# Precompile for BN256 HashToCurve Algorithms

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###### 正文

## 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 and EIP-1108 increased usage of cryptographic operations in the EVM by reducing the gas costs. In particular, the cost reduction from EIP-1108 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.

At this time, we are able to perform addition, scalar multiplication, and pairing checks in BN256. Reducing these costs in EIP-1108 made 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 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; 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
with modifications based on Wahby and Boneh's
paper.
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
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.
Algorithms for computing `inverse`

, `is_square`

, and `sqrt`

in finite field GF(fieldPrime^2) can be found
here.

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, the 128-bit security promised by the BN256 elliptic curve no longer applies; this was mentioned in the Cloudflare BN256 library. There has been some discussion on the exact security decrease from this advancement; see these two papers 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, 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.