Genus-1 curves over large-characteristic fields

Doubling-oriented Doche–Icart–Kohel curves: y^2=x^3+ax^2+16a*x

Standard coordinates: x=X/Z, y=Y/ZZ, ZZ=Z^2

Tripling-oriented Doche–Icart–Kohel curves: y^2=x^3+3a(x+1)^2

Standard coordinates: x=X/Z^2, y=Y/Z^3, ZZ=Z^2

Edwards curves: x^2+y^2=c^2(1+dx^2*y^2)

Inverted coordinates: x=Z/X, y=Z/Y
Projective coordinates: x=X/Z, y=Y/Z

Hessian curves: x^3+y^3+1=3dx*y

Extended coordinates: x=X/Z, y=Y/Z, XX=X*X, YY=Y*Y, ZZ=Z*Z, XY=2*X*Y, XZ=2*X*Z, YZ=2*Y*Z
Projective coordinates: x=X/Z, y=Y/Z

Jacobi intersections: s^2+c^2=1, a*s^2+d^2=1

Extended coordinates: s=S/Z, c=C/Z, d=D/Z, SC=S*C, DZ=D*Z
Projective coordinates: s=S/Z, c=C/Z, d=D/Z

Jacobi quartics: y^2=x^4+2ax^2+1

Doubling-oriented XXYZZ coordinates: x=X/Z, y=Y/ZZ, XX=X^2, ZZ=Z^2, a^2+c^2=1
Doubling-oriented XXYZZR coordinates: x=X/Z, y=Y/ZZ, XX=X^2, ZZ=Z^2, R=2*X*Z, a^2+c^2=1
Doubling-oriented XYZ coordinates: x=X/Z, y=Y/Z^2, a^2+c^2=1
XXYZZ coordinates: x=X/Z, y=Y/ZZ, XX=X^2, ZZ=Z^2
XXYZZR coordinates: x=X/Z, y=Y/ZZ, XX=X^2, ZZ=Z^2, R=2*X*Z
XYZ coordinates: x=X/Z, y=Y/Z^2

Montgomery curves: by^2=x^3+ax^2+x

XZ coordinates: x=X/Z

Short Weierstrass curves: y^2=x^3+a*x+b

Jacobian coordinates with a4=-3: x=X/Z^2, y=Y/Z^3, a=-3
Jacobian coordinates: x=X/Z^2, y=Y/Z^3
Projective coordinates with a4=-1: x=X/Z, y=Y/Z, a=-1
Projective coordinates with a4=-3: x=X/Z, y=Y/Z, a=-3
Projective coordinates: x=X/Z, y=Y/Z
XYZZ coordinates with a4=-3: x=X/ZZ, y=Y/ZZZ, ZZ^3=ZZZ^2, a=-3
XYZZ coordinates: x=X/ZZ, y=Y/ZZZ, ZZ^3=ZZZ^2

The EFD currently includes 316 explicit formulas for addition, doubling, tripling, and scaling of points in 22 representations on 8 shapes of genus-1 curves over large-characteristic fields.

Speed tables

The following speed tables distinguish between eighteen different operations:

DBL: Doubling. P1 |-> P1+P1.
ADD: Addition. P1,P2 |-> P1+P2.
reADD: Readdition; i.e., addition of an input that has been added before, when all reusable intermediate results are cached. P1,P2 |-> P1+P2 when all results depending solely on P2 are cached. Classic example: typical formulas to compute (X1:Y1:Z1) + (X2:Y2:Z2) in Jacobian coordinates begin by computing Z1^2, Z1^3, Z2^2, and Z2^3; caching Z2^2 and Z2^3 saves 1M+1S in a readdition of (X2:Y2:Z2).
preADD: Preaddition; i.e., addition compatible with the fastest readdition. Sometimes ADD is faster than reADD because there are faster addition formulas providing slower readdition.
mADD: Mixed addition; i.e., addition of an input that has been scaled to have Z-coordinate 1. P1,P2 |-> P1+P2 when Z(P2)=1.
mreADD: Mixed readdition.
mpreADD: Mixed preaddition.
mDBL: Mixed doubling; i.e., doubling of an input that has been scaled to have Z-coordinate 1. P1 |-> P1+P1 when Z(P1)=1.
mmADD: Addition of two inputs that have been scaled to have Z-coordinate 1. P1,P2 |-> P1+P2 when Z(P1)=1 and Z(P2)=1.
mmreADD: Addition of two inputs that have been scaled to have Z-coordinate 1, where the second input has been added before.
mmpreADD.
SUNI: Strongly unified addition; i.e., addition that also works without change for doubling.
TPL: Tripling. P1 |-> P1+P1+P1.
DADD: Differential addition, i.e., addition of points whose difference is known. P3-P2,P2,P3 |-> P3+P2.
mDADD: Mixed differential addition, i.e., addition of points whose difference is known and whose difference has Z-coordinate 1. P3-P2,P2,P3 |-> P3+P2 when Z(P3-P2)=1.
LADD: A ladder step; i.e., a differential addition and a doubling of one summand. P3-P2,P2,P3 |-> P2+P2,P3+P2.
mLADD: A mixed ladder step; i.e., a mixed differential addition and a doubling of one summand. P3-P2,P2,P3 |-> P2+P2,P3+P2 when Z(P3-P2)=1.
SCALE: Scaling to have Z-coordinate 1.

The tables are sorted in decreasing order of DBL cost; ties are broken in decreasing order of ADD cost; ties are broken in decreasing order of reADD cost; etc.

Smallest multiplication counts assuming I=100M, S=1M, *param=0M, add=0M, *const=0M:

Curve shape, representation DBL ADD reADD preADD mADD mreADD mpreADD mDBL mmADD mmreADD mmpreADD SUNI TPL DADD mDADD LADD mLADD SCALE
Short Weierstrass, projective 11 14 14 14 11 11 11 8 7 7 7 17 102

Short Weierstrass, projective with a4=-1 11 14 14 14 11 11 11 8 7 7 7 16 102

Short Weierstrass, XYZZ 10 14 14 14 14 14 14 10 14 14 14 104

Short Weierstrass, projective with a4=-3 10 14 14 14 11 11 11 8 7 7 7 17 102

Tripling-oriented Doche–Icart–Kohel, standard 9 17 16 17 11 11 11 6 6 6 6 12 104

Short Weierstrass, Jacobian 9 16 14 16 11 11 11 6 6 6 6 15 104

Short Weierstrass, XYZZ with a4=-3 9 14 14 14 14 14 14 9 14 14 14 104

Hessian, extended 9 12 12 12 11 11 11 9 11 11 11 105

Short Weierstrass, Jacobian with a4=-3 8 16 14 16 11 11 11 6 6 6 6 14 104

Jacobi quartic, doubling-oriented XYZ 8 13 11 14 11 11 11 5 7 7 7 13 103

Jacobi quartic, XYZ 8 13 11 14 11 11 11 5 7 7 7 13 103

Hessian, projective 8 12 12 12 10 10 10 6 8 7 8 14 102

Doubling-oriented Doche–Icart–Kohel, standard 7 17 17 17 12 12 12 6 8 8 8 103

Jacobi intersection, projective 7 14 12 14 12 11 12 6 10 9 10 14 14 103

Jacobi intersection, extended 7 12 12 12 11 11 11 7 11 11 11 12 16 104

Edwards, projective 7 11 11 11 9 9 9 6 7 7 7 11 13 102

Jacobi quartic, doubling-oriented XXYZZ 7 11 10 11 9 9 9 6 9 9 9 11 14 104

Jacobi quartic, XXYZZ 7 11 10 11 9 9 9 6 9 9 9 11 14 104

Jacobi quartic, XXYZZR 7 10 10 10 9 9 9 7 9 9 9 10 15 103

Jacobi quartic, doubling-oriented XXYZZR 7 10 10 10 9 9 9 7 9 9 9 10 15 103

Edwards, inverted 7 10 10 10 9 9 9 6 7 7 7 10 13 102

Montgomery, XZ 4 4 6 5 10 9 101

Curve shape, representation	DBL	ADD	reADD	preADD	mADD	mreADD	mpreADD	mDBL	mmADD	mmreADD	mmpreADD	SUNI	TPL	DADD	mDADD	LADD	mLADD	SCALE
Short Weierstrass, projective	11	14	14	14	11	11	11	8	7	7	7	17						102
Short Weierstrass, projective with a4=-1	11	14	14	14	11	11	11	8	7	7	7	16						102
Short Weierstrass, XYZZ	10	14	14	14	14	14	14	10	14	14	14							104
Short Weierstrass, projective with a4=-3	10	14	14	14	11	11	11	8	7	7	7	17						102
Tripling-oriented Doche–Icart–Kohel, standard	9	17	16	17	11	11	11	6	6	6	6		12					104
Short Weierstrass, Jacobian	9	16	14	16	11	11	11	6	6	6	6		15					104
Short Weierstrass, XYZZ with a4=-3	9	14	14	14	14	14	14	9	14	14	14							104
Hessian, extended	9	12	12	12	11	11	11	9	11	11	11							105
Short Weierstrass, Jacobian with a4=-3	8	16	14	16	11	11	11	6	6	6	6		14					104
Jacobi quartic, doubling-oriented XYZ	8	13	11	14	11	11	11	5	7	7	7	13						103
Jacobi quartic, XYZ	8	13	11	14	11	11	11	5	7	7	7	13						103
Hessian, projective	8	12	12	12	10	10	10	6	8	7	8		14					102
Doubling-oriented Doche–Icart–Kohel, standard	7	17	17	17	12	12	12	6	8	8	8							103
Jacobi intersection, projective	7	14	12	14	12	11	12	6	10	9	10	14	14					103
Jacobi intersection, extended	7	12	12	12	11	11	11	7	11	11	11	12	16					104
Edwards, projective	7	11	11	11	9	9	9	6	7	7	7	11	13					102
Jacobi quartic, doubling-oriented XXYZZ	7	11	10	11	9	9	9	6	9	9	9	11	14					104
Jacobi quartic, XXYZZ	7	11	10	11	9	9	9	6	9	9	9	11	14					104
Jacobi quartic, XXYZZR	7	10	10	10	9	9	9	7	9	9	9	10	15					103
Jacobi quartic, doubling-oriented XXYZZR	7	10	10	10	9	9	9	7	9	9	9	10	15					103
Edwards, inverted	7	10	10	10	9	9	9	6	7	7	7	10	13					102
Montgomery, XZ	4							4						6	5	10	9	101

Smallest multiplication counts assuming I=100M, S=0.8M, *param=0M, add=0M, *const=0M:

Curve shape, representation DBL ADD reADD preADD mADD mreADD mpreADD mDBL mmADD mmreADD mmpreADD SUNI TPL DADD mDADD LADD mLADD SCALE
Short Weierstrass, projective 9.8 13.6 13.6 13.6 10.6 10.6 10.6 7.0 6.6 6.6 6.6 15.8 102.0

Short Weierstrass, projective with a4=-1 9.8 13.6 13.6 13.6 10.6 10.6 10.6 7.0 6.6 6.6 6.6 15.4 102.0

Short Weierstrass, projective with a4=-3 9.4 13.6 13.6 13.6 10.6 10.6 10.6 7.0 6.6 6.6 6.6 15.8 102.0

Short Weierstrass, XYZZ 9.2 13.6 13.6 13.6 13.6 13.6 13.6 9.2 13.6 13.6 13.6 103.8

Short Weierstrass, XYZZ with a4=-3 8.6 13.6 13.6 13.6 13.6 13.6 13.6 8.6 13.6 13.6 13.6 103.8

Hessian, projective 7.8 12.0 10.8 15.6 10.0 10.0 10.0 5.4 8.0 7.0 8.0 12.8 102.0

Hessian, extended 7.8 10.8 10.8 10.8 9.8 9.8 9.8 7.8 9.8 9.8 9.8 104.6

Tripling-oriented Doche–Icart–Kohel, standard 7.6 15.8 14.8 15.8 10.2 10.2 10.2 5.0 5.6 5.6 5.6 10.8 103.8

Short Weierstrass, Jacobian 7.4 15.0 13.2 15.0 10.2 10.2 10.2 5.0 5.6 5.6 5.6 13.0 103.8

Short Weierstrass, Jacobian with a4=-3 7.0 15.0 13.2 15.0 10.2 10.2 10.2 5.0 5.6 5.6 5.6 12.6 103.8

Jacobi quartic, XYZ 6.8 12.4 10.4 12.8 10.4 10.4 10.4 4.2 6.6 6.6 6.6 12.4 102.8

Jacobi quartic, doubling-oriented XYZ 6.6 12.4 10.4 12.8 10.4 10.4 10.4 4.2 6.6 6.6 6.6 12.4 102.8

Edwards, projective 6.2 10.8 10.8 10.8 9.0 9.0 9.0 5.4 6.8 6.8 6.8 10.8 12.2 102.0

Jacobi quartic, doubling-oriented XXYZZ 6.2 10.2 9.4 10.2 8.4 8.4 8.4 5.0 8.4 8.4 8.4 10.2 12.8 103.6

Jacobi quartic, XXYZZ 6.2 10.2 9.4 10.2 8.4 8.4 8.4 5.0 8.4 8.4 8.4 10.2 12.8 103.6

Edwards, inverted 6.2 9.8 9.8 9.8 8.8 8.8 8.8 5.4 7.0 7.0 7.0 9.8 12.2 102.0

Jacobi quartic, XXYZZR 6.2 9.4 9.4 9.4 8.4 8.4 8.4 5.8 8.2 8.2 8.2 9.4 12.8 102.8

Jacobi quartic, doubling-oriented XXYZZR 6.2 9.4 9.4 9.4 8.4 8.4 8.4 5.8 8.2 8.2 8.2 9.4 12.8 102.8

Doubling-oriented Doche–Icart–Kohel, standard 6.0 16.0 16.0 16.0 11.2 11.2 11.2 5.0 7.2 7.2 7.2 102.8

Jacobi intersection, projective 6.0 13.8 11.8 13.8 11.8 10.8 11.8 5.2 9.6 8.8 9.6 13.8 12.0 103.0

Jacobi intersection, extended 6.0 11.8 11.8 11.8 10.8 10.8 10.8 6.0 10.8 10.8 10.8 11.8 14.0 104.0

Montgomery, XZ 3.6 3.6 5.6 4.6 9.2 8.2 101.0

Curve shape, representation	DBL	ADD	reADD	preADD	mADD	mreADD	mpreADD	mDBL	mmADD	mmreADD	mmpreADD	SUNI	TPL	DADD	mDADD	LADD	mLADD	SCALE
Short Weierstrass, projective	9.8	13.6	13.6	13.6	10.6	10.6	10.6	7.0	6.6	6.6	6.6	15.8						102.0
Short Weierstrass, projective with a4=-1	9.8	13.6	13.6	13.6	10.6	10.6	10.6	7.0	6.6	6.6	6.6	15.4						102.0
Short Weierstrass, projective with a4=-3	9.4	13.6	13.6	13.6	10.6	10.6	10.6	7.0	6.6	6.6	6.6	15.8						102.0
Short Weierstrass, XYZZ	9.2	13.6	13.6	13.6	13.6	13.6	13.6	9.2	13.6	13.6	13.6							103.8
Short Weierstrass, XYZZ with a4=-3	8.6	13.6	13.6	13.6	13.6	13.6	13.6	8.6	13.6	13.6	13.6							103.8
Hessian, projective	7.8	12.0	10.8	15.6	10.0	10.0	10.0	5.4	8.0	7.0	8.0		12.8					102.0
Hessian, extended	7.8	10.8	10.8	10.8	9.8	9.8	9.8	7.8	9.8	9.8	9.8							104.6
Tripling-oriented Doche–Icart–Kohel, standard	7.6	15.8	14.8	15.8	10.2	10.2	10.2	5.0	5.6	5.6	5.6		10.8					103.8
Short Weierstrass, Jacobian	7.4	15.0	13.2	15.0	10.2	10.2	10.2	5.0	5.6	5.6	5.6		13.0					103.8
Short Weierstrass, Jacobian with a4=-3	7.0	15.0	13.2	15.0	10.2	10.2	10.2	5.0	5.6	5.6	5.6		12.6					103.8
Jacobi quartic, XYZ	6.8	12.4	10.4	12.8	10.4	10.4	10.4	4.2	6.6	6.6	6.6	12.4						102.8
Jacobi quartic, doubling-oriented XYZ	6.6	12.4	10.4	12.8	10.4	10.4	10.4	4.2	6.6	6.6	6.6	12.4						102.8
Edwards, projective	6.2	10.8	10.8	10.8	9.0	9.0	9.0	5.4	6.8	6.8	6.8	10.8	12.2					102.0
Jacobi quartic, doubling-oriented XXYZZ	6.2	10.2	9.4	10.2	8.4	8.4	8.4	5.0	8.4	8.4	8.4	10.2	12.8					103.6
Jacobi quartic, XXYZZ	6.2	10.2	9.4	10.2	8.4	8.4	8.4	5.0	8.4	8.4	8.4	10.2	12.8					103.6
Edwards, inverted	6.2	9.8	9.8	9.8	8.8	8.8	8.8	5.4	7.0	7.0	7.0	9.8	12.2					102.0
Jacobi quartic, XXYZZR	6.2	9.4	9.4	9.4	8.4	8.4	8.4	5.8	8.2	8.2	8.2	9.4	12.8					102.8
Jacobi quartic, doubling-oriented XXYZZR	6.2	9.4	9.4	9.4	8.4	8.4	8.4	5.8	8.2	8.2	8.2	9.4	12.8					102.8
Doubling-oriented Doche–Icart–Kohel, standard	6.0	16.0	16.0	16.0	11.2	11.2	11.2	5.0	7.2	7.2	7.2							102.8
Jacobi intersection, projective	6.0	13.8	11.8	13.8	11.8	10.8	11.8	5.2	9.6	8.8	9.6	13.8	12.0					103.0
Jacobi intersection, extended	6.0	11.8	11.8	11.8	10.8	10.8	10.8	6.0	10.8	10.8	10.8	11.8	14.0					104.0
Montgomery, XZ	3.6							3.6						5.6	4.6	9.2	8.2	101.0

Smallest multiplication counts assuming I=100M, S=0.67M, *param=0M, add=0M, *const=0M:

Curve shape, representation DBL ADD reADD preADD mADD mreADD mpreADD mDBL mmADD mmreADD mmpreADD SUNI TPL DADD mDADD LADD mLADD SCALE
Short Weierstrass, projective 9.02 13.34 13.34 13.34 10.34 10.34 10.34 6.35 6.34 6.34 6.34 15.02 102.00

Short Weierstrass, projective with a4=-1 9.02 13.34 13.34 13.34 10.34 10.34 10.34 6.35 6.34 6.34 6.34 15.01 102.00

Short Weierstrass, projective with a4=-3 9.01 13.34 13.34 13.34 10.34 10.34 10.34 6.35 6.34 6.34 6.34 15.02 102.00

Short Weierstrass, XYZZ 8.68 13.34 13.34 13.34 13.34 13.34 13.34 8.68 13.34 13.34 13.34 103.67

Short Weierstrass, XYZZ with a4=-3 8.34 13.34 13.34 13.34 13.34 13.34 13.34 8.34 13.34 13.34 13.34 103.67

Hessian, projective 7.02 12.00 10.02 14.04 10.00 10.00 10.00 5.01 8.00 7.00 8.00 12.02 102.00

Hessian, extended 7.02 10.02 10.02 10.02 9.02 9.02 9.02 7.02 9.02 9.02 9.02 104.34

Tripling-oriented Doche–Icart–Kohel, standard 6.69 15.02 14.02 15.02 9.68 9.68 9.68 4.35 5.34 5.34 5.34 10.02 103.67

Short Weierstrass, Jacobian 6.36 14.35 12.68 14.35 9.68 9.68 9.68 4.35 5.34 5.34 5.34 11.70 103.67

Short Weierstrass, Jacobian with a4=-3 6.35 14.35 12.68 14.35 9.68 9.68 9.68 4.35 5.34 5.34 5.34 11.69 103.67

Jacobi quartic, XYZ 6.02 12.01 10.01 12.02 10.01 10.01 10.01 3.68 6.34 6.34 6.34 12.01 102.67

Jacobi quartic, doubling-oriented XYZ 5.69 12.01 10.01 12.02 10.01 10.01 10.01 3.68 6.34 6.34 6.34 12.01 102.67

Edwards, projective 5.68 10.35 10.35 10.35 9.00 9.00 9.00 5.01 6.67 6.67 6.67 10.35 11.68 102.00

Jacobi quartic, XXYZZ 5.68 9.68 9.01 9.68 8.01 8.01 8.01 4.35 8.01 8.01 8.01 9.68 11.37 103.34

Edwards, inverted 5.68 9.67 9.67 9.67 8.67 8.67 8.67 5.01 7.00 7.00 7.00 9.67 11.68 102.00

Jacobi quartic, XXYZZR 5.68 9.01 9.01 9.01 8.01 8.01 8.01 5.02 7.68 7.68 7.68 9.01 11.37 102.67

Jacobi quartic, doubling-oriented XXYZZ 5.36 9.68 9.01 9.68 8.01 8.01 8.01 4.35 8.01 8.01 8.01 9.68 11.37 103.34

Jacobi quartic, doubling-oriented XXYZZR 5.36 9.01 9.01 9.01 8.01 8.01 8.01 5.02 7.68 7.68 7.68 9.01 11.37 102.67

Doubling-oriented Doche–Icart–Kohel, standard 5.35 15.35 15.35 15.35 10.68 10.68 10.68 4.35 6.68 6.68 6.68 102.67

Jacobi intersection, projective 5.35 13.67 11.67 13.67 11.67 10.67 11.67 4.68 9.34 8.67 9.34 13.67 10.70 103.00

Jacobi intersection, extended 5.35 11.67 11.67 11.67 10.67 10.67 10.67 5.35 10.67 10.67 10.67 11.67 12.70 104.00

Montgomery, XZ 3.34 3.34 5.34 4.34 8.68 7.68 101.00

Curve shape, representation	DBL	ADD	reADD	preADD	mADD	mreADD	mpreADD	mDBL	mmADD	mmreADD	mmpreADD	SUNI	TPL	DADD	mDADD	LADD	mLADD	SCALE
Short Weierstrass, projective	9.02	13.34	13.34	13.34	10.34	10.34	10.34	6.35	6.34	6.34	6.34	15.02						102.00
Short Weierstrass, projective with a4=-1	9.02	13.34	13.34	13.34	10.34	10.34	10.34	6.35	6.34	6.34	6.34	15.01						102.00
Short Weierstrass, projective with a4=-3	9.01	13.34	13.34	13.34	10.34	10.34	10.34	6.35	6.34	6.34	6.34	15.02						102.00
Short Weierstrass, XYZZ	8.68	13.34	13.34	13.34	13.34	13.34	13.34	8.68	13.34	13.34	13.34							103.67
Short Weierstrass, XYZZ with a4=-3	8.34	13.34	13.34	13.34	13.34	13.34	13.34	8.34	13.34	13.34	13.34							103.67
Hessian, projective	7.02	12.00	10.02	14.04	10.00	10.00	10.00	5.01	8.00	7.00	8.00		12.02					102.00
Hessian, extended	7.02	10.02	10.02	10.02	9.02	9.02	9.02	7.02	9.02	9.02	9.02							104.34
Tripling-oriented Doche–Icart–Kohel, standard	6.69	15.02	14.02	15.02	9.68	9.68	9.68	4.35	5.34	5.34	5.34		10.02					103.67
Short Weierstrass, Jacobian	6.36	14.35	12.68	14.35	9.68	9.68	9.68	4.35	5.34	5.34	5.34		11.70					103.67
Short Weierstrass, Jacobian with a4=-3	6.35	14.35	12.68	14.35	9.68	9.68	9.68	4.35	5.34	5.34	5.34		11.69					103.67
Jacobi quartic, XYZ	6.02	12.01	10.01	12.02	10.01	10.01	10.01	3.68	6.34	6.34	6.34	12.01						102.67
Jacobi quartic, doubling-oriented XYZ	5.69	12.01	10.01	12.02	10.01	10.01	10.01	3.68	6.34	6.34	6.34	12.01						102.67
Edwards, projective	5.68	10.35	10.35	10.35	9.00	9.00	9.00	5.01	6.67	6.67	6.67	10.35	11.68					102.00
Jacobi quartic, XXYZZ	5.68	9.68	9.01	9.68	8.01	8.01	8.01	4.35	8.01	8.01	8.01	9.68	11.37					103.34
Edwards, inverted	5.68	9.67	9.67	9.67	8.67	8.67	8.67	5.01	7.00	7.00	7.00	9.67	11.68					102.00
Jacobi quartic, XXYZZR	5.68	9.01	9.01	9.01	8.01	8.01	8.01	5.02	7.68	7.68	7.68	9.01	11.37					102.67
Jacobi quartic, doubling-oriented XXYZZ	5.36	9.68	9.01	9.68	8.01	8.01	8.01	4.35	8.01	8.01	8.01	9.68	11.37					103.34
Jacobi quartic, doubling-oriented XXYZZR	5.36	9.01	9.01	9.01	8.01	8.01	8.01	5.02	7.68	7.68	7.68	9.01	11.37					102.67
Doubling-oriented Doche–Icart–Kohel, standard	5.35	15.35	15.35	15.35	10.68	10.68	10.68	4.35	6.68	6.68	6.68							102.67
Jacobi intersection, projective	5.35	13.67	11.67	13.67	11.67	10.67	11.67	4.68	9.34	8.67	9.34	13.67	10.70					103.00
Jacobi intersection, extended	5.35	11.67	11.67	11.67	10.67	10.67	10.67	5.35	10.67	10.67	10.67	11.67	12.70					104.00
Montgomery, XZ	3.34							3.34						5.34	4.34	8.68	7.68	101.00

Genus-1 curves over large-characteristic fields

Doubling-oriented Doche–Icart–Kohel curves: y^2=x^3+a*x^2+16*a*x

Tripling-oriented Doche–Icart–Kohel curves: y^2=x^3+3*a*(x+1)^2

Edwards curves: x^2+y^2=c^2*(1+d*x^2*y^2)

Hessian curves: x^3+y^3+1=3*d*x*y