Update EIP-2537: Rephrased subgroup check part

EIPS/eip-2537.md

@@ -338,9 +338,10 @@ There are no backward compatibility questions.
 
 ### Subgroup checks
 
-A subgroup check **is mandatory** during the pairing call. Implementations *should* use fast subgroup checks: at the time of writing, multiplication gas cost is based on the `double-and-add` multiplication method that has a clear "worst case" (all bits are equal to one). For pairing operations, it is expected that implementations use faster subgroup checks, e.g. by using the wNAF multiplication method for elliptic curves that is ~ `40%` cheaper with windows size equal to 4. (Tested empirically. Savings are due to lower hamming weight of the group order and even lower hamming weight for wNAF. Concretely, subgroup check for both G1 and G2 points in a pair are around `35000` combined).
-
+The check **is mandatory** during **ALL** the calls. Implementations *should* use fast subgroup checks. It can be stated that it **must** be performed by fast subgroup checks to achieve a speedup, resulting in a discount over naive implementations based on the `double-and-add` multiplication method, which has a clear "worst-case" scenario where all bits are equal to one.
+Before accepting an input, it is recommended to subject the input to the appropriate endomorphism test as described in https://eprint.iacr.org/2021/1130.pdf.
 
+The algorithms and set of parameters for fast subgroup checks are provided by a separate [document](../assets/eip-2537/fast_subgroup_checks.md)
 
 ### Field to curve mapping
 

assets/eip-2537/fast_subgroup_checks.md

@@ -0,0 +1,51 @@
+# Fast subgroup checks used by EIP-2537
+
+### Fields and Groups
+
+Field Fp is defined as the finite field of size `p` with elements represented as integers between 0 and p-1 (both inclusive). 
+
+Field Fp2 is defined as `Fp[X]/(X^2-nr2)` with elements  `el = c0 + c1 * v`, where `v` is the formal square root of `nr2` represented as integer pairs `(c0,c1)`.
+
+Group G1 is defined as a set of Fp pairs (points) `(x,y)` such that either `(x,y)` is  `(0,0)` or `x,y` satisfy the curve Fp equation.
+
+Group G2 is defined as a set of Fp2 pairs (points) `(x',y')` such that either `(x,y)` is `(0,0)` or `(x',y')` satisfy the curve Fp2 equation.
+
+## Curve parameters
+
+The set of parameters used by fast subgroup checks:
+
+```
+|x| (seed) = 15132376222941642752
+x is negative = true
+Cube root of unity modulo p - Beta = 793479390729215512621379701633421447060886740281060493010456487427281649075476305620758731620350
+r = 4002409555221667392624310435006688643935503118305586438271171395842971157480381377015405980053539358417135540939437 * v
+s = 2973677408986561043442465346520108879172042883009249989176415018091420807192182638567116318576472649347015917690530 + 1028732146235106349975324479215795277384839936929757896155643118032610843298655225875571310552543014690878354869257 * v
+```
+
+## Helper function to compute the conjugate over Fp2 - `conjugate`
+
+`conjugate(c0 + c1 * v) := c0 - c1 * v`
+
+## G1 endomorphism - `phi`
+
+The endomorphism `phi` transform the point from `(x,y)` to `(Beta*x,y)` where `Beta` is a precomputed cube root of unity modulo `p` given above in parameters sections:
+
+`phi((x,y)) := (Beta*x,y)`
+
+## G2 endomorphism - `psi`
+
+`psi((x,y)) := (conjugate(x)*r,conjugate(y)*s)`
+
+# The G1 case
+
+Before accepting a point `P` as input that purports to be a member of G1 subject the input to the following endomorphism test: `phi(P) + x^2*P = 0`
+
+
+# The G2 case
+
+Before accepting a point `P` as input that purports to be a member of G2 subject the input to the following endomorphism test: `psi(P) + x*P = 0`
+
+# Resources
+
+* https://eprint.iacr.org/2021/1130.pdf, sec.4
+* https://eprint.iacr.org/2022/352.pdf, sec. 4.2