Update EIP-196: Fix coordinate description

EIPS/eip-196.md

@@ -1,14 +1,14 @@
 ---
 eip: 196
 title: Precompiled contracts for addition and scalar multiplication on the elliptic curve alt_bn128
 author: Christian Reitwiessner <[email protected]>
 type: Standards Track
 category: Core
 status: Final
 created: 2017-02-02
 ---
 
 ## Simple Summary
 
 Precompiled contracts for elliptic curve operations are required in order to perform zkSNARK verification within the block gas limit.
 
@@ -31,7 +31,7 @@
 Address for MUL: 0x7
 
 The curve is defined by:
 ```
 Y^2 = X^3 + 3
 over the field F_p with
 p = 21888242871839275222246405745257275088696311157297823662689037894645226208583
@@ -39,7 +39,7 @@
 
 ### Encoding
 
-Field elements and scalars are encoded as 32 byte big-endian numbers. Curve points are encoded as two field elements `(x, y)`, where the point at infinity is encoded as `(0, 0)`.
+Field elements and scalars are encoded as 32 byte big-endian numbers. Curve points are encoded as two field elements `(x, y)` representing the affine coordinates, where the point at infinity is encoded as `(0, 0)`.
 
 Tuples of objects are encoded as their concatenation.
 
@@ -51,12 +51,12 @@
 
 Invalid input: For both contracts, if any input point does not lie on the curve or any of the field elements (point coordinates) is equal or larger than the field modulus p, the contract fails. The scalar can be any number between `0` and `2**256-1`.
 
 #### ADD
 Input: two curve points `(x, y)`.
 Output: curve point `x + y`, where `+` is point addition on the elliptic curve `alt_bn128` specified above.
 Fails on invalid input and consumes all gas provided.
 
 #### MUL
 Input: curve point and scalar `(x, s)`.
 Output: curve point `s * x`, where `*` is the scalar multiplication on the elliptic curve `alt_bn128` specified above.
 Fails on invalid input and consumes all gas.
@@ -93,12 +93,12 @@
 
 Implementation of these primitives are available here:
 
  - [libff](https://github.com/scipr-lab/libff/blob/master/libff/algebra/curves/alt_bn128/alt_bn128_g1.cpp) (C++)
  - [bn](https://github.com/zcash/bn/blob/master/src/groups/mod.rs) (Rust)
 
 In both codebases, a specific group on the curve alt_bn128 is used and is called G1.
 
  - [Python](https://github.com/ethereum/py_pairing/blob/master/py_ecc/bn128/bn128_curve.py) - probably most self-contained and best readable.
 
 ## Copyright
 

EIPS/eip-197.md

@@ -8,7 +8,7 @@
 created: 2017-02-06
 ---
 
 ## Simple Summary
 
 Precompiled contracts for elliptic curve pairing operations are required in order to perform zkSNARK verification within the block gas limit.
 
@@ -57,7 +57,7 @@
 The group `G_1` is defined on the curve `Y^2 = X^3 + 3` over the field `F_p` with `p = 21888242871839275222246405745257275088696311157297823662689037894645226208583` with generator `P1 = (1, 2)`.
 
 The group `G_2` is defined on the curve `Y^2 = X^3 + 3/(i+9)` over a different field `F_p^2 = F_p[i] / (i^2 + 1)` (p is the same as above) with generator
 ```
 P2 = (
   11559732032986387107991004021392285783925812861821192530917403151452391805634 * i +
   10857046999023057135944570762232829481370756359578518086990519993285655852781,
@@ -75,7 +75,7 @@
 
 Elements `a * i + b` of `F_p^2` are encoded as two elements of `F_p`, `(a, b)`.
 
-Elliptic curve points are encoded as a Jacobian pair `(X, Y)` where the point at infinity is encoded as `(0, 0)`.
+Elliptic curve points are encoded as as two field elements `(x, y)` representing the affine coordinates where the point at infinity is encoded as `(0, 0)`.
 
 Note that the number `k` is derived from the input length.
 
@@ -112,16 +112,16 @@
 (2) `e` is non-degenerate
 
 Now observe that
 ```
 log_P1(a1) * log_P2(b1) + ... + log_P1(ak) * log_P2(bk) = 0 (in F_q)
 ```
 if and only if
 ```
 e(P1, P2)^(log_P1(a1) * log_P2(b1) + ... + log_P1(ak) * log_P2(bk)) = 1 (in G_T)
 ```
 
 Furthermore, the left hand side of this equation is equal to
 ```
 e(log_P1(a1) * P1, log_P2(b1) * P2) * ... * e(log_P1(ak) * P1, log_P2(bk) * P2)
 = e(a1, b1) * ... * e(ak, bk)
 ```