--- eip: 1829 title: Precompile for Elliptic Curve Linear Combinations author: Remco Bloemen discussions-to: https://ethereum-magicians.org/t/ewasm-precompile-for-general-elliptic-curve-math/2581 status: Draft type: Standards Track category: Core created: 2019-03-06 --- # Precompile for Elliptic Curve Linear Combinations ## Simple Summary Currently the EVM only supports *secp261k1* in a limited way through `ecrecover` and *altbn128* through two pre-compiles. There are draft proposals to add more curves. There are many more elliptic curve that have useful application for integration with existing systems or newly developed curves for zero-knownledge proofs. This EIP adds a precompile that allows whole classes of curves to be used. ## Abstract A precompile that takes a curve and computes a linear combination of curve points. ## Motivation ## Specification Given integers `m, α` and `β`, scalars `s_i`, and curve points `A_i` construct the elliptic curve ``` y² = x³ + α ⋅ x + β mod m ``` and compute the following ``` C = s₀ ⋅ A₀ + s₁ ⋅ A₁ + ⋯ + s_n ⋅ A_n ``` aka *linear combination*, *inner product*, *multi-multiplication* or even *multi-exponentiation*. ``` (Cx, Cy) := ecmul(m, α, β, s0, Ax0, As0, s1, Ax1, As1, ...) ``` ### Gas cost ``` BASE_GAS = ... ADD_GAS = ... MUL_GAS = ... ``` The total gas cost is `BASE_GAS` plus `ADD_GAS` for each `s_i` that is `1` and `MUL_GAS` for each `s_i > 1` (`s_i = 0` is free). ### Encoding of points Encode as `(x, y')` where `s` is the indicates the wheter `y` or `-y` is to be taken. It follows SEC 1 v 1.9 2.3.4, except uncompressed points (`y' = 0x04`) are not supported. | `y'` | `(x, y)` | |--------|-----| | `0x00` | Point at infinity | | `0x02` | Solution with `y` even | | `0x03` | Solution with `y` odd | Conversion from affine coordinates to compressed coordinates is trivial: `y' = 0x02 | (y & 0x01)`. ### Special cases **Coordinate recovery.** Set `s₀ = 1`. The output will be the recovered coordinates of `A₀`. **On-curve checking.** Do coordinate recovery and compare `y` coordinate. **Addition.** Set `s₀ = s₁ = 1`, the output will be `A₀ + A₁`. **Doubling.** Set `s₀ = 2`. The output will be `2 ⋅ A₀`. (Note: under current gas model this may be more costly than self-addition!) **Scalar multiplication.** Set only `s₀` and `A₀`. **Modular square root.** Set `α = s₀ = A = 0` the output will have `Cy² = β mod m`. ### Edge cases * Non-prime moduli or too small modulus * Field elements larger than modulus * Curve has singular points (`4 α³ + 27 β² = 0`) * Invalid sign bytes * x coordinate not on curve * Returning the point at infinity * (Please add if you spot more) ## Rationale **Generic Field and Curve.** Many important optimizations are independent of the field and curve used. Some missed specific optimizations are: * Reductions specific to the binary structure of the field prime. * Precomputation of Montgomery factors. * Precomputation of multiples of certain popular points like the generator. * Special point addition/doubling [formulas][formulas] for `α = -3`, `α = -1`, `α = 0`, `β = 0`. [formulas]: http://www.hyperelliptic.org/EFD/g1p/auto-shortw.html TODO: The special cases for `α` and `β` might be worth implementing and offered a gas discount. **Compressed Coordinates.** Compressed coordinates allow contract to work with only `x` coordinates and sign bytes. It also prevents errors around points not being on-curve. Conversion to compressed coordinates is trivial. **Linear Combination.** We could instead have a simple multiply `C = r ⋅ A`. In this case we would need a separate pre-compile for addition. In addtion, a linear combination allows for optimizations that like Shamir's trick that are not available in a single scalar multiplication. ECDSA requires `s₀ ⋅ A₀ + s₁ ⋅ A₁` and would benfit from this. The BN254 (aka alt_bn8) multiplication operation introduced by the [EIP-196][eip196] precompile only handles a single scalar multiplication. The missed performance is such that for two or more points it is cheaper to use EVM, as pratically demonstrated by [Weierstrudel][ws]. [eip196]: https://eips.ethereum.org/EIPS/eip-196 [ws]: https://medium.com/aztec-protocol/huffing-for-crypto-with-weierstrudel-9c9568c06901 **Variable Time Math.** When called during a transaction, there is no assumption of privacy and no mittigations for side-channel attacks are necessary. **Prime Fields.** This EIP is for fields of large characteristic. It does not cover Binary fields and other fields of non-prime characteristic. **256-bit modulus.** This EIP is for field moduli less than `2^{256}`. This covers many of the popular curves while still having all parameters fit in a single EVM word. TODO: Consider a double-word version. 512 bits would cover all known curves except E-521. In particular it will cover the NIST P-384 curve used by the Estonian e-Identity and the BLS12-381 curve used by [ZCash Sappling][sappling]. [sappling]: https://z.cash/blog/new-snark-curve/ **Short Weierstrass Curves.** This EIP is for fields specified in short Weierstrass form. While any curve can be converted to short Weierstrass form through a [substitution of variables][cov], this misses out on the performance advantages of those specific forms. [cov]: https://safecurves.cr.yp.to/equation.html ## Backwards Compatibility ## Test Cases ## Implementation There will be a reference implementation in Rust based on the existing libraries (in particular those by ZCash and The Matter Inc.). The reference implementation will be production grade and compile to a native library with a C api and a webassembly version. Node developers are encouraged to use the reference implementation and can use either the rust library, the native C bindings or the webassembly module. Node developers can of course always decide to implement their own. ## References This EIP overlaps in scope with * [EIP-196](https://eips.ethereum.org/EIPS/eip-196): ecadd, ecmul for altbn128 * [EIP issue 603](https://github.com/ethereum/EIPs/issues/603): ecadd, ecmul for SECP256k1. * [EIP 665](https://eips.ethereum.org/EIPS/eip-665): ECDSA verify for ED25519. * [EIP 1108](https://eips.ethereum.org/EIPS/eip-1108): Optimize ecadd and ecmul for altbn128. ## Copyright Copyright and related rights waived via [CC0](https://creativecommons.org/publicdomain/zero/1.0/).