├── src └── lib.rs ├── .gitignore ├── Cargo.toml ├── LICENSE-MIT ├── README.md └── LICENSE-APACHE /src/lib.rs: -------------------------------------------------------------------------------- 1 | #![doc = include_str!("../README.md")] -------------------------------------------------------------------------------- /.gitignore: -------------------------------------------------------------------------------- 1 | /target 2 | Cargo.lock 3 | .vscode/ 4 | -------------------------------------------------------------------------------- /Cargo.toml: -------------------------------------------------------------------------------- 1 | [package] 2 | name = "ark-algebra-intro" 3 | version = "0.3.1-alpha.0" 4 | edition = "2018" 5 | description = "Introduction to `arkworks` algebra APIs" 6 | repository = "https://github.com/Pratyush/algebra-intro" 7 | documentation = "https://docs.rs/ark-algebra-intro/" 8 | keywords = ["finite-fields", "elliptic-curves", "pairings"] 9 | categories = [ "cryptography" ] 10 | include = ["Cargo.toml", "src", "README.md", "LICENSE-APACHE", "LICENSE-MIT"] 11 | license = "MIT/Apache-2.0" 12 | 13 | [dependencies] 14 | 15 | ark-std = { version = "0.3" } 16 | ark-ff = { version = "0.3" } 17 | ark-ec = { version = "0.3" } 18 | ark-serialize = { version = "0.3" } 19 | ark-bls12-381 = { version = "0.3", features = [ "curve" ] } 20 | rand = { version = "0.8", features = [ "std", "std_rng" ] } 21 | num-bigint = { version = "0.4" } 22 | -------------------------------------------------------------------------------- /LICENSE-MIT: -------------------------------------------------------------------------------- 1 | The MIT License (MIT) 2 | 3 | Permission is hereby granted, free of charge, to any person obtaining a copy 4 | of this software and associated documentation files (the "Software"), to deal 5 | in the Software without restriction, including without limitation the rights 6 | to use, copy, modify, merge, publish, distribute, sublicense, and/or sell 7 | copies of the Software, and to permit persons to whom the Software is 8 | furnished to do so, subject to the following conditions: 9 | 10 | The above copyright notice and this permission notice shall be included in 11 | all copies or substantial portions of the Software. 12 | 13 | THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR 14 | IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 15 | FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE 16 | AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER 17 | LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, 18 | OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN 19 | THE SOFTWARE. 20 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Introduction to `arkworks` algebra APIs 2 | 3 | The [`arkworks` ecosystem](https://arkworks.rs) is a set of state-of-the-art Rust libraries that collectively provide tools to program zkSNARKs. 4 | zkHack puzzles will be using `arkworks` libraries for elliptic curve and finite field arithmetic. This document is a helpful cheat-sheet to get started with using these libraries. 5 | 6 | ## Finite field arithmetic 7 | 8 | There are three important traits when working with finite fields: [`Field`], 9 | [`SquareRootField`], and [`PrimeField`]. Let's explore these via examples. 10 | 11 | ### [`Field`] 12 | 13 | The [`Field`] trait provides a generic interface for any finite field. 14 | Types implementing [`Field`] support common field operations 15 | such as addition, subtraction, multiplication, and inverses. 16 | 17 | ```rust 18 | use ark_ff::Field; 19 | // We'll use a field associated with the BLS12-381 pairing-friendly 20 | // group for this example. 21 | use ark_bls12_381::Fq2 as F; 22 | // `ark-std` is a utility crate that enables `arkworks` libraries 23 | // to easily support `std` and `no_std` workloads, and also re-exports 24 | // useful crates that should be common across the entire ecosystem, such as `rand`. 25 | use ark_std::{One, UniformRand}; 26 | 27 | let mut rng = ark_std::rand::thread_rng(); 28 | // Let's sample uniformly random field elements: 29 | let a = F::rand(&mut rng); 30 | let b = F::rand(&mut rng); 31 | 32 | // We can add... 33 | let c = a + b; 34 | // ... subtract ... 35 | let d = a - b; 36 | // ... double elements ... 37 | assert_eq!(c + d, a.double()); 38 | 39 | // ... multiply ... 40 | let e = c * d; 41 | // ... square elements ... 42 | assert_eq!(e, a.square() - b.square()); 43 | 44 | // ... and compute inverses ... 45 | assert_eq!(a.inverse().unwrap() * a, F::one()); // have to to unwrap, as `a` could be zero. 46 | ``` 47 | 48 | ### [`SquareRootField`] 49 | 50 | In some cases, it is important to take square roots of field elements 51 | (e.g.: for point compression of elliptic curve elements.) 52 | To support this, users can implement the [`SquareRootField`] trait for their field type. This 53 | provides access to the following methods: 54 | 55 | ```rust 56 | use ark_ff::{Field, SquareRootField}; 57 | // As before, we'll use a field associated with the BLS12-381 pairing-friendly 58 | // group for this example. 59 | use ark_bls12_381::Fq2 as F; 60 | use ark_std::{One, UniformRand}; 61 | 62 | let mut rng = ark_std::rand::thread_rng(); 63 | // Let's try to sample a random square via rejection sampling: 64 | let mut a = F::rand(&mut rng); 65 | while a.legendre().is_qnr() { // A square is also called a *quadratic residue* 66 | a = F::rand(&mut rng); 67 | } 68 | 69 | // Since `a` is a square, we can compute its square root: 70 | let b = a.sqrt().unwrap(); 71 | assert_eq!(b.square(), a); 72 | 73 | // Let's sample a random *non-square* 74 | let mut a = F::rand(&mut rng); 75 | while a.legendre().is_qr() { 76 | a = F::rand(&mut rng); 77 | } 78 | // The square root should not exist: 79 | assert_eq!(a.sqrt(), None); 80 | ``` 81 | 82 | ### [`PrimeField`] 83 | 84 | If the field is of prime order, then users can choose 85 | to implement the [`PrimeField`] trait for it. This provides access to the following 86 | additional APIs: 87 | 88 | ```rust 89 | use ark_ff::{Field, PrimeField, FpParameters, BigInteger}; 90 | // Now we'll use the prime field underlying the BLS12-381 G1 curve. 91 | use ark_bls12_381::Fq as F; 92 | use ark_std::{One, Zero, UniformRand}; 93 | 94 | let mut rng = ark_std::rand::thread_rng(); 95 | let a = F::rand(&mut rng); 96 | // We can access the prime modulus associated with `F`: 97 | let modulus = ::Params::MODULUS; 98 | assert_eq!(a.pow(&modulus), a); 99 | 100 | // We can convert field elements to integers in the range [0, MODULUS - 1]: 101 | let one: num_bigint::BigUint = F::one().into(); 102 | assert_eq!(one, num_bigint::BigUint::one()); 103 | 104 | // We can construct field elements from an arbitrary sequence of bytes: 105 | let n = F::from_le_bytes_mod_order(&modulus.to_bytes_le()); 106 | assert_eq!(n, F::zero()); 107 | ``` 108 | 109 | ## Elliptic curve arithmetic 110 | 111 | There are two traits that are important when working with elliptic curves 112 | over finite fields: [`ProjectiveCurve`], and [`AffineCurve`]. Both traits 113 | represent the same curve, but provide different underlying representations. 114 | In particular, a [`ProjectiveCurve`] representation of a curve point is generally 115 | more efficient for arithmetic, but does not provide a unique representative 116 | for a curve point. An [`AffineCurve`] representation, on the other hand, *is* unique, 117 | but is slower for most arithmetic operations. Let's explore how and when to use 118 | these: 119 | 120 | ```rust 121 | use ark_ec::{ProjectiveCurve, AffineCurve}; 122 | use ark_ff::{PrimeField, Field}; 123 | // We'll use the BLS12-381 G1 curve for this example. 124 | use ark_bls12_381::{G1Projective as G, G1Affine as GAffine, Fr as ScalarField}; 125 | use ark_std::{Zero, UniformRand}; 126 | 127 | let mut rng = ark_std::rand::thread_rng(); 128 | // Let's sample uniformly random field elements: 129 | let a = G::rand(&mut rng); 130 | let b = G::rand(&mut rng); 131 | 132 | // We can add... 133 | let c = a + b; 134 | // ... subtract ... 135 | let d = a - b; 136 | // ... and double elements. 137 | assert_eq!(c + d, a.double()); 138 | // We can also negate elements... 139 | let e = -a; 140 | assert_eq!(e + a, G::zero()); 141 | 142 | // ...and multiply group elements by elements of the corresponding scalar field 143 | let scalar = ScalarField::rand(&mut rng); 144 | let e = c.mul(&scalar.into_repr()); // into_repr() converts the scalar into a `BigInteger`. 145 | let f = e.mul(&scalar.inverse().unwrap().into_repr()); 146 | assert_eq!(f, c); 147 | 148 | // Finally, we can also convert curve points in projective coordinates to affine coordinates. 149 | let c_aff = c.into_affine(); 150 | // Most group operations are slower in affine coordinates, but adding an affine point 151 | // to a projective one is slightly more efficient. 152 | let d = c.add_mixed(&c_aff); 153 | assert_eq!(d, c.double()); 154 | 155 | // This efficiency also translates into more efficient scalar multiplication routines. 156 | let e_from_aff = c_aff.mul(scalar.into_repr()); 157 | assert_eq!(e, e_from_aff); 158 | 159 | // Finally, while not recommended, users can directly construct group elements 160 | // from the x and y coordinates. This is useful when implementing algorithms 161 | // like hash-to-curve. 162 | let e_affine = e.into_affine(); 163 | let e_x = e_affine.x; 164 | let e_y = e_affine.y; 165 | let is_at_infinity = e_affine.is_zero(); 166 | let new_e = GAffine::new(e_x, e_y, is_at_infinity); 167 | assert_eq!(e_affine, new_e); 168 | // Users should check that the new point is on the curve and is in the prime-order group: 169 | assert!(new_e.is_on_curve()); 170 | assert!(new_e.is_in_correct_subgroup_assuming_on_curve()); 171 | ``` 172 | 173 | ## Pairings 174 | 175 | [`PairingEngine`] is the primary trait for working with pairings. It contains 176 | associated types and methods that are relevant to pairing operations: 177 | 178 | ```rust 179 | use ark_ec::{ProjectiveCurve, AffineCurve, PairingEngine}; 180 | use ark_ff::{PrimeField, Field}; 181 | // We'll use the BLS12-381 pairing-friendly group for this example. 182 | use ark_bls12_381::{Bls12_381, G1Projective as G1, G2Projective as G2, G1Affine, G2Affine, Fr as ScalarField}; 183 | use ark_std::{Zero, UniformRand}; 184 | 185 | let mut rng = ark_std::rand::thread_rng(); 186 | // Let's sample uniformly random field elements: 187 | let a: G1Affine = G1::rand(&mut rng).into(); 188 | let b: G2Affine = G2::rand(&mut rng).into(); 189 | // We can compute the pairing of `a` and `b`: 190 | let c = Bls12_381::pairing(a, b); 191 | 192 | // We can also compute the pairing partwise: 193 | // First, we compute the Miller loop: 194 | let c_ml = Bls12_381::miller_loop(&[(a.into(), b.into())]); 195 | let c_fe = Bls12_381::final_exponentiation(&c_ml).unwrap(); 196 | assert_eq!(c, c_fe); 197 | ``` 198 | 199 | ## Serialization 200 | 201 | Most types in the `arkworks` ecosystem implement the [`CanonicalSerialize`] 202 | and [`CanonicalDeserialize`] traits. These traits enable converting these types 203 | to canonical byte representations that are suitable for disk storage and network 204 | communication. They also enable support for point compression. 205 | 206 | ```rust 207 | use ark_ec::AffineCurve; 208 | // We'll use the BLS12-381 pairing-friendly group for this example. 209 | use ark_bls12_381::{G1Projective as G1, G2Projective as G2, G1Affine, G2Affine}; 210 | use ark_serialize::{CanonicalSerialize, CanonicalDeserialize}; 211 | use ark_std::UniformRand; 212 | 213 | let mut rng = ark_std::rand::thread_rng(); 214 | // Let's sample uniformly random field elements: 215 | let a: G1Affine = G1::rand(&mut rng).into(); 216 | let b: G2Affine = G2::rand(&mut rng).into(); 217 | 218 | // We can serialize with compression... 219 | let mut compressed_bytes = Vec::new(); 220 | a.serialize(&mut compressed_bytes).unwrap(); 221 | // ...and without: 222 | let mut uncompressed_bytes = Vec::new(); 223 | a.serialize_uncompressed(&mut uncompressed_bytes).unwrap(); 224 | 225 | // We can reconstruct our points from the compressed serialization... 226 | let a_compressed = G1Affine::deserialize(&*compressed_bytes).unwrap(); 227 | 228 | // ... and from the uncompressed one: 229 | let a_uncompressed = G1Affine::deserialize_uncompressed(&*uncompressed_bytes).unwrap(); 230 | 231 | assert_eq!(a_compressed, a); 232 | assert_eq!(a_uncompressed, a); 233 | 234 | // If we trust the origin of the serialization 235 | // (eg: if the serialization was stored on authenticated storage), 236 | // then we can skip some validation checks: 237 | let a_unchecked = G1Affine::deserialize_unchecked(&*uncompressed_bytes).unwrap(); 238 | assert_eq!(a_unchecked, a); 239 | ``` 240 | 241 | [`Field`]: https://docs.rs/ark-ff/0.3.0/ark_ff/fields/trait.Field.html 242 | [`PrimeField`]: https://docs.rs/ark-ff/0.3.0/ark_ff/fields/trait.PrimeField.html 243 | [`SquareRootField`]: https://docs.rs/ark-ff/0.3.0/ark_ff/fields/trait.SquareRootField.html 244 | [`AffineCurve`]: https://docs.rs/ark-ec/0.3.0/ark_ec/trait.AffineCurve.html 245 | [`ProjectiveCurve`]: https://docs.rs/ark-ec/0.3.0/ark_ec/trait.ProjectiveCurve.html 246 | [`PairingEngine`]: https://docs.rs/ark-ec/0.3.0/ark_ec/trait.PairingEngine.html 247 | [`CanonicalSerialize`]: https://docs.rs/ark-serialize/0.3.0/ark_serialize/trait.CanonicalSerialize.html 248 | [`CanonicalDeserialize`]: https://docs.rs/ark-serialize/0.3.0/ark_serialize/trait.CanonicalDeserialize.html 249 | -------------------------------------------------------------------------------- /LICENSE-APACHE: -------------------------------------------------------------------------------- 1 | Apache License 2 | Version 2.0, January 2004 3 | http://www.apache.org/licenses/ 4 | 5 | TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION 6 | 7 | 1. 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