EIP-2537: Precompile for BLS12-381 curve operations Source

AuthorAlex Vlasov
Discussions-Tohttps://ethereum-magicians.org/t/eip2537-bls12-precompile-discussion-thread/4187
StatusDraft
TypeStandards Track
CategoryCore
Created2020-02-21

Simple Summary

This precompile adds operation on BLS12-381 curve as a precompile in a set necessary to efficiently perform operations such as BLS signature verification and perform SNARKs verifications.

Abstract

If block.number >= X we introduce nine separate precompiles to perform the following operations:

  • BLS12_G1ADD - to perform point addition on a curve defined over prime field
  • BLS12_G1MUL - to perform point multiplication on a curve defined over prime field
  • BLS12_G1MULTIEXP - to perform multiexponentiation on a curve defined over prime field
  • BLS12_G2ADD - to perform point addition on a curve twist defined over quadratic extension of the base field
  • BLS12_G2MUL - to perform point multiplication on a curve twist defined over quadratic extension of the base field
  • BLS12_G2MULTIEXP - to perform multiexponentiation on a curve twist defined over quadratic extension of the base field
  • BLS12_PAIRING - to perform a pairing operations between a set of pairs of (G1, G2) points
  • BLS12_MAP_FP_TO_G1 - maps base field element into the G1 point
  • BLS12_MAP_FP2_TO_G2 - maps extension field element into the G2 point

Mapping functions are implemented according to IEFT specification version v7(!) using an simplified SWU method. It does NOT perform mapping of the byte string into field element that can be implemented in many different ways and can be efficiently performed in EVM, but only does field arithmetic to map field element into curve point. Such functionality is required for signature schemes.

Multiexponentiation operation is included to efficiently aggregate public keys or individual signer’s signatures during BLS signature verification.

Proposed addresses table

Precompile Address
BLS12_G1ADD 0x0a
BLS12_G1MUL 0x0b
BLS12_G1MULTIEXP 0x0c
BLS12_G2ADD 0x0d
BLS12_G2MUL 0x0e
BLS12_G2MULTIEXP 0x0f
BLS12_PAIRING 0x10
BLS12_MAP_FP_TO_G1 0x11
BLS12_MAP_FP2_TO_G2 0x12

Motivation

Motivation of this precompile is to add a cryptographic primitive that allows to get 120+ bits of security for operations over pairing friendly curve compared to the existing BN254 precompile that only provides 80 bits of security.

Specification

Curve parameters:

BLS12 curve is fully defined by the following set of parameters (coefficient A=0 for all BLS12 curves):

Base field modulus = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab
B coefficient = 0x000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004
Main subgroup order = 0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001
Extension tower
Fp2 construction:
Fp quadratic non-residue = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaaa
Fp6/Fp12 construction:
Fp2 cubic non-residue c0 = 0x000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001
Fp2 cubic non-residue c1 = 0x000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001
Twist parameters:
Twist type: M
B coefficient for twist c0 = 0x000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004
B coefficient for twist c1 = 0x000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004
Generators:
G1:
X = 0x17f1d3a73197d7942695638c4fa9ac0fc3688c4f9774b905a14e3a3f171bac586c55e83ff97a1aeffb3af00adb22c6bb
Y = 0x08b3f481e3aaa0f1a09e30ed741d8ae4fcf5e095d5d00af600db18cb2c04b3edd03cc744a2888ae40caa232946c5e7e1
G2:
X c0 = 0x024aa2b2f08f0a91260805272dc51051c6e47ad4fa403b02b4510b647ae3d1770bac0326a805bbefd48056c8c121bdb8
X c1 = 0x13e02b6052719f607dacd3a088274f65596bd0d09920b61ab5da61bbdc7f5049334cf11213945d57e5ac7d055d042b7e
Y c0 = 0x0ce5d527727d6e118cc9cdc6da2e351aadfd9baa8cbdd3a76d429a695160d12c923ac9cc3baca289e193548608b82801
Y c1 = 0x0606c4a02ea734cc32acd2b02bc28b99cb3e287e85a763af267492ab572e99ab3f370d275cec1da1aaa9075ff05f79be
Pairing parameters:
|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, although as described below gas cost of decompression is larger than gas cost of supplying decompressed point data in calldata.

Fine points and encoding of base elements

Field elements encoding:

To encode points involved in the operation one has to encode elements of the base field and the extension field.

Base field element (Fp) is encoded as 64 bytes by performing BigEndian encoding of the corresponding (unsigned) integer (top 16 bytes are always zeroes). 64 bytes are chosen to have 32 byte aligned ABI (representable as e.g. bytes32[2] or uint256[2]). Corresponding integer must be less than field modulus.

For elements of the quadratic extension field (Fp2) encoding is byte concatenation of individual encoding of the coefficients totaling in 128 bytes for a total encoding. For an Fp2 element in a form el = c0 + c1 * v where v is formal quadratic non-residue and c0 and c1 are Fp elements the corresponding byte encoding will be encode(c0) || encode(c1) where || means byte concatenation (or one can use bytes32[4] or uint256[4] in terms of Solidity types).

Note on the top 16 bytes being zero: it’s required that encoded element is “in a field” that means strictly < modulus. In BigEndian encoding it automatically means that for a modulus that is just 381 bit long top 16 bytes in 64 bytes encoding are zeroes and it must be checked if only a subslice of input data is used for actual decoding.

If encodings do not follow this spec anywhere during parsing in the precompile the precompile must return an error.

Encoding of points in G1/G2:

Points in either G1 (in base field) or in G2 (in extension field) are encoded as byte concatenation of encodings of the x and y affine coordinates. Total encoding length for G1 point is thus 128 bytes and for G2 point is 256 bytes.

Point of infinity encoding:

Also referred as “zero point”. For BLS12 curves point with coordinates (0, 0) (formal zeroes in Fp or Fp2) is not on the curve, so encoding of such point (0, 0) is used as a convention to encode point of infinity.

Encoding of scalars for multiplication operation:

Scalar for multiplication operation is encoded as 32 bytes by performing BigEndian encoding of the corresponding (unsigned) integer. Corresponding integer is not required to be less than or equal than main subgroup size.

Behavior on empty inputs:

Certain operations have variable length input, such as multiexponentiations (takes a list of pairs (point, scalar)), or pairing (takes a list of (G1, G2) points). While their behavior is well-defined (from arithmetic perspective) on empty inputs, this EIP discourages such use cases and variable input length operations must return an error if input is empty.

ABI for operations

ABI for G1 addition

G1 addition call expects 256 bytes as an input that is interpreted as byte concatenation of two G1 points (128 bytes each). Output is an encoding of addition operation result - single G1 point (128 bytes).

Error cases:

  • Either of points being not on the curve must result in error
  • Field elements encoding rules apply (obviously)
  • Input has invalid length
ABI for G1 multiplication

G1 multiplication call expects 160 bytes as an input that is interpreted as byte concatenation of encoding of G1 point (128 bytes) and encoding of a scalar value (32 bytes). Output is an encoding of multiplication operation result - single G1 point (128 bytes).

Error cases:

  • Point being not on the curve must result in error
  • Field elements encoding rules apply (obviously)
  • Input has invalid length
ABI for G1 multiexponentiation

G1 multiexponentiation call expects 160*k bytes as an input that is interpreted as byte concatenation of k slices each of them being a byte concatenation of encoding of G1 point (128 bytes) and encoding of a scalar value (32 bytes). Output is an encoding of multiexponentiation operation result - single G1 point (128 bytes).

Error cases:

  • Any of G1 points being not on the curve must result in error
  • Field elements encoding rules apply (obviously)
  • Input has invalid length
  • Input is empty
ABI for G2 addition

G2 addition call expects 512 bytes as an input that is interpreted as byte concatenation of two G2 points (256 bytes each). Output is an encoding of addition operation result - single G2 point (256 bytes).

Error cases:

  • Either of points being not on the curve must result in error
  • Field elements encoding rules apply (obviously)
  • Input has invalid length
ABI for G2 multiplication

G2 multiplication call expects 288 bytes as an input that is interpreted as byte concatenation of encoding of G2 point (256 bytes) and encoding of a scalar value (32 bytes). Output is an encoding of multiplication operation result - single G2 point (256 bytes).

Error cases:

  • Point being not on the curve must result in error
  • Field elements encoding rules apply (obviously)
  • Input has invalid length
ABI for G2 multiexponentiation

G2 multiexponentiation call expects 288*k bytes as an input that is interpreted as byte concatenation of k slices each of them being a byte concatenation of encoding of G2 point (256 bytes) and encoding of a scalar value (32 bytes). Output is an encoding of multiexponentiation operation result - single G2 point (256 bytes).

Error cases:

  • Any of G2 points being not on the curve must result in error
  • Field elements encoding rules apply (obviously)
  • Input has invalid length
  • Input is empty
ABI for pairing

Pairing call expects 384*k bytes as an inputs that is interpreted as byte concatenation of k slices. Each slice has the following structure:

  • 128 bytes of G1 point encoding
  • 256 bytes of G2 point encoding

Output is a 32 bytes where first 31 bytes are equal to 0x00 and the last byte is 0x01 if pairing result is equal to multiplicative identity in a pairing target field and 0x00 otherwise.

Error cases:

  • Any of G1 or G2 points being not on the curve must result in error
  • Any of G1 or G2 points are not in the correct subgroup
  • Field elements encoding rules apply (obviously)
  • Input has invalid length
  • Input is empty
ABI for mapping Fp element to G1 point

Field-to-curve call expects 64 bytes an an input that is interpreted as a an element of the base field. Output of this call is 128 bytes and is G1 point following respective encoding rules.

Error cases:

  • Input has invalid length
  • Input is not a valid field element
ABI for mapping Fp2 element to G2 point

Field-to-curve call expects 128 bytes an an input that is interpreted as a an element of the quadratic extension field. Output of this call is 256 bytes and is G2 point following respective encoding rules.

Error cases:

  • Input has invalid length
  • Input is not a valid field element

Gas burinig on error

Following the current state of all other precompiles if call to one of the precompiles in this EIP results in an error then all the gas supplied along with a CALL or STATICCALL is burned.

DDoS protection

Sane implementation of this EIP should not contain infinite cycles (it is possible and not even hard to implement all the functionality without while cycles) and gas schedule accurately reflects a time spent on computations of the corresponding function (precompiles pricing reflects an amount of gas consumed in the worst case where such case exists).

Gas schedule

Assuming a constant 30 MGas/second following prices are suggested.

G1 addition

500 gas

G1 multiplication

12000 gas

G2 addition

800 gas

G2 multiplication

45000 gas

G1/G2 Multiexponentiation

Multiexponentiations are expected to be performed by the Peppinger algorithm (we can also say that is must be performed by Peppinger algorithm to have a speedup that results in a discount over naive implementation by multiplying each pair separately and adding the results). For this case there was a table prepared for discount in case of k <= 128 points in the multiexponentiation with a discount cup max_discount for k > 128.

To avoid non-integer arithmetic call cost is calculated as k * multiplication_cost * discount / multiplier where multiplier = 1000, k is a number of (scalar, point) pairs for the call, multiplication_cost is a corresponding single multiplication call cost for G1/G2.

Discounts table as a vector of pairs [k, discount]:

[[1, 1200], [2, 888], [3, 764], [4, 641], [5, 594], [6, 547], [7, 500], [8, 453], [9, 438], [10, 423], [11, 408], [12, 394], [13, 379], [14, 364], [15, 349], [16, 334], [17, 330], [18, 326], [19, 322], [20, 318], [21, 314], [22, 310], [23, 306], [24, 302], [25, 298], [26, 294], [27, 289], [28, 285], [29, 281], [30, 277], [31, 273], [32, 269], [33, 268], [34, 266], [35, 265], [36, 263], [37, 262], [38, 260], [39, 259], [40, 257], [41, 256], [42, 254], [43, 253], [44, 251], [45, 250], [46, 248], [47, 247], [48, 245], [49, 244], [50, 242], [51, 241], [52, 239], [53, 238], [54, 236], [55, 235], [56, 233], [57, 232], [58, 231], [59, 229], [60, 228], [61, 226], [62, 225], [63, 223], [64, 222], [65, 221], [66, 220], [67, 219], [68, 219], [69, 218], [70, 217], [71, 216], [72, 216], [73, 215], [74, 214], [75, 213], [76, 213], [77, 212], [78, 211], [79, 211], [80, 210], [81, 209], [82, 208], [83, 208], [84, 207], [85, 206], [86, 205], [87, 205], [88, 204], [89, 203], [90, 202], [91, 202], [92, 201], [93, 200], [94, 199], [95, 199], [96, 198], [97, 197], [98, 196], [99, 196], [100, 195], [101, 194], [102, 193], [103, 193], [104, 192], [105, 191], [106, 191], [107, 190], [108, 189], [109, 188], [110, 188], [111, 187], [112, 186], [113, 185], [114, 185], [115, 184], [116, 183], [117, 182], [118, 182], [119, 181], [120, 180], [121, 179], [122, 179], [123, 178], [124, 177], [125, 176], [126, 176], [127, 175], [128, 174]]

max_discount = 174

Pairing operation

Cost of the pairing operation is 43000*k + 65000 where k is a number of pairs.

Fp-to-G1 mappign operation

Fp -> G1 mapping is 5500 gas.

Fp2-to-G2 mappign operation

Fp2 -> G2 mapping is 75000 gas

Gas schedule clarifications for the variable-length input

For multiexponentiation and pairing functions gas cost depends on the input length. The current state of how gas schedule is implemented in major clients (at the time of writing) is that gas cost function does not perform any validation of the length of the input and never returns an error. So we present a list of rules how gas cost functions must be implemented to ensure consistency between clients and safety.

Gas schedule clarifications for G1/G2 Multiexponentiation

Define a constant LEN_PER_PAIR that is equal to 160 for G1 operation and to 288 for G2 operation. Define a function discount(k) following the rules in the corresponding section, where k is number of pairs.

The following pseudofunction reflects how gas should be calculated:

  k = floor(len(input) / LEN_PER_PAIR);
  if k == 0 {
    return 0;
  }

  gas_cost = k * multiplication_cost * discount(k) / multiplier;

  return gas_cost;

We use floor division to get number of pairs. If length of the input is not divisible by LEN_PER_PAIR we still produce some result, but later on precompile will return an error. Also, case when k = 0 is safe: CALL or STATICCALL cost is non-zero, and case with formal zero gas cost is already used in Blake2f precompile. In any case, main precompile routine must produce an error on such an input because it violated encoding rules.

Gas schedule clarifications for pairing

Define a constant LEN_PER_PAIR = 384;

The following pseudofunction reflects how gas should be calculated:

  k = floor(len(input) / LEN_PER_PAIR);

  gas_cost = 23000*k + 115000;

  return gas_cost;

We use floor division to get number of pairs. If length of the input is not divisible by LEN_PER_PAIR we still produce some result, but later on precompile will return an error (precompile routine must produce an error on such an input because it violated encoding rules).

Rationale

Motivation section covers a total motivation to have operations over BLS12-381 curve available. We also extend a rationale for move specific fine points.

Multiexponentiation as a separate call

Explicit separate multiexponentiation operation that allows one to save execution time (so gas) by both the algorithm used (namely Peppinger algorithm) and (usually forgotten) by the fact that CALL operation in Ethereum is expensive (at the time of writing), so one would have to pay non-negigible overhead if e.g. for multiexponentiation of 100 points would have to call the multipication precompile 100 times and addition for 99 times (roughly 138600 would be saved).

Backwards Compatibility

There are no backward compatibility questions.

Important notes

Subgroup checks

Subgroup check is mandatory during the pairing call. Implementations should use fast subgroup checks: at the time of writing multiplication gas cost is based on double-and-add multiplication method that has a clear “worst case” (all bits are equal to one). For pairing operation it’s expected that implementation uses faster subgroup check, e.g. by using wNAF multiplication method for elliptic curves that is ~ 40% cheaper with windows size equal to 4. (Tested empirically. Savings are due to lower hamming weight of the group order and even lower hamming weight for wNAF. Concretely, subgroup check for both G1 and G2 points in a pair are around 35000 combined).

Field to curve mapping

Set of parameters for SWU mapping method is provided by IETF.

One should pay particular attention to the following fine points during implementation:

Once again, hash to field is NOT a part of this EIP as it can be implemented in EVM and with different strategies.

Test Cases

Due to the large test parameters space we first provide properties that various operations must satisfy. We use additive notation for point operations, capital letters (P, Q) for points, small letters (a, b) for scalars. Generator for G1 is labeled as G, generator for G2 is labeled as H, otherwise we assume random point on a curve in a correct subgroup. 0 means either scalar zero or point of infinity. 1 means either scalar one or multiplicative identity. group_order is a main subgroup order. e(P, Q) means pairing operation where P is in G1, Q is in G2.

Requeired properties for basic ops (add/multiply):

  • Commutativity: P + Q = Q + P
  • Additive negation: P + (-P) = 0
  • Doubling P + P = 2*P
  • Subgroup check: group_order * P = 0
  • Trivial multiplication check: 1 * P = P
  • Multiplication by zero: 0 * P = 0
  • Multiplication by the unnormalized scalar (scalar + group_order) * P = scalar * P

Required properties for pairing operation:

  • Degeneracy e(P, 0*Q) = e(0*P, Q) = 1
  • Bilinearity e(a*P, b*Q) = e(a*b*P, Q) = e(P, a*b*Q) (internal test, not visible through ABI)

Test vector for all operations are expanded in this csv files in repo.

Benchmarking test cases

Here one can find inputs (encoded with ABI of from this spec) that can be considered “worst cases” for “double-and-add” multiplication algorithm, and some cases for pairing call. Those are purely for convenience of initial benchmarking of the full ABI without manual test generation.

G1 addition example input = 
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
G2 addition example input = 
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
G1 mul double and add worst case = 
0000000000000000000000000000000017f1d3a73197d7942695638c4fa9ac0fc3688c4f9774b905a14e3a3f171bac586c55e83ff97a1aeffb3af00adb22c6bb0000000000000000000000000000000008b3f481e3aaa0f1a09e30ed741d8ae4fcf5e095d5d00af600db18cb2c04b3edd03cc744a2888ae40caa232946c5e7e1ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
G2 mul double and add worst case = 
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
Pairing case for 2 pairs = 
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
Pairing case for 4 pairs = 
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Pairing case for 6 pairs = 
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Implementation

There is a various choice of existing implementations of the curve operations. It may require extra work to add an ABI:

Security Considerations

Strictly following the spec will eliminate security implications or consensus implications in a contrast to the previous BN254 precompile.

Important topic is a “constant time” property for performed operations. We explicitly state that this precompile IS NOT REQUIRED to perform all the operations using constant time algorithms.

Copyright and related rights waived via CC0.

Citation

Please cite this document as:

Alex Vlasov, "EIP-2537: Precompile for BLS12-381 curve operations [DRAFT]," Ethereum Improvement Proposals, no. 2537, February 2020. [Online serial]. Available: https://eips.ethereum.org/EIPS/eip-2537.