# BLS signature verification **Warning: This document is pending academic review and should not yet be considered secure.** ## Table of contents - [BLS signature verification](#bls-signature-verification) - [Table of contents](#table-of-contents) - [Curve parameters](#curve-parameters) - [Point representations](#point-representations) - [G1 points](#g1-points) - [G2 points](#g2-points) - [Helpers](#helpers) - [`hash_to_G2`](#hash_to_g2) - [`modular_squareroot`](#modular_squareroot) - [Aggregation operations](#aggregation-operations) - [`bls_aggregate_pubkeys`](#bls_aggregate_pubkeys) - [`bls_aggregate_signatures`](#bls_aggregate_signatures) - [Signature verification](#signature-verification) - [`bls_verify`](#bls_verify) - [`bls_verify_multiple`](#bls_verify_multiple) ## Curve parameters The BLS12-381 curve parameters are defined [here](https://z.cash/blog/new-snark-curve). ## Point representations We represent points in the groups G1 and G2 following [zkcrypto/pairing](https://github.com/zkcrypto/pairing/tree/master/src/bls12_381). We denote by `q` the field modulus and by `i` the imaginary unit. ### G1 points A point in G1 is represented as a 384-bit integer `z` decomposed as a 381-bit integer `x` and three 1-bit flags in the top bits: * `x = z % 2**381` * `a_flag = (z % 2**382) // 2**381` * `b_flag = (z % 2**383) // 2**382` * `c_flag = (z % 2**384) // 2**383` Respecting bit ordering, `z` is decomposed as `(c_flag, b_flag, a_flag, x)`. We require: * `x < q` * `c_flag == 1` * if `b_flag == 1` then `a_flag == x == 0` and `z` represents the point at infinity * if `b_flag == 0` then `z` represents the point `(x, y)` where `y` is the valid coordinate such that `(y * 2) // q == a_flag` ### G2 points A point in G2 is represented as a pair of 384-bit integers `(z1, z2)`. We decompose `z1` as above into `x1`, `a_flag1`, `b_flag1`, `c_flag1` and `z2` into `x2`, `a_flag2`, `b_flag2`, `c_flag2`. We require: * `x1 < q` and `x2 < q` * `a_flag2 == b_flag2 == c_flag2 == 0` * `c_flag1 == 1` * if `b_flag1 == 1` then `a_flag1 == x1 == x2 == 0` and `(z1, z2)` represents the point at infinity * if `b_flag1 == 0` then `(z1, z2)` represents the point `(x1 * i + x2, y)` where `y` is the valid coordinate such that the imaginary part `y_im` of `y` satisfies `(y_im * 2) // q == a_flag1` ## Helpers ### `hash_to_G2` ```python G2_cofactor = 305502333931268344200999753193121504214466019254188142667664032982267604182971884026507427359259977847832272839041616661285803823378372096355777062779109 q = 4002409555221667393417789825735904156556882819939007885332058136124031650490837864442687629129015664037894272559787 def hash_to_G2(message: bytes32, domain: uint64) -> [uint384]: # Initial candidate x coordinate x_re = int.from_bytes(hash(message + bytes8(domain) + b'\x01'), 'big') x_im = int.from_bytes(hash(message + bytes8(domain) + b'\x02'), 'big') x_coordinate = Fq2([x_re, x_im]) # x = x_re + i * x_im # Test candidate y coordinates until a one is found while 1: y_coordinate_squared = x_coordinate ** 3 + Fq2([4, 4]) # The curve is y^2 = x^3 + 4(i + 1) y_coordinate = modular_squareroot(y_coordinate_squared) if y_coordinate is not None: # Check if quadratic residue found return multiply_in_G2((x_coordinate, y_coordinate), G2_cofactor) x_coordinate += Fq2([1, 0]) # Add 1 and try again ``` ### `modular_squareroot` `modular_squareroot(x)` returns a solution `y` to `y**2 % q == x`, and `None` if none exists. If there are two solutions the one with higher imaginary component is favored; if both solutions have equal imaginary component the one with higher real component is favored (note that this is equivalent to saying that the single solution with either imaginary component > p/2 or imaginary component zero and real component > p/2 is favored). The following is a sample implementation; implementers are free to implement modular square roots as they wish. Note that `x2 = -x1` is an _additive modular inverse_ so real and imaginary coefficients remain in `[0 .. q-1]`. `coerce_to_int(element: Fq) -> int` is a function that takes Fq element `element` (ie. integers `mod q`) and converts it to a regular integer. ```python Fq2_order = q ** 2 - 1 eighth_roots_of_unity = [Fq2([1,1]) ** ((Fq2_order * k) // 8) for k in range(8)] def modular_squareroot(value: Fq2) -> Fq2: candidate_squareroot = value ** ((Fq2_order + 8) // 16) check = candidate_squareroot ** 2 / value if check in eighth_roots_of_unity[::2]: x1 = candidate_squareroot / eighth_roots_of_unity[eighth_roots_of_unity.index(check) // 2] x2 = -x1 x1_re, x1_im = coerce_to_int(x1.coeffs[0]), coerce_to_int(x1.coeffs[1]) x2_re, x2_im = coerce_to_int(x2.coeffs[0]), coerce_to_int(x2.coeffs[1]) return x1 if (x1_im > x2_im or (x1_im == x2_im and x1_re > x2_re)) else x2 return None ``` ## Aggregation operations ### `bls_aggregate_pubkeys` Let `bls_aggregate_pubkeys(pubkeys: List[Bytes48]) -> Bytes48` return `pubkeys[0] + .... + pubkeys[len(pubkeys)-1]`, where `+` is the elliptic curve addition operation over the G1 curve. ### `bls_aggregate_signatures` Let `bls_aggregate_signatures(signatures: List[Bytes96]) -> Bytes96` return `signatures[0] + .... + signatures[len(signatures)-1]`, where `+` is the elliptic curve addition operation over the G2 curve. ## Signature verification In the following `e` is the pairing function and `g` is the G1 generator with the following coordinates (see [here](https://github.com/zkcrypto/pairing/tree/master/src/bls12_381#g1)): ```python g_x = 3685416753713387016781088315183077757961620795782546409894578378688607592378376318836054947676345821548104185464507 g_y = 1339506544944476473020471379941921221584933875938349620426543736416511423956333506472724655353366534992391756441569 g = Fq2([g_x, g_y]) ``` ### `bls_verify` Let `bls_verify(pubkey: Bytes48, message: Bytes32, signature: Bytes96, domain: uint64) -> bool`: * Verify that `pubkey` is a valid G1 point. * Verify that `signature` is a valid G2 point. * Verify that `e(pubkey, hash_to_G2(message, domain)) == e(g, signature)`. ### `bls_verify_multiple` Let `bls_verify_multiple(pubkeys: List[Bytes48], messages: List[Bytes32], signature: Bytes96, domain: uint64) -> bool`: * Verify that each `pubkey` in `pubkeys` is a valid G1 point. * Verify that `signature` is a valid G2 point. * Verify that `len(pubkeys)` equals `len(messages)` and denote the length `L`. * Verify that `e(pubkeys[0], hash_to_G2(messages[0], domain)) * ... * e(pubkeys[L-1], hash_to_G2(messages[L-1], domain)) == e(g, signature)`.