ethereum · nicksavers · Mar 7, 2019 · Mar 6, 2019 · Mar 6, 2019 · Mar 6, 2019
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+---
+eip: 1829
+title: Precompile for Elliptic Curve Linear Combinations
+author: Remco Bloemen <Recmo@0x.org>
+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
+<!--"If you can't explain it simply, you don't understand it well enough." Provide a simplified and layman-accessible explanation of the EIP.-->
+
+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 short (~200 word) description of the technical issue being addressed.-->
+
+A precompile that takes a curve and computes a linear combination of curve points.
+
+## Motivation
+<!--The motivation is critical for EIPs that want to change the Ethereum protocol. It should clearly explain why the existing protocol specification is inadequate to address the problem that the EIP solves. EIP submissions without sufficient motivation may be rejected outright.-->
+
+## Specification
+<!--The technical specification should describe the syntax and semantics of any new feature. The specification should be detailed enough to allow competing, interoperable implementations for any of the current Ethereum platforms (go-ethereum, parity, cpp-ethereum, ethereumj, ethereumjs, and [others](https://github.com/ethereum/wiki/wiki/Clients)).-->
+
+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
+<!--The rationale fleshes out the specification by describing what motivated the design and why particular design decisions were made. It should describe alternate designs that were considered and related work, e.g. how the feature is supported in other languages. The rationale may also provide evidence of consensus within the community, and should discuss important objections or concerns raised during discussion.-->
+
+**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
+<!--All EIPs that introduce backwards incompatibilities must include a section describing these incompatibilities and their severity. The EIP must explain how the author proposes to deal with these incompatibilities. EIP submissions without a sufficient backwards compatibility treatise may be rejected outright.-->
+
+## Test Cases
+<!--Test cases for an implementation are mandatory for EIPs that are affecting consensus changes. Other EIPs can choose to include links to test cases if applicable.-->
+
+## Implementation
+<!--The implementations must be completed before any EIP is given status "Final", but it need not be completed before the EIP is accepted. While there is merit to the approach of reaching consensus on the specification and rationale before writing code, the principle of "rough consensus and running code" is still useful when it comes to resolving many discussions of API details.-->
+
+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/).
+