From 90e2a57614f0c3fff4102bc80ffa89949b2b2960 Mon Sep 17 00:00:00 2001 From: vbuterin Date: Mon, 28 Jan 2019 10:28:05 -0600 Subject: [PATCH] Modular squareroot clarification --- specs/bls_signature.md | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) diff --git a/specs/bls_signature.md b/specs/bls_signature.md index 4dd479a6a..034de479b 100644 --- a/specs/bls_signature.md +++ b/specs/bls_signature.md @@ -86,13 +86,15 @@ def hash_to_G2(message: bytes32, domain: uint64) -> [uint384]: ### `modular_squareroot` -`modular_squareroot(x)` returns a solution `y` to `y**2 % q == x`, and `None` if none exists. If there are two solutions the one with higher imaginary component is favored; if both solutions have equal imaginary component the one with higher real component is favored. +`modular_squareroot(x)` returns a solution `y` to `y**2 % q == x`, and `None` if none exists. If there are two solutions the one with higher imaginary component is favored; if both solutions have equal imaginary component the one with higher real component is favored (note that this is equivalent to saying that the single solution with either imaginary component > p/2 or imaginary component zero and real component > p/2 is favored). + +The following is a sample implementation; implementers are free to implement modular square roots as they wish. Note that `x2 = -x1` is an _additive modular inverse_ so real and imaginary coefficients remain in `[0 .. q-1]` ```python Fq2_order = q ** 2 - 1 eighth_roots_of_unity = [Fq2([1,1]) ** ((Fq2_order * k) // 8) for k in range(8)] -def modular_squareroot(value: int) -> int: +def modular_squareroot(value: Fq2) -> Fq2: candidate_squareroot = value ** ((Fq2_order + 8) // 16) check = candidate_squareroot ** 2 / value if check in eighth_roots_of_unity[::2]: