This CIP proposes an extension of the current plutus functions to provide support for basic operations over BLS12_381 curve to the plutus language. We expose a candidate implementation, and describe clearly the benefits that this would bring. In a nutshell, pairing friendly curves will enable a large number of cryptographic primitives that will be essential for the scalability of Cardano.
Pairing Friendly Curves are a type of curves that provide the functionality of computing pairings. A pairing is a binary function that maps two points from two groups to a third element in a third target group. For a more in-depth introduction to pairings, we recommend reading Pairings for Beginners or Pairings for Cryptographers. For level of detail required in this document, it is sufficient to understand that a pairing is a map: e:G1 X G2 -> GT, which satisfies the following properties:
- Bilinearity: for all a,b in F^*_q, for all P in G1, Q in G2: e(aP,bQ)=e(P,Q)^(ab)
- Non-degeneracy: e != 1
- Computability: There exists an efficient algorithm to compute e
where G1, G2 and GT are three distinct groups of order a prime q. Given that all three groups have the same order, we can refer to the scalars of each group using the same type/structure. Pairing computation is an expensive operation. However, they enable a range of interesting cryptographic primitives which can be used for scaling Cardano and many other use cases. We now provide a list of use cases of pairings as well as an estimate operation count to understand the number of 'expensive' operations that need to be computed for each of them (a preliminary benchmark can be found in Section 'Costing of function').
- Sidechains are a crucial component for the scalability of Cardano, and its interoperability with other
chains/tokens/smart contracts. However, sidechains need to periodically commit their state to the Cardano mainnet to
provide the same security guarantees as the latter. This periodical commitment is performed through a threshold
signature by the dynamic committee of the Sidechain. The most interesting construction for medium sized committees
is ATMS, presented in the paper Proof of Stake Sidechains,
in section 5.2 and requires pairings. We have yet not found an efficient solution that does not require pairings.
ATMS is a k-of-t threshold signature scheme (meaning that we need at least k signers to participate). Let
n BLS12_381G1Element -> BLS12_381G1Element`
* `BLS12_381_G1_mult :: Integer -> BLS12_381G1Element -> BLS12_381G1Element`
* `BLS12_381_G1_neg :: BLS12_381G1Element -> BLS12_381G1Element`
* `BLS12_381_H2G1 :: ByteString -> BLS12_381G1Element`
* `BLS12_381_G2_add :: BLS12_381G2Element -> BLS12_381G2Element -> BLS12_381G2Element`
* `BLS12_381_G2_mult :: Integer -> BLS12_381G2Element -> BLS12_381G2Element`
* `BLS12_381_G2_neg :: BLS12_381G2Element -> BLS12_381G2Element`
* `BLS12_381_H2G2 :: ByteString -> BLS12_381G2Element`
* `BLS12_381_GT_mul :: Blst12381GTElement -> Blst12381GTElement -> Blst12381GTElement`
- Pairing operations:
BLS12_381_ppairing_ml :: BLS12_381G1Element -> BLS12_381G2Element -> BLS12_381GTElement
BLS12_381_final_verify :: BLS12_381GTElement -> BLS12_381GTElement -> boolThis performs the final exponentiation (see section
An important note on GT elementsbelow).
On top of the elliptic curve operations, we also need to include deserialization functions, and equality definitions among the G1 and G2 types.
serialiseG1 :: Bls12_381G1Element -> ByteString
deserialiseG1 :: ByteString -> Bls12_381G1Element
serialiseG2 :: Bls12_381G2Element -> ByteString
deserialiseG2 :: ByteString -> Bls12_381G2Element
deserialiseGT :: ByteString -> Bls12_381GTElement
- Equality functions:
eq :: BLS12_381G1Element -> BLS12_381G1Element -> bool
eq :: BLS12_381G2Element -> BLS12_381G2Element -> bool
This makes a total of 18 new functions and three new types.
We follow the ZCash Bls12-381 specification for the serialization of elements:
- Fq elements are encoded in big-endian form. They occupy 48 bytes in this form.
- Fq2 elements are encoded in big-endian form, meaning that the Fq2 element c0 + c1 * u is represented by the Fq element c1 followed by the Fq element c0. This means Fq2 elements occupy 96 bytes in this form.
- The group G1 uses Fq elements for coordinates. The group G2 uses Fq2 elements for coordinates.
- G1 and G2 elements are encoded in compressed form (just the x-coordinate). G1 elements occupy 48 bytes and G2 elements occupy 96 bytes.
The most-significant three bits of a G1 or G2 encoding should be masked away before the coordinate(s) are interpreted. These bits are used to unambiguously represent the underlying element:
- The most significant bit, when set, indicates that the point is in compressed form.
- The second-most significant bit indicates that the point is at infinity. If this bit is set, the remaining bits of the group element's encoding should be set to zero.
- The third-most significant bit is set if (and only if) this point is in compressed form, and it is not the point at infinity and its y-coordinate is the lexicographically largest of the two associated with the encoded x-coordinate.
We include the serialisation of the generator of G1 and the generator of G2:
- generator G1:
[151, 241, 211, 167, 49, 151, 215, 148, 38, 149, 99, 140, 79, 169, 172, 15, 195, 104, 140, 79, 151, 116, 185,
5, 161,78, 58, 63, 23, 27, 172, 88, 108, 85, 232, 63, 249, 122, 26, 239, 251, 58, 240, 10, 219, 34, 198, 187]
- generator G2:
[147, 224, 43, 96, 82, 113, 159, 96, 125, 172, 211, 160, 136, 39, 79, 101, 89, 107, 208, 208, 153, 32, 182,
26, 181, 218, 97, 187, 220, 127, 80, 73, 51, 76, 241, 18, 19, 148, 93, 87, 229, 172, 125, 5, 93, 4, 43, 126,
2, 74, 162, 178, 240, 143, 10, 145, 38, 8, 5, 39, 45, 197, 16, 81, 198, 228, 122, 212, 250, 64, 59, 2, 180,
81, 11, 100, 122, 227, 209, 119, 11, 172, 3, 38, 168, 5, 187, 239, 212, 128, 86, 200, 193, 33, 189, 184]
An important note on GT elements
We intentionally limit what can be done with the GT element. In fact, the only way that one can generate a
GT element is using the
BLS12_381_ppairing_ml function. Then these elements can only be used for operations among
them (mult) and a final equality check (denote
final_verify). In other words, GT elements are only there to eventually
perform some equality checks. We thought of including the inverse
function to allow division, but given that we simply allow for equality checks, if one needs to divide by
A can simply move to the other side of the equality sign. These limitations allow us for a performance
trick, which is also used for the verification of BLS signatures. In a nutshell, a pairing is divided into two
operations: (i) Miller loop, and (ii) final exponentiation. Both are expensive, but what we can do is first
compute the miller loop, which already maps two points from G1 and G2 to GT. Then we can use this result
to perform the operations over these points (mults). Finally, when we want to check for equality, we invert
one of the two points (or equivalently in this case we compute the conjugate), and multiply the result by the
other point. Only now we compute the final exponentiation
and verify that the result is the identity element. In other words:
- the 'partial pairing' function,
BLS12_381_ppairing_ml(stands for partial pairing miller loop) simply computes the miller loop
- the equality check function,
final_verify, first computes an inverse, then a multiplication, a final exponentiation and checks that the element is the identity.
Using the estimates exposed in the section
Costing of function, we can see how this representation of
GT elements is beneficial. Assume that we want to check the following relation:
e(A,B) * e(C,D) =? e(E,F) * e(G,H). The miller look takes around 330us, and the final exponentiation
around 370us. A full pairing would be 700us, and therefore checking this relation would cost around
2,8ms. If, instead, we break down the pairing into a miller loop and a final exponentiation, and only
compute the latter once, the cost of the relation above would be 330 * 4 + 370 = 1,7ms.
For this reason it is important to limit what can be done with GTElements, as the pairing really is not the full pairing operation, but only the miller loop.
- BLST library, providing the algebraic operations.
- cardano-base with the haskell FFI to the BLST library.
Other libraries of interest
- Ethereum support for BLS12_381. Not directly relevant as this is an Ethereum Improvement Proposal for a precompiled solidity contracts.
Comparison with existing function
We present what would be the alternatives of using pairings in the different use cases presented above.
- Sidechain bridges using the current technology would rely on either of the two possibilities:
- Require the bridge committee to interact during signature, or to rely on a precomputation phase. Current solutions only support non-robust signature schemes, meaning that if one signer misbehaves, the whole signature procedure needs to be restarted. This could seriously hinder sidechains.
- Non-aggregation of signatures. This would result in a linear "checkpoint certificate" with respect to the number of signers (both in communication and computation complexity). Basically, all committee members need to submit their signature, and the smart contract needs to verify all ed25519 signatures.
- Zero Knowledge Proofs cannot be verified with current functions available in Plutus. There exists proofs that can be instantiated over non-pairing friendly curves, but these result in logarithmic sized proofs and linear verification with respect to the computation to prove, while solutions that rely on pairings can be represented more concisely, and are cheaper to verify.
Reason for exposing curve operations API
One might be concerned of why we are exposing such low-level primitives, instead of exposing higher level protocol
functions, such as
VerifyZKP. The motivation behind that is because pairings can enable a
big number of use cases, and covering all of those can considerably extend the list of required functions.
BLS12 curve is fully defined by the following set of parameters (coefficient A=0 for all BLS12 curves). Taken from EIP 2537:
Base field modulus = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab
B coefficient = 0x000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004
Main subgroup order = 0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001
Fp quadratic non-residue = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaaa
Fp2 cubic non-residue c0 = 0x000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001
Fp2 cubic non-residue c1 = 0x000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001
Twist type: M
B coefficient for twist c0 = 0x000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004
B coefficient for twist c1 = 0x000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004
X = 0x17f1d3a73197d7942695638c4fa9ac0fc3688c4f9774b905a14e3a3f171bac586c55e83ff97a1aeffb3af00adb22c6bb
Y = 0x08b3f481e3aaa0f1a09e30ed741d8ae4fcf5e095d5d00af600db18cb2c04b3edd03cc744a2888ae40caa232946c5e7e1
X c0 = 0x024aa2b2f08f0a91260805272dc51051c6e47ad4fa403b02b4510b647ae3d1770bac0326a805bbefd48056c8c121bdb8
X c1 = 0x13e02b6052719f607dacd3a088274f65596bd0d09920b61ab5da61bbdc7f5049334cf11213945d57e5ac7d055d042b7e
Y c0 = 0x0ce5d527727d6e118cc9cdc6da2e351aadfd9baa8cbdd3a76d429a695160d12c923ac9cc3baca289e193548608b82801
Y c1 = 0x0606c4a02ea734cc32acd2b02bc28b99cb3e287e85a763af267492ab572e99ab3f370d275cec1da1aaa9075ff05f79be
|x| (miller loop scalar) = 0xd201000000010000
x is negative = true
One should note that base field modulus is equal to 3 mod 4 that allows an efficient square root extraction.
The reason for choosing BLS12_381 over BN256 curve is that the former is claimed to provide 128 bits of security, while the latter was reduced to 100 bits of security after the extended number field sieve (a new algorithm to compute the discrete logarithm) was shown to reduce the security of these curves.
An EIP for precompiles of curve BLS12_381 already exists, but has been stagnant for a while. Nonetheless, Zcash, MatterLabs and Consensys support BLS12_381 curve, so it is certainly widely used in the space.
Further reading regarding curve BLS12_381 can be found here and the references thereof cited.
Path to Proposed
To move this draft to proposed, we need to argue the trustworthiness of
blst library, cost all functions and
include these functions to Plutus. Furthermore, before proceeding to
Proposed, we will provide a Haskell implementation
of one of the algorithms that we want to do on-chain, implemented in terms of the primitives we are going to provide.
This will help in benchmarking the viability of using these primitives on main-net.
Trustworthiness of implementations
Costing of function
We performed some benchmarks on the curve operations that can be used to cost each of the functions. We performed
the benchmarks on a 2,7 GHz Quad-Core Intel Core i7, using the
blst library. Further benchmarks are required
to provide final costings of functions.
Deserialization functions check that elements are part of the prime order subgroup. The pairing evaluation only
computes the miller loop, and the final verify of GTs computes an inversion in GT, a multiplication, a final
exponentiation and a check wrt the identity element (more info in section An important note on GT elements).
- Group operations:
- G1 addition: 806 ns
- G1 multiplication: 20,5 us
- G1 negation: 12 ns
- G1 Hash to group: 61,8 us
- G2 addition: 1,6 us
- G2 multiplication: 40,5 us
- G2 negation: 18 ns
- G2 Hash to group: 167,2 us
- GT multiplication: 2 us
- Pairing operations:
- Miller loop: 330,2 us
- Final verify: 371.2 us
- G1: 63,8 us
- G2: 77 us
- Equality checks:
- G1: 228 ns
- G2: 656 ns
IOHK internal. We currently have implemented the FFI binding in
Haskell implementation of ATMS
We will provide a Haskell implementation of ATMS verification to understand the complexity of such a procedure.
Path to Active
Release in upcoming update.