16575888417

Committed 28 Jul 2025 05:28PM UTC coverage: 97.719% (+2.1%) from 95.58%

Build # 16575888417

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Pull #15

github

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web-flow

Commit Message

Merge 6503eb405 into d917b1701

Pull Request Pull Request #15: Improve bernstein math

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/src/Curves/BernsteinCurveFactory.php

<?php

namespace Famoser\Elliptic\Curves;

use Famoser\Elliptic\Math\MathInterface;
use Famoser\Elliptic\Math\Primitives\PrimeField;
use Famoser\Elliptic\Primitives\BirationalMap;
use Famoser\Elliptic\Primitives\Curve;
use Famoser\Elliptic\Primitives\CurveType;
use Famoser\Elliptic\Primitives\Point;

/**
 * Montgomery curves from https://datatracker.ietf.org/doc/html/rfc7748.
 * Popularized for fast elliptic curve cryptography in https://cr.yp.to/ecdh/curve25519-20060209.pdf
 *
 * Defined using evaluated values for easy and fast consumption.
 * See unit tests to check that these variables indeed correspond to their definition (e.g., that p = 2^255 - 19).
 */
class BernsteinCurveFactory
{
    public static function curve25519(): Curve
    {
        /**
         * sage:
         * F = GF(2^255 - 19)
         * e = EllipticCurve(F, [0, 486662, 0, 1, 0])
         * u = e(9, 43114425171068552920764898935933967039370386198203806730763910166200978582548)
         */

        // p = 2^255 - 19
        $p = gmp_init('7FFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFED', 16);
        $a = gmp_init(486662);
        $b = gmp_init(1);

        $u = gmp_init(9);
        // see https://www.rfc-editor.org/errata/eid4730; the v in the RFC is incorrect
        $v = gmp_init('431144251710685529207648989359339670393703861982038067307639101
66200978582548', 10);
        $P = new Point($u, $v);

        // order = 2^252 + 0x14def9dea2f79cd65812631a5cf5d3ed
        $order = gmp_init('10000000 00000000 00000000 00000000 14DEF9DE A2F79CD6 5812631A 5CF5D3ED', 16);
        $cofactor = gmp_init(8);

        return new Curve(CurveType::Montgomery, $p, $a, $b, $P, $order, $cofactor);
    }

    public static function curve25519ToEdwards25519(): BirationalMap
    {
        $p = gmp_init('7FFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFED', 16);
        $field = new PrimeField($p);

        // sqrt(-486664)
        // - calculated using sage: sqrt(GF(2^255-19)(-486664), all=True)
        // - then chosen the second value as it correctly converts the base points
        $squareRootOfMinus486664 = gmp_init('F26EDF46 0A006BBD 27B08DC0 3FC4F7EC 5A1D3D14 B7D1A82C C6E04AAF F457E06', 16);

        // (x, y) = (sqrt(-486664)*u/v, (u-1)/(u+1))
        $map = static function (MathInterface $math, Point $point) use ($field, $squareRootOfMinus486664) {
            $x = $field->mul(
                $field->mul($squareRootOfMinus486664, $point->x),
                /** @phpstan-ignore-next-line */
                $field->invert($point->y)
            );

            // if x is 0, short-cut to resolve y = 1 (following the normal flow would choose the other y^2 root)
            if (gmp_cmp($x, 0) === 0) {
                return $math->getInfinity();
            }

            $y = $field->mul(
                $field->sub($point->x, gmp_init(1)),
                /** @phpstan-ignore-next-line */
                $field->invert(
                    $field->add($point->x, gmp_init(1))
                )
            );

            return new Point($x, $y);
        };

        // (u, v) = ((1+y)/(1-y), sqrt(-486664)*u/x)
        $reverse = static function (MathInterface $math, Point $point) use ($field, $squareRootOfMinus486664) {
            $divisor = $field->sub(gmp_init(1), $point->y);
            if (gmp_cmp($divisor, 0) === 0) {
                return $math->getInfinity();
            }
            if (gmp_cmp($point->x, 0) === 0) {
                return $math->getInfinity();
            }

            $u = $field->mul(
                $field->add(gmp_init(1), $point->y),
                /** @phpstan-ignore-next-line */
                $field->invert($divisor)
            );

            $v = $field->mul(
                $field->mul($squareRootOfMinus486664, $u),
                /** @phpstan-ignore-next-line */
                $field->invert($point->x)
            );

            return new Point($u, $v);
        };

        return new BirationalMap($map, $reverse);
    }

    public static function edwards25519(): Curve
    {
        // p = 2^255 - 19
        $p = gmp_init('7FFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFED', 16);
        // a = -1 mod p
        $a = gmp_init('7FFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFEC', 16);
        $d = gmp_init('370957059346694393431380835087545651895421138798432190163887855330
      85940283555', 10);

        $x = gmp_init('151122213495354007725011514095885315114540126930418572060461132
      83949847762202');
        $y = gmp_init('463168356949264781694283940034751631413079938662562256157830336
      03165251855960', 10);
        $P = new Point($x, $y);

        // order = 2^252 + 0x14def9dea2f79cd65812631a5cf5d3ed
        $order = gmp_init('10000000 00000000 00000000 00000000 14DEF9DE A2F79CD6 5812631A 5CF5D3ED', 16);
        $cofactor = gmp_init(8);

        return new Curve(CurveType::TwistedEdwards, $p, $a, $d, $P, $order, $cofactor);
    }

    public static function curve448(): Curve
    {
        /**
         * sage:
         * F = GF(2^448 - 2^224 - 1)
         * e = EllipticCurve(F, [0, 156326, 0, 1, 0])
         * u = e(5, 355293926785568175264127502063783334808976399387714271831880898435169088786967410002932673765864550910142774147268105838985595290606362)
         */

        // p = 2^448 - 2^224 - 1
        $p = gmp_init('FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF', 16);
        $a = gmp_init(156326);
        $b = gmp_init(1);

        $u = gmp_init(5);
        $v = gmp_init('355293926785568175264127502063783334808976399387714271831880898
      435169088786967410002932673765864550910142774147268105838985595290
      606362', 10);
        $P = new Point($u, $v);

        // order = 2^446 - 0x8335dc163bb124b65129c96fde933d8d723a70aadc873d6d54a7bb0d
        $order = gmp_init('3FFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF 7CCA23E9 C44EDB49 AED63690 216CC272 8DC58F55 2378C292 AB5844F3', 16);
        $cofactor = gmp_init(4);

        return new Curve(CurveType::Montgomery, $p, $a, $b, $P, $order, $cofactor);
    }

    public static function curve448ToEdwards(): BirationalMap
    {
        $p = gmp_init('FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF', 16);
        $field = new PrimeField($p);

        // sqrt(156324)
        // - calculated using sage: sqrt(GF(2^448-2^224-1)(156324), all=True)
        // - then chosen the first value as it correctly converts the base points
        $squareRootOf15634 = gmp_init('45B2C5F7 D649EED0 77ED1AE4 5F44D541 43E34F71 4B71AA96 C945AF01 2D182975 0734CDE9 FADDBDA4 C066F7ED 54419CA5 2C85DE1E 8AAE4E6C', 16);

        // (x, y) = (sqrt(156324)*u/v, (1+u)/(1-u))
        $map = static function (MathInterface $math, Point $point) use ($field, $squareRootOf15634) {
            $x = $field->mul(
                $field->mul($squareRootOf15634, $point->x),
                /** @phpstan-ignore-next-line */
                $field->invert($point->y)
            );
            // note: if x === 0, then y === 1 (as opposed to the other map from curve25519 to edwards25519
            $y = $field->mul(
                $field->add(gmp_init(1), $point->x),
                /** @phpstan-ignore-next-line */
                $field->invert(
                    $field->sub(gmp_init(1), $point->x)
                )
            );

            return new Point($x, $y);
        };

        // (u, v) = ((y-1)/(y+1), sqrt(156324)*u/x)
        $reverse = static function (MathInterface $math, Point $point) use ($field, $squareRootOf15634) {
            $divisor = $field->add($point->y, gmp_init(1));
            if (gmp_cmp($divisor, 0) === 0) {
                return $math->getInfinity();
            }
            if (gmp_cmp($point->x, 0) === 0) {
                return $math->getInfinity();
            }

            $u = $field->mul(
                $field->sub($point->y, gmp_init(1)),
                /** @phpstan-ignore-next-line */
                $field->invert($divisor)
            );

            $v = $field->mul(
                $field->mul($squareRootOf15634, $u),
                /** @phpstan-ignore-next-line */
                $field->invert($point->x)
            );

            return new Point($u, $v);
        };

        return new BirationalMap($map, $reverse);
    }

    public static function curve448Edwards(): Curve
    {
        // p = 2^448 - 2^224 - 1
        $p = gmp_init('FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF', 16);
        // a = 1 mod p
        $a = gmp_init('1', 10);
        $d = gmp_init('611975850744529176160423220965553317543219696871016626328968936415
      087860042636474891785599283666020414768678979989378147065462815545
      017', 10);

        $x = gmp_init('345397493039729516374008604150537410266655260075183290216406970
      281645695073672344430481787759340633221708391583424041788924124567
      700732', 10);
        $y = gmp_init('363419362147803445274661903944002267176820680343659030140745099
      590306164083365386343198191849338272965044442230921818680526749009
      182718', 10);
        $P = new Point($x, $y);

        // order = 2^446 - 0x8335dc163bb124b65129c96fde933d8d723a70aadc873d6d54a7bb0d
        $order = gmp_init('3FFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF 7CCA23E9 C44EDB49 AED63690 216CC272 8DC58F55 2378C292 AB5844F3', 16);
        $cofactor = gmp_init(4);

        return new Curve(CurveType::Edwards, $p, $a, $d, $P, $order, $cofactor);
    }

    public static function edwards448(): Curve
    {
        // p = 2^448 - 2^224 - 1
        $p = gmp_init('FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF', 16);
        // a = 1 mod p
        $a = gmp_init('1', 10);
        $d = gmp_init('FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFF6756 ', 16);

        $x = gmp_init('224580040295924300187604334099896036246789641632564134246125461
      686950415467406032909029192869357953282578032075146446173674602635
      247710', 10);
        $y = gmp_init('298819210078481492676017930443930673437544040154080242095928241
      372331506189835876003536878655418784733982303233503462500531545062
      832660', 10);
        $P = new Point($x, $y);

        // order = 2^446 - 0x8335dc163bb124b65129c96fde933d8d723a70aadc873d6d54a7bb0d
        $order = gmp_init('3FFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF 7CCA23E9 C44EDB49 AED63690 216CC272 8DC58F55 2378C292 AB5844F3', 16);
        $cofactor = gmp_init(4);

        return new Curve(CurveType::Edwards, $p, $a, $d, $P, $order, $cofactor);
    }
}

1	<?php
2
3	namespace Famoser\Elliptic\Curves;
4
5	use Famoser\Elliptic\Math\MathInterface;
6	use Famoser\Elliptic\Math\Primitives\PrimeField;
7	use Famoser\Elliptic\Primitives\BirationalMap;
8	use Famoser\Elliptic\Primitives\Curve;
9	use Famoser\Elliptic\Primitives\CurveType;
10	use Famoser\Elliptic\Primitives\Point;
11
12	/**
13	* Montgomery curves from https://datatracker.ietf.org/doc/html/rfc7748.
14	* Popularized for fast elliptic curve cryptography in https://cr.yp.to/ecdh/curve25519-20060209.pdf
15	*
16	* Defined using evaluated values for easy and fast consumption.
17	* See unit tests to check that these variables indeed correspond to their definition (e.g., that p = 2^255 - 19).
18	*/
19	class BernsteinCurveFactory
20	{
21	public static function curve25519(): Curve	21✔
22	{
23	/**
24	* sage:
25	* F = GF(2^255 - 19)
26	* e = EllipticCurve(F, [0, 486662, 0, 1, 0])
27	* u = e(9, 43114425171068552920764898935933967039370386198203806730763910166200978582548)
28	*/
29
30	// p = 2^255 - 19
31	$p = gmp_init('7FFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFED', 16);	21✔
32	$a = gmp_init(486662);	21✔
33	$b = gmp_init(1);	21✔
34
35	$u = gmp_init(9);	21✔
36	// see https://www.rfc-editor.org/errata/eid4730; the v in the RFC is incorrect
37	$v = gmp_init('431144251710685529207648989359339670393703861982038067307639101	21✔
38	66200978582548', 10);	21✔
39	$P = new Point($u, $v);	21✔
40
41	// order = 2^252 + 0x14def9dea2f79cd65812631a5cf5d3ed
42	$order = gmp_init('10000000 00000000 00000000 00000000 14DEF9DE A2F79CD6 5812631A 5CF5D3ED', 16);	21✔
43	$cofactor = gmp_init(8);	21✔
44
45	return new Curve(CurveType::Montgomery, $p, $a, $b, $P, $order, $cofactor);	21✔
46	}
47
48	public static function curve25519ToEdwards25519(): BirationalMap	20✔
49	{
50	$p = gmp_init('7FFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFED', 16);	12✔
51	$field = new PrimeField($p);	12✔
52
53	// sqrt(-486664)
54	// - calculated using sage: sqrt(GF(2^255-19)(-486664), all=True)
55	// - then chosen the second value as it correctly converts the base points
56	$squareRootOfMinus486664 = gmp_init('F26EDF46 0A006BBD 27B08DC0 3FC4F7EC 5A1D3D14 B7D1A82C C6E04AAF F457E06', 16);	12✔
57
58	// (x, y) = (sqrt(-486664)*u/v, (u-1)/(u+1))
59	$map = static function (MathInterface $math, Point $point) use ($field, $squareRootOfMinus486664) {	20✔
60	$x = $field->mul(	10✔
61	$field->mul($squareRootOfMinus486664, $point->x),	10✔
62	/** @phpstan-ignore-next-line */
63	$field->invert($point->y)	10✔
64	);	10✔
65
66	// if x is 0, short-cut to resolve y = 1 (following the normal flow would choose the other y^2 root)
67	if (gmp_cmp($x, 0) === 0) {	10✔
68	return $math->getInfinity();	1✔
69	}
70
71	$y = $field->mul(	10✔
72	$field->sub($point->x, gmp_init(1)),	10✔
73	/** @phpstan-ignore-next-line */
74	$field->invert(	10✔
75	$field->add($point->x, gmp_init(1))	10✔
76	)	10✔
77	);	10✔
78
79	return new Point($x, $y);	10✔
80	};	20✔
81
82	// (u, v) = ((1+y)/(1-y), sqrt(-486664)*u/x)
83	$reverse = static function (MathInterface $math, Point $point) use ($field, $squareRootOfMinus486664) {	20✔
84	$divisor = $field->sub(gmp_init(1), $point->y);	10✔
85	if (gmp_cmp($divisor, 0) === 0) {	10✔
86	return $math->getInfinity();	1✔
87	}
88	if (gmp_cmp($point->x, 0) === 0) {	10✔
NEW 89	return $math->getInfinity();	×
90	}
91
92	$u = $field->mul(	10✔
93	$field->add(gmp_init(1), $point->y),	10✔
94	/** @phpstan-ignore-next-line */
95	$field->invert($divisor)	10✔
96	);	10✔
97
98	$v = $field->mul(	10✔
99	$field->mul($squareRootOfMinus486664, $u),	10✔
100	/** @phpstan-ignore-next-line */
101	$field->invert($point->x)	10✔
102	);	10✔
103
104	return new Point($u, $v);	10✔
105	};	20✔
106
107	return new BirationalMap($map, $reverse);	12✔
108	}
109
110	public static function edwards25519(): Curve	4✔
111	{
112	// p = 2^255 - 19
113	$p = gmp_init('7FFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFED', 16);	4✔
114	// a = -1 mod p
115	$a = gmp_init('7FFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFEC', 16);	4✔
116	$d = gmp_init('370957059346694393431380835087545651895421138798432190163887855330	4✔
117	85940283555', 10);	4✔
118
119	$x = gmp_init('151122213495354007725011514095885315114540126930418572060461132	4✔
120	83949847762202');	4✔
121	$y = gmp_init('463168356949264781694283940034751631413079938662562256157830336	4✔
122	03165251855960', 10);	4✔
123	$P = new Point($x, $y);	4✔
124
125	// order = 2^252 + 0x14def9dea2f79cd65812631a5cf5d3ed
126	$order = gmp_init('10000000 00000000 00000000 00000000 14DEF9DE A2F79CD6 5812631A 5CF5D3ED', 16);	4✔
127	$cofactor = gmp_init(8);	4✔
128
129	return new Curve(CurveType::TwistedEdwards, $p, $a, $d, $P, $order, $cofactor);	4✔
130	}
131
132	public static function curve448(): Curve	6✔
133	{
134	/**
135	* sage:
136	* F = GF(2^448 - 2^224 - 1)
137	* e = EllipticCurve(F, [0, 156326, 0, 1, 0])
138	* u = e(5, 355293926785568175264127502063783334808976399387714271831880898435169088786967410002932673765864550910142774147268105838985595290606362)
139	*/
140
141	// p = 2^448 - 2^224 - 1
142	$p = gmp_init('FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF', 16);	6✔
143	$a = gmp_init(156326);	6✔
144	$b = gmp_init(1);	6✔
145
146	$u = gmp_init(5);	6✔
147	$v = gmp_init('355293926785568175264127502063783334808976399387714271831880898	6✔
148	435169088786967410002932673765864550910142774147268105838985595290
149	606362', 10);	6✔
150	$P = new Point($u, $v);	6✔
151
152	// order = 2^446 - 0x8335dc163bb124b65129c96fde933d8d723a70aadc873d6d54a7bb0d
153	$order = gmp_init('3FFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF 7CCA23E9 C44EDB49 AED63690 216CC272 8DC58F55 2378C292 AB5844F3', 16);	6✔
154	$cofactor = gmp_init(4);	6✔
155
156	return new Curve(CurveType::Montgomery, $p, $a, $b, $P, $order, $cofactor);	6✔
157	}
158
159	public static function curve448ToEdwards(): BirationalMap	9✔
160	{
161	$p = gmp_init('FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF', 16);	1✔
162	$field = new PrimeField($p);	1✔
163
164	// sqrt(156324)
165	// - calculated using sage: sqrt(GF(2^448-2^224-1)(156324), all=True)
166	// - then chosen the first value as it correctly converts the base points
167	$squareRootOf15634 = gmp_init('45B2C5F7 D649EED0 77ED1AE4 5F44D541 43E34F71 4B71AA96 C945AF01 2D182975 0734CDE9 FADDBDA4 C066F7ED 54419CA5 2C85DE1E 8AAE4E6C', 16);	1✔
168
169	// (x, y) = (sqrt(156324)*u/v, (1+u)/(1-u))
170	$map = static function (MathInterface $math, Point $point) use ($field, $squareRootOf15634) {	9✔
171	$x = $field->mul(	9✔
172	$field->mul($squareRootOf15634, $point->x),	9✔
173	/** @phpstan-ignore-next-line */
174	$field->invert($point->y)	9✔
175	);	9✔
176	// note: if x === 0, then y === 1 (as opposed to the other map from curve25519 to edwards25519
177	$y = $field->mul(	9✔
178	$field->add(gmp_init(1), $point->x),	9✔
179	/** @phpstan-ignore-next-line */
180	$field->invert(	9✔
181	$field->sub(gmp_init(1), $point->x)	9✔
182	)	9✔
183	);	9✔
184
185	return new Point($x, $y);	9✔
186	};	9✔
187
188	// (u, v) = ((y-1)/(y+1), sqrt(156324)*u/x)
189	$reverse = static function (MathInterface $math, Point $point) use ($field, $squareRootOf15634) {	9✔
190	$divisor = $field->add($point->y, gmp_init(1));	9✔
191	if (gmp_cmp($divisor, 0) === 0) {	9✔
NEW 192	return $math->getInfinity();	×
193	}
194	if (gmp_cmp($point->x, 0) === 0) {	9✔
195	return $math->getInfinity();	2✔
196	}
197
198	$u = $field->mul(	9✔
199	$field->sub($point->y, gmp_init(1)),	9✔
200	/** @phpstan-ignore-next-line */
201	$field->invert($divisor)	9✔
202	);	9✔
203
204	$v = $field->mul(	9✔
205	$field->mul($squareRootOf15634, $u),	9✔
206	/** @phpstan-ignore-next-line */
207	$field->invert($point->x)	9✔
208	);	9✔
209
210	return new Point($u, $v);	9✔
211	};	9✔
212
213	return new BirationalMap($map, $reverse);	1✔
214	}
215
216	public static function curve448Edwards(): Curve	2✔
217	{
218	// p = 2^448 - 2^224 - 1
219	$p = gmp_init('FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF', 16);	2✔
220	// a = 1 mod p
221	$a = gmp_init('1', 10);	2✔
222	$d = gmp_init('611975850744529176160423220965553317543219696871016626328968936415	2✔
223	087860042636474891785599283666020414768678979989378147065462815545
224	017', 10);	2✔
225
226	$x = gmp_init('345397493039729516374008604150537410266655260075183290216406970	2✔
227	281645695073672344430481787759340633221708391583424041788924124567
228	700732', 10);	2✔
229	$y = gmp_init('363419362147803445274661903944002267176820680343659030140745099	2✔
230	590306164083365386343198191849338272965044442230921818680526749009
231	182718', 10);	2✔
232	$P = new Point($x, $y);	2✔
233
234	// order = 2^446 - 0x8335dc163bb124b65129c96fde933d8d723a70aadc873d6d54a7bb0d
235	$order = gmp_init('3FFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF 7CCA23E9 C44EDB49 AED63690 216CC272 8DC58F55 2378C292 AB5844F3', 16);	2✔
236	$cofactor = gmp_init(4);	2✔
237
238	return new Curve(CurveType::Edwards, $p, $a, $d, $P, $order, $cofactor);	2✔
239	}
240
241	public static function edwards448(): Curve	1✔
242	{
243	// p = 2^448 - 2^224 - 1
244	$p = gmp_init('FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF', 16);	1✔
245	// a = 1 mod p
246	$a = gmp_init('1', 10);	1✔
247	$d = gmp_init('FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFF6756 ', 16);	1✔
248
249	$x = gmp_init('224580040295924300187604334099896036246789641632564134246125461	1✔
250	686950415467406032909029192869357953282578032075146446173674602635
251	247710', 10);	1✔
252	$y = gmp_init('298819210078481492676017930443930673437544040154080242095928241	1✔
253	372331506189835876003536878655418784733982303233503462500531545062
254	832660', 10);	1✔
255	$P = new Point($x, $y);	1✔
256
257	// order = 2^446 - 0x8335dc163bb124b65129c96fde933d8d723a70aadc873d6d54a7bb0d
258	$order = gmp_init('3FFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF 7CCA23E9 C44EDB49 AED63690 216CC272 8DC58F55 2378C292 AB5844F3', 16);	1✔
259	$cofactor = gmp_init(4);	1✔
260
261	return new Curve(CurveType::Edwards, $p, $a, $d, $P, $order, $cofactor);	1✔
262	}
263	}

famoser / elliptic / 16575888417

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