specificaiton

* Update example solidity

* Update eip-1753.md

* Update eip-1753.md

* Update eip-1753.md

* Update eip-1753.md

* Update eip-1753.md

* Update eip-1753.md

* Do not touch eip-X

* Fix typo

* EIP-1895: Improvements (ethereum#2153)

* EIP-1895: Better constants and address range description

Address @Arachnid's comments about presenting constants and how an address value is limited.

* EIP-1985: Fix a typo

* EIP-1985: Use "instructions" instead of "opcodes"

* eip-225: mark as final (ethereum#1954)

* set EIP875 to last call and correct errors (ethereum#1549)

* Automatically merged updates to draft EIP(s) 1571 (ethereum#2168)


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* Automatically merged updates to draft EIP(s) 999 (ethereum#2170)


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* Automatically merged updates to draft EIP(s) 2159 (ethereum#2174)


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* Automatically merged updates to draft EIP(s) 1884 (ethereum#2175)


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* Automatically merged updates to draft EIP(s) 2028 (ethereum#2176)


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* EIP-1901: Add OpenRPC Service Discovery To JSON-RPC Services (ethereum#1901)

* EIP-????: Add OpenRPC Service Discovery To JSON-RPC Services

* fix:

* fix(openrpc): add openrpc.json to eip assets

* fix(openrpc): reference eip asset

* Update EIPS/eip-1901.md - document link

* Delete openrpc.json

* Update eip-1901: move link to spec to top

* fix(1901):  preamble + author github usernames

* fix(1901): add copyright info

* fix(1901): add link to eth json rpc spec

* fix(1901): type and category in preamble + spelling

* fix(1901): motivation section and use vendored assets

* fix(1901): add link to multi-geth OpenRPC discovery

* fix(1901): change ordering + small fixes

* fix(1901): typo

* Automatically merged updates to draft EIP(s) 1108 (ethereum#2177)


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* Automatically merged updates to draft EIP(s) 1102 (ethereum#2178)


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* EIP-1900 Decentralized Type System for EVM (ethereum#1900)

* Automatically merged updates to draft EIP(s) 1884 (ethereum#2180)


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ilanolkies pushed a commit to ilanolkies/EIPs that referenced this pull request Nov 12, 2019
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