From c50e893c003cf0117f3546140f51374d9b664015 Mon Sep 17 00:00:00 2001 From: chgormanMH <57261628+chgormanMH@users.noreply.github.com> Date: Sun, 1 Nov 2020 00:27:23 -0400 Subject: [PATCH] New EIP to add precompile for BN256 hash-to-curve algorithms. (#3068) This PR adds an EIP for precompile contracts for BN256 hash-to-curve algorithms. --- EIPS/eip-3068.md | 297 ++++++++++++++++++ .../2012-685_Square_Root_Even_Ext.pdf | Bin 0 -> 433498 bytes assets/eip-3068/2015-1027_exTNFS.pdf | Bin 0 -> 477103 bytes .../2016-1102_Assessing_NFS_Advances.pdf | Bin 0 -> 364232 bytes assets/eip-3068/2017-334.pdf | Bin 0 -> 493800 bytes assets/eip-3068/2019-403_BLS12_H2C.pdf | Bin 0 -> 629241 bytes assets/eip-3068/latincrypt12.pdf | Bin 0 -> 444936 bytes assets/eip-3068/madnet.pdf | Bin 0 -> 659334 bytes assets/eip-3068/weilsigs.pdf | Bin 0 -> 299722 bytes 9 files changed, 297 insertions(+) create mode 100644 EIPS/eip-3068.md create mode 100644 assets/eip-3068/2012-685_Square_Root_Even_Ext.pdf create mode 100644 assets/eip-3068/2015-1027_exTNFS.pdf create mode 100644 assets/eip-3068/2016-1102_Assessing_NFS_Advances.pdf create mode 100644 assets/eip-3068/2017-334.pdf create mode 100644 assets/eip-3068/2019-403_BLS12_H2C.pdf create mode 100644 assets/eip-3068/latincrypt12.pdf create mode 100644 assets/eip-3068/madnet.pdf create mode 100644 assets/eip-3068/weilsigs.pdf diff --git a/EIPS/eip-3068.md b/EIPS/eip-3068.md new file mode 100644 index 00000000000000..6a27b8bbf416dc --- /dev/null +++ b/EIPS/eip-3068.md @@ -0,0 +1,297 @@ +--- +eip: 3068 +title: Precompile for BN256 HashToCurve Algorithms +author: Dr. Christopher Gorman (@chgormanMH) +discussions-to: https://ethereum-magicians.org/t/pre-compile-for-bls/3973 +status: Draft +type: Standards Track +category: Core +created: 2020-10-23 +requires: 198, 1108 +--- + +## Simple Summary +This EIP defines a hash-to-curve precompile for use in BN256 +and would allow for cheaper BLS signature verification. + +## Abstract +There is currently no inexpensive way to perform BLS signature +verification for arbitrary messages. +This stems from the fact that there is no precompiled contract +in the EVM for a hash-to-curve algorithm for the BN256 elliptic curve. +The gas cost of calling a deterministic hash-to-curve algorithm +written in Solidity is approximately that of one pairing check, +although the latter requires an order of magnitude +more computation. +This EIP remedies this by implementing a hash-to-curve algorithm +for the BN256 G1 curve, which would reduce the cost of +signature verification to essentially that of the pairing check +precompiled contract. +We also include a hash-to-curve algorithm for the BN256 G2 group. + +## Motivation +The precompiled contracts in +[EIP-198](./eip-198.md) and +[EIP-1108](./eip-1108.md) +increased usage of cryptographic operations in the EVM +by reducing the gas costs. +In particular, the cost reduction from +[EIP-1108](./eip-1108.md) +helps increase the use of SNARKs in Ethereum +via an elliptic curve pairing check; +however, a hash-to-curve algorithm enabling arbitrary +BLS signature verification on BN256 in the EVM was noticeably missing. +There is interest in having a precompiled contract which would allow +for signature verification, as noted +[here](https://ethereum-magicians.org/t/pre-compile-for-bls/3973). + +At this time, we are able to perform addition, scalar multiplication, +and pairing checks in BN256. +Reducing these costs in +[EIP-1108](./eip-1108.md) +made [ETHDKG](https://github.com/PhilippSchindler/ethdkg), +a distributed key generation protocol in Ethereum, +less expensive. +ETHDKG by itself is useful; however, what it is lacking is +the ability to verify arbitrary BLS signatures. +Creating group signatures by aggregating partial signatures +is one goal of a DKG protocol. +The DKG enables the computation of partial signatures to be +combined into a group signature offline, but there is no +easy way to verify partial signatures or group signatures +in the EVM. + +In order to perform BLS signature validation, a hash-to-curve +algorithm is required. +While the MapToGroup method initially discussed in the original BLS +[paper](../assets/eip-3068/weilsigs.pdf) +works in practice, the nondeterministic nature of the algorithm +leaves something to be desired as we would like to bound +the overall computational cost in the EVM. +A deterministic method for mapping to BN curves is given +[here](../assets/eip-3068/latincrypt12.pdf); +in the paper, Fouque and Tibouchi proved their mapping +was indifferentiable from a random oracle. +This gives us the desired algorithm. + +## Specification +Here is the pseudocode for the `HashToG1` function: + +``` +function HashToG1(msg) + fieldElement0 = HashToBase(msg, 0x00, 0x01) + fieldElement1 = HashToBase(msg, 0x02, 0x03) + curveElement0 = BaseToG1(fieldElement0) + curveElement1 = BaseToG1(fieldElement1) + g1Element = ECAdd(curveElement0, curveElement1) + return g1Element +end function +``` + +Here is the pseudocode for `HashToBase`; +`msg` is the byte slice to be hashed while `dsp1` and `dsp2` +are domain separation parameters. +`fieldPrime` is the prime of the underlying field. + +``` +function HashToBase(msg, dsp1, dsp2) + hashResult0 = uint256(Keccak256(dsp1||msg)) + hashResult1 = uint256(Keccak256(dsp2||msg)) + constant = 2^256 mod fieldPrime + fieldElement0 = hashResult0*constant mod fieldPrime + fieldElement1 = hashResult1 mod fieldPrime + fieldElement = fieldElement0 + fieldElement1 mod fieldPrime + return fieldElement +end function +``` + +Here is the pseudocode for `BaseToG1`. +All of these operations are performed in the finite field. +`inverse` computes the multiplicative inverse in the underlying +finite field; we have the convention `inverse(0) == 0`. +`is_square(a)` computes the Legendre symbol of the element, +returning 1 if `a` is a square, -1 if `a` is not a square, +and 0 if `a` is 0. +`sqrt` computes the square root of the element in the finite +field; a square root is assumed to exist. +`sign0` returns the sign of the finite field element. + +``` +function BaseToG1(t) + # All operations are done in the finite field GF(fieldPrime) + # Here, the elliptic curve satisfies the equation + # y^2 == g(x) == x^3 + curveB + constant1 = (-1 + sqrt(-3))/2 + constant2 = -3 + constant3 = 1/3 + constant4 = g(1) + s = (constant4 + t^2)^3 + alpha = inverse(t^2*(constant4 + t^2)) + x1 = constant1 - constant2*t^4*alpha + x2 = -1 - x1 + x3 = 1 - constant3*s*alpha + a1 = x1^3 + curveB + a2 = x2^3 + curveB + residue1 = is_square(a1) + residue2 = is_square(a2) + index = (residue1 - 1)*(residue2 - 3)/4 + 1 + coef1 = ConstantTimeEquality(1, index) + coef2 = ConstantTimeEquality(2, index) + coef3 = ConstantTimeEquality(3, index) + x = coef1*x1 + coef2*x2 + coef3*x3 + y = sign0(t)*sqrt(x^3 + curveB) + return (x, y) +end function + +function sign0(t) + if t <= (fieldPrime-1)/2 + return 1 + else + return fieldPrime-1 + end if +end function + +function ConstantTimeEquality(a, b) + # This function operates in constant time + if a == b + return 1 + else + return 0 + end if +end function +``` + +In `HashToG2`, we first map to the underlying twist curve +and then clear the cofactor to map to G2. +Here is the pseudocode for `HashToG2`: + +``` +function HashToG2(msg) + fieldElement00 = HashToBase(msg, 0x04, 0x05) + fieldElement01 = HashToBase(msg, 0x06, 0x07) + fieldElement10 = HashToBase(msg, 0x08, 0x09) + fieldElement11 = HashToBase(msg, 0x0a, 0x0b) + fieldElement0 = (fieldElement00, fieldElement01) + fieldElement1 = (fieldElement10, fieldElement11) + twistElement0 = BaseToTwist(fieldElement0) + twistElement1 = BaseToTwist(fieldElement1) + twistElement = ECAdd(twistElement0, twistElement1) + g2Element = ClearCofactor(twistElement) + return g2Element +end function + +function ClearCofactor(twistElement) + return ECMul(twistElement, cofactor) +end function +``` + +Here is the pseudocode for `BaseToTwist`. + +``` +function BaseToTwist(t) + # All operations are done in the finite field GF(fieldPrime^2) + # Here, the twist curve satisfies the equation + # y^2 == g'(x) == x^3 + curveBPrime + constant1 = (-1 + sqrt(-3))/2 + constant2 = -3 + constant3 = 1/3 + constant4 = g'(1) + s = (constant4 + t^2)^3 + alpha = inverse(t^2*(constant4 + t^2)) + x1 = constant1 - constant2*t^4*alpha + x2 = -1 - x1 + x3 = 1 - constant3*s*alpha + a1 = x1^3 + curveBPrime + a2 = x2^3 + curveBPrime + residue1 = is_square(a1) + residue2 = is_square(a2) + index = (residue1 - 1)*(residue2 - 3)/4 + 1 + coef1 = ConstantTimeEquality(1, index) + coef2 = ConstantTimeEquality(2, index) + coef3 = ConstantTimeEquality(3, index) + x = coef1*x1 + coef2*x2 + coef3*x3 + y = sign0(t)*sqrt(x^3 + curveBPrime) + return (x, y) +end function +``` + +## Rationale +The BaseToG1 algorithm is based on the original Fouque and Tibouchi +[paper](../assets/eip-3068/latincrypt12.pdf) +with modifications based on Wahby and Boneh's +[paper](../assets/eip-3068/2019-403_BLS12_H2C.pdf). +There is freedom in choosing the HashToBase function +and this could easily be changed. +Within HashToBase, the particular hashing algorithm +(Keccak256 in our case) could also be modified. +It may be desired to change the call to `sign0` +at the end of BaseToG1 and BaseToTwist with `is_square`, +as this would result in the same deterministic map to curve from the +Fouque and Tibouchi +[paper](../assets/eip-3068/latincrypt12.pdf) +and ensure HashToG1 is indifferentiable from a random oracle; +they proved this result in their paper. +It may be possible to show that switching the `is_square` +call with `sign0` does not affect indifferentiability, +although this has not been proven. + +The HashToG2 algorithm follows from the Wahby and Boneh +[paper](../assets/eip-3068/2019-403_BLS12_H2C.pdf). +Algorithms for computing `inverse`, `is_square`, and `sqrt` +in finite field GF(fieldPrime^2) can be found +[here](../assets/eip-3068/2012-685_Square_Root_Even_Ext.pdf). + +We now discuss the potential gas cost for the HashToG1 +and HashToG2 operations. +On a local machine, ECMul was clocked at 68 microseconds +per operation. +The same machine clocked HashToG1 at 94 microseconds per operation +when hashing 32 bytes into G1 and 105 microseconds per operation +when hashing 1024 bytes into G1. +Given that it currently costs 6000 gas for ECMul, this gives us +an estimated gas cost for HashToG1 at `8500 + len(bytes)`. +Similarly, HashToG2 was clocked at 886 microseconds per operation +when hashing 32 bytes into G2 and 912 microseconds per operation when +hashing 1024 bytes into G2. +This allows us to estimate the gas cost at `80000 + 3*len(bytes)`. + +## Backwards Compatibility +There are no backward compatibility concerns. + +## Test Cases +TBD + +## Implementation +TBD + +## Security Considerations +Due to recent [work](../assets/eip-3068/2015-1027_exTNFS.pdf), the +128-bit security promised by the BN256 elliptic curve no longer applies; +this was mentioned in the Cloudflare BN256 +[library](https://github.com/cloudflare/bn256). +There has been some discussion on the exact security decrease +from this advancement; see these +[two](../assets/eip-3068/2016-1102_Assessing_NFS_Advances.pdf) +[papers](../assets/eip-3068/2017-334.pdf) +for different estimates. +The more conservative estimate puts the security of BN256 at +100-bits. +While this is likely still out of reach for many adversaries, +it should give us pause. +This reduced security was noted in the recent MadNet +[whitepaper](../assets/eip-3068/madnet.pdf), +and this security concern was partially mitigated by +requiring Secp256k1 signatures of the partial group signatures +in order for those partial signatures to be valid. +Full disclosure: the author of this EIP works for MadHive, +assisted in the development of MadNet, and +helped write the MadNet whitepaper. + +The security concerns of the BN256 elliptic curve +affect any operation using pairing check because it is +related to the elliptic curve pairing; +they are independent of this EIP. + +## Copyright +Copyright and related rights waived via +[CC0](https://creativecommons.org/publicdomain/zero/1.0/). diff --git a/assets/eip-3068/2012-685_Square_Root_Even_Ext.pdf b/assets/eip-3068/2012-685_Square_Root_Even_Ext.pdf new file mode 100644 index 0000000000000000000000000000000000000000..077e093b4927dcf37cff12b1565833f4a638f74d GIT binary patch literal 433498 zcmb@NV~}Obx~{9bY_kiqY}>9b+qP{RT~*zMF59+k+qP}{_F8+bd)JP0Zp1zjXTtBLq0?-3&4J@F!xdC)`rY3d{rUn2_IyrqO zCu0X20G+V0i@Bk(!Z$%}0531Jv5nCm_p<&g2Efe?O($h+W9noEV5Dbb0sMJE(}|i} z{SF4uiCXD984DX5+8P-{{|*V?2F(;m#63s^d_ut#20N&z(N9aq6^*rg4F3c={M0e;LoT zDRq+XQ|^8;c&{7!#cjf^i7MW#W{VMpr0b*g`3`scwd3J3?>lh?yJT6Ca>{iRkac{X zeGl}n`>SW_pBfdveu#*kW{B4FW=yo7eO*TupmRrmAN(K}Laj#y$;Ty6dRr)Ms9nw4 z2~btnLWhbxuump++JP7d;kFOL*S+pF95v?flsYky}ur^Ha! z+9!0qpxC9J#89UOPkl-=n*c=9sQ_nMH4;BA3^mxaXw2TVZ(z@4-m=Tif<=VV;0tH8 zkcd&~t5x-YZl=%ss`@I@oKJ|P84a`9e*74(ESFTHW=AMTk?E5_)!Vs>8`_C8-ZX1= zJznoUuMx{VrDA+aY{0Ueb&|e}Brj#JEPHuMKS3CeAan+Ywd4)jz}s^_9b_A+n{mVV zG)y+0zD1OIqqJ-Du)YtE;GWR`8sV-C=HeS@ZL56-um2*)HiEZJdO0Va@=>(1k@@*> z@v(S`w=hZ>R*}d(oTHnVR&D0|INN^sG>w&3#f`4c>Y6x#q3)J^(g+^IoR~A!k))RF z7&K&{%rAN|yX!G~FS_u)2%3iiEP{L}*d=eAMy%80vFRk2aeG88JAvS0rUcRY9+1bE zdLVjkJrMb5no0(0f(6aj2K>$$Ml1>Cy> z{Q&!kvh}^XbST%CM)IJJhEc6hwnedS%D0X4 zBvL~^q8CN9jG6$3VL0rxHnCQ0w`8vZ8Q`6ycGXnUH#&zsDfpAz@#})1n>Z^`IN{5e z#phV}B26TWEu&b)-Za`6z;V^bA{G>o=0K_cF0-LFI5_aVi0CiQ@Xq- z^8pF?EqciZC-sj8r`q43?0VYHQ-N8#rs^CSCXo5nAADhN_EH11KsRXovp2r6 z#*XWK^v~+}US5?$0J&*Dj)cOyXWtc>x6^Q4M$z%FC94J8#s~poZu|S{#O31cGY$ph zu9ve$o-q~8Fh-6b<3Z=@YY8Px-~1=&LCC8ldkV{HPOwSzEuqz%57aCQEv#S3xLnw* zAv}$S1=@{BM%#-xMyFSsBJ_Nhj9i9u_gt6LK$nft+WGL~%`Ni4)cvGl3XZ9LTDisT zMApzZ=w(~l8~m^OUEnFk5-UR zy=P@l8~3v|Slfq=@22UNPp=50R@`ztYLKn!Rzr$2sU+x!8f`ZKtO-NIr`ts@?Hex4 z{^~M~qM2NwI9c8n!}8W=m$_Ams>aDzQ9xWS($d3T*j_amij#Z%wePn}u#sPxwzD=j z_o6&JdBvD2i!Lt$c>5-cEDQrEGx-sDZXmfQ5?((_8nU%f2VW`nn3(8M~is z*t7{Nw-L|1JH$tr{nW*4|0W!Mlo@$5A@5_q+Xy_F{uO2NJ*}TgC!gUENXVaj;{Dam znv3W;3_F?H%!nQjJNZnRD0Uykb#GT_d>s3e?zoaSD@iV3$=s9I2AtwMe2_99Z4TEe z&YZnke%%l-y3U?xarXIFn~a}qbd8!uqK7vD`O^afh@&~VeH>5ba#LEW!7S=u*Pj{s z$uTX1ZkL%y?tiK^Sss5DHM(UX%0@A6YPYWyMM*MG7#6~$oUWSH!q8@tW_MuNCgwIu z>QeJ7tcwfm2abY`=D=H&qQWBafY`OB&Qqx;r-qzb->O;n>~RGMgTs0l152Ct7G9kDQQdzb^L+@W(4 z6p&{^>q>e|Ax~sxwPc12L|zh{G{!PTgb%OcMZtqe$m0N73`LeeTxyq~224I;Y_)az z_2_m4Ih-6b?xriC6&XYU2;cmQ!;B@Bgg5`!aP7sf3~?jf6Y29`usvQj5*n-K=hIsO zTv_nV?!SKekCgtr%ZgD0J#j8a!28-TQp{P_(d0@cnCW%i< z1Lpe>riVAv&XYltbSX0}fQ8`UnXqekfQ*dIuR@8t3s@e~h&oM5qzX z1gIk1>y(fs`RkyftUuOTaM@vH!3S>hzevnB_)sm>Kz#vomvD;(zg8%Sb?bHt>bVAk zyMgfy%#pJUWq7dq*?(VYVZ$ev!6uWdnu*&hc)=C@4FcJwZ`3}?=LgxQnvR#V^f{a4 zYYfbNM&ifnOzY!f|*XAJnQOiYcT>4a=; 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