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alt_bn128_gt.cairo
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alt_bn128_gt.cairo
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from bigint import BigInt3, nondet_bigint3, bigint_mul
from alt_bn128_field import (
fq_zero,
is_zero,
FQ12,
nondet_fq12,
fq12_eq_zero,
fq12_sum,
fq12_diff,
fq12_is_zero,
fq12_zero,
unreducedFQ12,
)
from alt_bn128_g1 import G1Point
from alt_bn128_g2 import g2, G2Point
struct GTPoint {
x: FQ12,
y: FQ12,
}
// TODO Importing this from bn128_field yields a revoked variable (slope) issue
func fq12_mul{range_check_ptr}(a: FQ12, b: FQ12) -> (res: FQ12) {
%{
import sys, os
cwd = os.getcwd()
sys.path.append(cwd)
from utils.bn128_field import FQ, FQ12
from utils.bn128_utils import parse_fq12, print_g12
a = FQ12(list(map(FQ, parse_fq12(ids.a))))
b = FQ12(list(map(FQ, parse_fq12(ids.b))))
value = res = list(map(lambda x: x.n, (a*b).coeffs))
# print("a*b =", value)
%}
let (res) = nondet_fq12();
// TODO CHECKS
return (res=res);
}
// ### ADDITION, MULTIPLICATION
func gt_doubling_slope{range_check_ptr}(pt: GTPoint) -> (slope: FQ12) {
%{
from utils.bn128_field import FQ, FQ12
from utils.bn128_utils import parse_fq12
# Compute the slope.
x = FQ12(list(map(FQ, parse_fq12(ids.pt.x))))
y = FQ12(list(map(FQ, parse_fq12(ids.pt.y))))
slope = (3 * x ** 2) / (2 * y)
value = list(map(lambda x: x.n, slope.coeffs))
%}
let (slope: FQ12) = nondet_fq12();
// TODO VERIFY
return (slope=slope);
}
func gt_slope{range_check_ptr}(pt0: GTPoint, pt1: GTPoint) -> (slope: FQ12) {
%{
from utils.bn128_field import FQ, FQ12
from utils.bn128_utils import parse_fq12
# Compute the slope.
x0 = FQ12(list(map(FQ, parse_fq12(ids.pt0.x))))
y0 = FQ12(list(map(FQ, parse_fq12(ids.pt0.y))))
x1 = FQ12(list(map(FQ, parse_fq12(ids.pt1.x))))
y1 = FQ12(list(map(FQ, parse_fq12(ids.pt1.y))))
slope = (y0 - y1) / (x0 - x1)
value = list(map(lambda x: x.n, slope.coeffs))
%}
let (slope) = nondet_fq12();
// TODO verify
return (slope,);
}
// Given a point 'pt' on the elliptic curve, computes pt + pt.
func gt_double{range_check_ptr}(pt: GTPoint) -> (res: GTPoint) {
let (x_is_zero) = fq12_eq_zero(pt.x);
if (x_is_zero == 1) {
return (res=pt);
}
let (slope: FQ12) = gt_doubling_slope(pt);
let (slope_sqr: FQ12) = fq12_mul(slope, slope);
%{
from utils.bn128_field import FQ, FQ12
from utils.bn128_utils import parse_fq12
# Compute the slope.
x = FQ12(list(map(FQ, parse_fq12(ids.pt.x))))
y = FQ12(list(map(FQ, parse_fq12(ids.pt.y))))
slope = FQ12(list(map(FQ, parse_fq12(ids.slope))))
res = slope ** 2 - x * 2
value = new_x = list(map(lambda x: x.n, res.coeffs))
%}
let (new_x: FQ12) = nondet_fq12();
%{
new_x = FQ12(list(map(FQ, parse_fq12(ids.new_x))))
res = slope * (x - new_x) - y
value = new_x = list(map(lambda x: x.n, res.coeffs))
%}
let (new_y: FQ12) = nondet_fq12();
// VERIFY
// verify_zero5(
// UnreducedBigInt5(
// d0=slope_sqr.d0 - new_x.d0 - 2 * pt.x.d0,
// d1=slope_sqr.d1 - new_x.d1 - 2 * pt.x.d1,
// d2=slope_sqr.d2 - new_x.d2 - 2 * pt.x.d2,
// d3=slope_sqr.d3,
// d4=slope_sqr.d4))
// let (x_diff_slope : UnreducedBigInt5) = bigint_mul(
// BigInt3(d0=pt.x.d0 - new_x.d0, d1=pt.x.d1 - new_x.d1, d2=pt.x.d2 - new_x.d2), slope)
// verify_zero5(
// UnreducedBigInt5(
// d0=x_diff_slope.d0 - pt.y.d0 - new_y.d0,
// d1=x_diff_slope.d1 - pt.y.d1 - new_y.d1,
// d2=x_diff_slope.d2 - pt.y.d2 - new_y.d2,
// d3=x_diff_slope.d3,
// d4=x_diff_slope.d4))
return (GTPoint(new_x, new_y),);
}
func fast_gt_add{range_check_ptr}(pt0: GTPoint, pt1: GTPoint) -> (res: GTPoint) {
let (pt0_x_is_zero) = fq12_eq_zero(pt0.x);
if (pt0_x_is_zero == 1) {
return (pt1,);
}
let (pt1_x_is_zero) = fq12_eq_zero(pt1.x);
if (pt1_x_is_zero == 1) {
return (pt1,);
}
let (slope: FQ12) = gt_slope(pt0, pt1);
let (slope_sqr: FQ12) = fq12_mul(slope, slope);
%{
from utils.bn128_field import FQ, FQ12
from utils.bn128_utils import parse_fq12
# Compute the slope.
x0 = FQ12(list(map(FQ, parse_fq12(ids.pt0.x))))
x1 = FQ12(list(map(FQ, parse_fq12(ids.pt1.x))))
y0 = FQ12(list(map(FQ, parse_fq12(ids.pt0.y))))
slope = FQ12(list(map(FQ, parse_fq12(ids.slope))))
res = slope ** 2 - x0 - x1
value = new_x = list(map(lambda x: x.n, res.coeffs))
%}
let (new_x: FQ12) = nondet_fq12();
%{
new_x = res
res = slope * (x0 - new_x) - y0
value = new_x = list(map(lambda x: x.n, res.coeffs))
%}
let (new_y: FQ12) = nondet_fq12();
// verify_zero5(
// UnreducedBigInt5(
// d0=slope_sqr.d0 - new_x.d0 - pt0.x.d0 - pt1.x.d0,
// d1=slope_sqr.d1 - new_x.d1 - pt0.x.d1 - pt1.x.d1,
// d2=slope_sqr.d2 - new_x.d2 - pt0.x.d2 - pt1.x.d2,
// d3=slope_sqr.d3,
// d4=slope_sqr.d4))
// let (x_diff_slope : UnreducedBigInt5) = bigint_mul(
// BigInt3(d0=pt0.x.d0 - new_x.d0, d1=pt0.x.d1 - new_x.d1, d2=pt0.x.d2 - new_x.d2), slope)
// verify_zero5(
// UnreducedBigInt5(
// d0=x_diff_slope.d0 - pt0.y.d0 - new_y.d0,
// d1=x_diff_slope.d1 - pt0.y.d1 - new_y.d1,
// d2=x_diff_slope.d2 - pt0.y.d2 - new_y.d2,
// d3=x_diff_slope.d3,
// d4=x_diff_slope.d4))
return (GTPoint(new_x, new_y),);
}
func gt_add{range_check_ptr}(pt0: GTPoint, pt1: GTPoint) -> (res: GTPoint) {
let (x_diff) = fq12_diff(pt0.x, pt1.x);
let (same_x: felt) = fq12_is_zero(x_diff);
if (same_x == 0) {
return fast_gt_add(pt0, pt1);
}
// We have pt0.x = pt1.x. This implies pt0.y = ±pt1.y.
// Check whether pt0.y = -pt1.y.
let (y_sum) = fq12_sum(pt0.x, pt0.y);
let (opposite_y: felt) = fq12_is_zero(y_sum);
if (opposite_y != 0) {
// pt0.y = -pt1.y.
// Note that the case pt0 = pt1 = 0 falls into this branch as well.
let (zero_12) = fq12_zero();
let ZERO_POINT = GTPoint(zero_12, zero_12);
return (ZERO_POINT,);
} else {
// pt0.y = pt1.y.
return gt_double(pt0);
}
}
// ### CASTING G1 INTO GT
func g1_to_gt{range_check_ptr}(pt: G1Point) -> (res: GTPoint) {
// Point should not be zero
alloc_locals;
let (x_iszero) = is_zero(pt.x);
let (y_iszero) = is_zero(pt.y);
assert x_iszero + y_iszero = 0;
let (zero: BigInt3) = fq_zero();
return (
GTPoint(
x=FQ12(pt.x, zero, zero, zero,
zero, zero, zero, zero,
zero, zero, zero, zero),
y=FQ12(pt.y, zero, zero, zero,
zero, zero, zero, zero,
zero, zero, zero, zero)),
);
}
// ### TWISTING G2 INTO GT
func twist{range_check_ptr}(P: G2Point) -> (res: GTPoint) {
let (zero: BigInt3) = fq_zero();
tempvar x0 = P.x.e0;
tempvar x1 = P.x.e1;
let xx = BigInt3(d0=x0.d0 - 9 * x1.d0, d1=x0.d1 - 9 * x1.d1, d2=x0.d2 - 9 * x1.d2);
let nxw2 = FQ12(zero, zero, xx, zero, zero, zero, zero, zero, x1, zero, zero, zero);
tempvar y0 = P.y.e0;
tempvar y1 = P.y.e1;
let yy = BigInt3(d0=y0.d0 - 9 * y1.d0, d1=y0.d1 - 9 * y1.d1, d2=y0.d2 - 9 * y1.d2);
let nyw3 = FQ12(zero, zero, zero, yy, zero, zero, zero, zero, zero, y1, zero, zero);
return (res=GTPoint(x=nxw2, y=nyw3));
}
// CONSTANTS
func g12{range_check_ptr}() -> (res: GTPoint) {
let g2_tmp: G2Point = g2();
let res: GTPoint = twist(g2_tmp);
return (res=res);
}
func gt_two() -> (res: GTPoint) {
return (
GTPoint(
FQ12(
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=66531434795446507742202402, d1=57810563030407162761699450, d2=3024423940099633003033660),
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=29266951114122318337060217, d1=8315677858884295077185307, d2=2436188124856487536975890),
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
),
FQ12(
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
BigInt3(d0=20729108619955071395783599, d1=33713532092400076519474348, d2=1387780998518836325215322), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
BigInt3(d0=37820632520797176012333394, d1=58429338205645183884307771, d2=1916850345724626333016760), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
)),
);
}
func gt_three() -> (res: GTPoint) {
return (
GTPoint(
FQ12(
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=60558478434004798536211741, d1=43863242049195550535444726, d2=489660925987493189701501),
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=68458519662193915565862533, d1=8714965904636858911272353, d2=1214966188589858263872793),
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
),
FQ12(
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
BigInt3(d0=62542305924506548566602652, d1=11947492361179029427546200, d2=2636020001628383667142327), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
BigInt3(d0=64470951246555278864748978, d1=61710966665510361729249574, d2=160010759829214388101887), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
)),
);
}
func gt_negone() -> (res: GTPoint) {
return (
GTPoint(
FQ12(
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=37098765567079062928113790, d1=75069397608736819304955002, d2=2716309570043849407818057),
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=50657168248156029357068994, d1=75996009454876762764004566, d2=1931027739743020521039371),
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
),
FQ12(
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
BigInt3(d0=31925659635663368785745730, d1=76797642075525941605650950, d2=1061992001333544670783866), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
BigInt3(d0=75234859396250709295523308, d1=58200249186681967413131230, d2=2974432145097327839591194), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
)),
);
}
func gt_negtwo() -> (res: GTPoint) {
return (
GTPoint(
FQ12(
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=66531434795446507742202402, d1=57810563030407162761699450, d2=3024423940099633003033660),
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=29266951114122318337060217, d1=8315677858884295077185307, d2=2436188124856487536975890),
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
),
FQ12(
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
BigInt3(d0=39464779894232690824419736, d1=71283675355909246543773941, d2=2268601696092355443562665), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
BigInt3(d0=22373255993390586207869941, d1=46567869242664139178940518, d2=1739532348886565435761227), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
)),
);
}
func gt_negthree() -> (res: GTPoint) {
return (
GTPoint(
FQ12(
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=60558478434004798536211741, d1=43863242049195550535444726, d2=489660925987493189701501),
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=68458519662193915565862533, d1=8714965904636858911272353, d2=1214966188589858263872793),
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
),
FQ12(
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
BigInt3(d0=75022835045017480834795947, d1=15678462631794026454506824, d2=1020362692982808101635661), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
BigInt3(d0=73094189722968750536649621, d1=43286240782798961333998714, d2=3496371934781977380676100), BigInt3(d0=0, d1=0, d2=0), BigInt3(d0=0, d1=0, d2=0),
)),
);
}