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
YZ coordinates with square d: r*y=Y/Z, c=1, d=r^²
Squared YZ coordinates with square d: r*y^²=Y/Z, c=1, d=r^²

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=0: x=X/Z^², y=Y/Z^³, a=0
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
Modified Jacobian coordinates: x=X/Z^², y=Y/Z^³, T=a*Z^⁴
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
W12 coordinates with a6=0: x=X/Z, y=Y/Z^², b=0
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
XZ coordinates: x=X/Z

Twisted Edwards curves: ax^²+y^²=1+dx^²*y^²

Extended coordinates with a=-1: x=X/Z, y=Y/Z, x*y=T/Z, a=-1
Extended coordinates: x=X/Z, y=Y/Z, x*y=T/Z
Inverted coordinates: x=Z/X, y=Z/Y
Projective coordinates: x=X/Z, y=Y/Z

Twisted Hessian curves: ax^³+y^³+1=dx*y

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

The EFD currently includes 471 explicit formulas for addition, doubling, tripling, and scaling of points in 33 representations on 10 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 preADD 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 with a4=-1 11 12 12 12 11 11 11 8 7 7 7 12 102

Short Weierstrass, projective 11 12 12 12 11 11 11 8 7 7 7 12 102

Short Weierstrass, XYZZ 10 14 14 14 10 10 10 7 6 6 6 104

Short Weierstrass, projective with a4=-3 10 12 12 12 11 11 11 8 7 7 7 12 102

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 10 10 10 7 6 6 6 104

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

Short Weierstrass, modified Jacobian 8 18 16 18 13 13 13 7 7 7 7

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

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

Jacobi quartic, doubling-oriented 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 12 14 102

Twisted Hessian, projective 8 11 11 11 11 11 11 8 11 11 11 11 12

Twisted Edwards, extended 8 9 9 9 8 8 8 7 7 7 7 9 14

Twisted Edwards, extended with a=-1 8 8 8 8 7 7 7 7 6 6 6 8 14

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

Short Weierstrass, W12 with a6=0 7 17 16 17 13 13 13 7 13 13 13

Short Weierstrass, Jacobian with a4=0 7 16 14 16 11 11 11 6 6 6 6 15 104

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

Twisted Edwards, projective 7 11 11 11 10 10 10 6 7 7 7 11 12

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

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

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

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

Jacobi quartic, 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

Twisted Edwards, inverted 7 10 10 10 9 9 9 6 7 7 7 10

Short Weierstrass, XZ 7 7 9 8 16 15

Montgomery, XZ 4 3 6 5 10 9 101

Edwards, squared YZ with square d 4 3 6 5 10 9 101

Edwards, YZ with square d 4 2 8 7 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 with a4=-1	11	12	12	12	11	11	11	8	7	7	7	12						102
Short Weierstrass, projective	11	12	12	12	11	11	11	8	7	7	7	12						102
Short Weierstrass, XYZZ	10	14	14	14	10	10	10	7	6	6	6							104
Short Weierstrass, projective with a4=-3	10	12	12	12	11	11	11	8	7	7	7	12						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	10	10	10	7	6	6	6							104
Hessian, extended	9	12	12	12	11	11	11	9	11	11	11							105
Short Weierstrass, modified Jacobian	8	18	16	18	13	13	13	7	7	7	7
Short Weierstrass, Jacobian with a4=-3	8	16	14	16	11	11	11	6	6	6	6		14					104
Jacobi quartic, XYZ	8	13	11	14	11	11	11	5	7	7	7	13						103
Jacobi quartic, doubling-oriented 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	12	14					102
Twisted Hessian, projective	8	11	11	11	11	11	11	8	11	11	11	11	12
Twisted Edwards, extended	8	9	9	9	8	8	8	7	7	7	7	9	14
Twisted Edwards, extended with a=-1	8	8	8	8	7	7	7	7	6	6	6	8	14
Doubling-oriented Doche–Icart–Kohel, standard	7	17	17	17	12	12	12	6	8	8	8							103
Short Weierstrass, W12 with a6=0	7	17	16	17	13	13	13	7	13	13	13
Short Weierstrass, Jacobian with a4=0	7	16	14	16	11	11	11	6	6	6	6		15					104
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
Twisted Edwards, projective	7	11	11	11	10	10	10	6	7	7	7	11	12
Edwards, projective	7	11	11	11	9	9	9	6	7	7	7	11	13					102
Jacobi quartic, XXYZZ	7	11	10	11	9	9	9	6	9	9	9	11	14					104
Jacobi quartic, doubling-oriented XXYZZ	7	11	10	11	9	9	9	6	9	9	9	11	14					104
Jacobi quartic, doubling-oriented XXYZZR	7	10	10	10	9	9	9	7	9	9	9	10	15					103
Jacobi quartic, 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
Twisted Edwards, inverted	7	10	10	10	9	9	9	6	7	7	7	10
Short Weierstrass, XZ	7							7						9	8	16	15
Montgomery, XZ	4							3						6	5	10	9	101
Edwards, squared YZ with square d	4							3						6	5	10	9	101
Edwards, YZ with square d	4							2						8	7	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 with a4=-1 9.8 12.0 12.0 12.0 10.6 10.6 10.6 7.0 6.6 6.6 6.6 12.0 102.0

Short Weierstrass, projective 9.8 12.0 12.0 12.0 10.6 10.6 10.6 7.0 6.6 6.6 6.6 12.0 102.0

Short Weierstrass, projective with a4=-3 9.4 12.0 12.0 12.0 10.6 10.6 10.6 7.0 6.6 6.6 6.6 12.0 102.0

Short Weierstrass, XYZZ 9.2 13.6 13.6 13.6 9.6 9.6 9.6 6.4 5.6 5.6 5.6 103.8

Short Weierstrass, XYZZ with a4=-3 8.6 13.6 13.6 13.6 9.6 9.6 9.6 6.4 5.6 5.6 5.6 103.8

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

Twisted Hessian, projective 7.6 11.0 11.0 11.0 11.0 11.0 11.0 7.6 11.0 11.0 11.0 11.0 10.8

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

Twisted Edwards, extended 7.2 9.0 9.0 9.0 8.0 8.0 8.0 6.2 7.0 7.0 7.0 9.0 13.4

Twisted Edwards, extended with a=-1 7.2 8.0 8.0 8.0 7.0 7.0 7.0 6.2 6.0 6.0 6.0 8.0 13.4

Short Weierstrass, modified Jacobian 7.0 16.6 14.8 16.6 11.8 11.8 11.8 6.0 6.2 6.2 6.2

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

Twisted Edwards, projective 6.2 10.8 10.8 10.8 9.8 9.8 9.8 5.2 6.8 6.8 6.8 10.8 11.4

Edwards, projective 6.2 10.8 10.6 12.4 9.0 9.0 9.0 5.4 6.8 6.8 6.8 10.8 12.2 102.0

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

Twisted 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

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

Short Weierstrass, Jacobian with a4=0 6.0 15.0 13.2 15.0 10.2 10.2 10.2 5.0 5.6 5.6 5.6 13.0 103.8

Jacobi intersection, projective 6.0 13.8 11.8 13.8 11.8 10.8 11.8 4.8 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

Jacobi quartic, XXYZZ 6.0 10.2 9.4 10.2 8.4 8.4 8.4 4.8 8.4 8.4 8.4 10.2 12.8 103.6

Jacobi quartic, doubling-oriented XXYZZ 6.0 10.2 9.4 10.2 8.4 8.4 8.4 4.8 8.4 8.4 8.4 10.2 12.8 103.6

Jacobi quartic, doubling-oriented XXYZZR 6.0 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, XXYZZR 6.0 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

Short Weierstrass, XZ 6.0 6.0 8.6 7.6 14.6 13.6

Short Weierstrass, W12 with a6=0 5.8 15.6 14.8 15.6 12.0 12.0 12.0 5.8 12.0 12.0 12.0

Montgomery, XZ 3.6 2.6 5.6 4.6 9.2 8.2 101.0

Edwards, squared YZ with square d 3.2 2.4 5.6 4.6 8.8 7.8 101.0

Edwards, YZ with square d 3.2 1.6 7.2 6.2 8.8 7.8 101.0

Curve shape, representation	DBL	ADD	reADD	preADD	mADD	mreADD	mpreADD	mDBL	mmADD	mmreADD	mmpreADD	SUNI	TPL	DADD	mDADD	LADD	mLADD	SCALE
Short Weierstrass, projective with a4=-1	9.8	12.0	12.0	12.0	10.6	10.6	10.6	7.0	6.6	6.6	6.6	12.0						102.0
Short Weierstrass, projective	9.8	12.0	12.0	12.0	10.6	10.6	10.6	7.0	6.6	6.6	6.6	12.0						102.0
Short Weierstrass, projective with a4=-3	9.4	12.0	12.0	12.0	10.6	10.6	10.6	7.0	6.6	6.6	6.6	12.0						102.0
Short Weierstrass, XYZZ	9.2	13.6	13.6	13.6	9.6	9.6	9.6	6.4	5.6	5.6	5.6							103.8
Short Weierstrass, XYZZ with a4=-3	8.6	13.6	13.6	13.6	9.6	9.6	9.6	6.4	5.6	5.6	5.6							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	12.0	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
Twisted Hessian, projective	7.6	11.0	11.0	11.0	11.0	11.0	11.0	7.6	11.0	11.0	11.0	11.0	10.8
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
Twisted Edwards, extended	7.2	9.0	9.0	9.0	8.0	8.0	8.0	6.2	7.0	7.0	7.0	9.0	13.4
Twisted Edwards, extended with a=-1	7.2	8.0	8.0	8.0	7.0	7.0	7.0	6.2	6.0	6.0	6.0	8.0	13.4
Short Weierstrass, modified Jacobian	7.0	16.6	14.8	16.6	11.8	11.8	11.8	6.0	6.2	6.2	6.2
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
Twisted Edwards, projective	6.2	10.8	10.8	10.8	9.8	9.8	9.8	5.2	6.8	6.8	6.8	10.8	11.4
Edwards, projective	6.2	10.8	10.6	12.4	9.0	9.0	9.0	5.4	6.8	6.8	6.8	10.8	12.2					102.0
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
Twisted 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
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
Short Weierstrass, Jacobian with a4=0	6.0	15.0	13.2	15.0	10.2	10.2	10.2	5.0	5.6	5.6	5.6		13.0					103.8
Jacobi intersection, projective	6.0	13.8	11.8	13.8	11.8	10.8	11.8	4.8	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
Jacobi quartic, XXYZZ	6.0	10.2	9.4	10.2	8.4	8.4	8.4	4.8	8.4	8.4	8.4	10.2	12.8					103.6
Jacobi quartic, doubling-oriented XXYZZ	6.0	10.2	9.4	10.2	8.4	8.4	8.4	4.8	8.4	8.4	8.4	10.2	12.8					103.6
Jacobi quartic, doubling-oriented XXYZZR	6.0	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, XXYZZR	6.0	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
Short Weierstrass, XZ	6.0							6.0						8.6	7.6	14.6	13.6
Short Weierstrass, W12 with a6=0	5.8	15.6	14.8	15.6	12.0	12.0	12.0	5.8	12.0	12.0	12.0
Montgomery, XZ	3.6							2.6						5.6	4.6	9.2	8.2	101.0
Edwards, squared YZ with square d	3.2							2.4						5.6	4.6	8.8	7.8	101.0
Edwards, YZ with square d	3.2							1.6						7.2	6.2	8.8	7.8	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 with a4=-1 9.02 12.00 12.00 12.00 10.34 10.34 10.34 6.35 6.34 6.34 6.34 12.00 102.00

Short Weierstrass, projective 9.02 12.00 12.00 12.00 10.34 10.34 10.34 6.35 6.34 6.34 6.34 12.00 102.00

Short Weierstrass, projective with a4=-3 9.01 12.00 12.00 12.00 10.34 10.34 10.34 6.35 6.34 6.34 6.34 12.00 102.00

Short Weierstrass, XYZZ 8.68 13.34 13.34 13.34 9.34 9.34 9.34 6.01 5.34 5.34 5.34 103.67

Short Weierstrass, XYZZ with a4=-3 8.34 13.34 13.34 13.34 9.34 9.34 9.34 6.01 5.34 5.34 5.34 103.67

Twisted Hessian, projective 7.34 11.00 11.00 11.00 11.00 11.00 11.00 7.34 11.00 11.00 11.00 11.00 10.02

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

Twisted Edwards, extended 6.68 9.00 9.00 9.00 8.00 8.00 8.00 5.68 7.00 7.00 7.00 9.00 13.01

Twisted Edwards, extended with a=-1 6.68 8.00 8.00 8.00 7.00 7.00 7.00 5.68 6.00 6.00 6.00 8.00 13.01

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, modified Jacobian 6.35 15.69 14.02 15.69 11.02 11.02 11.02 5.35 5.68 5.68 5.68

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

Twisted Edwards, projective 5.68 10.67 10.67 10.67 9.67 9.67 9.67 4.68 6.67 6.67 6.67 10.67 11.01

Edwards, projective 5.68 10.35 10.34 12.01 9.00 9.00 9.00 5.01 6.67 6.67 6.67 10.35 11.68 102.00

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

Twisted 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

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

Short Weierstrass, Jacobian with a4=0 5.35 14.35 12.68 14.35 9.68 9.68 9.68 4.35 5.34 5.34 5.34 11.70 103.67

Jacobi intersection, projective 5.35 13.67 11.67 13.67 11.67 10.67 11.67 4.02 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

Jacobi quartic, XXYZZ 5.35 9.68 9.01 9.68 8.01 8.01 8.01 4.02 8.01 8.01 8.01 9.68 11.37 103.34

Jacobi quartic, doubling-oriented XXYZZ 5.35 9.68 9.01 9.68 8.01 8.01 8.01 4.02 8.01 8.01 8.01 9.68 11.37 103.34

Jacobi quartic, doubling-oriented XXYZZR 5.35 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, XXYZZR 5.35 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

Short Weierstrass, XZ 5.35 5.35 8.34 7.34 13.69 12.69

Short Weierstrass, W12 with a6=0 5.02 14.69 14.02 14.69 11.35 11.35 11.35 5.02 11.35 11.35 11.35

Montgomery, XZ 3.34 2.34 5.34 4.34 8.68 7.68 101.00

Edwards, squared YZ with square d 2.68 2.01 5.34 4.34 8.02 7.02 101.00

Edwards, YZ with square d 2.68 1.34 6.68 5.68 8.02 7.02 101.00

Curve shape, representation	DBL	ADD	reADD	preADD	mADD	mreADD	mpreADD	mDBL	mmADD	mmreADD	mmpreADD	SUNI	TPL	DADD	mDADD	LADD	mLADD	SCALE
Short Weierstrass, projective with a4=-1	9.02	12.00	12.00	12.00	10.34	10.34	10.34	6.35	6.34	6.34	6.34	12.00						102.00
Short Weierstrass, projective	9.02	12.00	12.00	12.00	10.34	10.34	10.34	6.35	6.34	6.34	6.34	12.00						102.00
Short Weierstrass, projective with a4=-3	9.01	12.00	12.00	12.00	10.34	10.34	10.34	6.35	6.34	6.34	6.34	12.00						102.00
Short Weierstrass, XYZZ	8.68	13.34	13.34	13.34	9.34	9.34	9.34	6.01	5.34	5.34	5.34							103.67
Short Weierstrass, XYZZ with a4=-3	8.34	13.34	13.34	13.34	9.34	9.34	9.34	6.01	5.34	5.34	5.34							103.67
Twisted Hessian, projective	7.34	11.00	11.00	11.00	11.00	11.00	11.00	7.34	11.00	11.00	11.00	11.00	10.02
Hessian, projective	7.02	12.00	10.02	14.04	10.00	10.00	10.00	5.01	8.00	7.00	8.00	12.00	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
Twisted Edwards, extended	6.68	9.00	9.00	9.00	8.00	8.00	8.00	5.68	7.00	7.00	7.00	9.00	13.01
Twisted Edwards, extended with a=-1	6.68	8.00	8.00	8.00	7.00	7.00	7.00	5.68	6.00	6.00	6.00	8.00	13.01
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, modified Jacobian	6.35	15.69	14.02	15.69	11.02	11.02	11.02	5.35	5.68	5.68	5.68
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
Twisted Edwards, projective	5.68	10.67	10.67	10.67	9.67	9.67	9.67	4.68	6.67	6.67	6.67	10.67	11.01
Edwards, projective	5.68	10.35	10.34	12.01	9.00	9.00	9.00	5.01	6.67	6.67	6.67	10.35	11.68					102.00
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
Twisted 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
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
Short Weierstrass, Jacobian with a4=0	5.35	14.35	12.68	14.35	9.68	9.68	9.68	4.35	5.34	5.34	5.34		11.70					103.67
Jacobi intersection, projective	5.35	13.67	11.67	13.67	11.67	10.67	11.67	4.02	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
Jacobi quartic, XXYZZ	5.35	9.68	9.01	9.68	8.01	8.01	8.01	4.02	8.01	8.01	8.01	9.68	11.37					103.34
Jacobi quartic, doubling-oriented XXYZZ	5.35	9.68	9.01	9.68	8.01	8.01	8.01	4.02	8.01	8.01	8.01	9.68	11.37					103.34
Jacobi quartic, doubling-oriented XXYZZR	5.35	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, XXYZZR	5.35	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
Short Weierstrass, XZ	5.35							5.35						8.34	7.34	13.69	12.69
Short Weierstrass, W12 with a6=0	5.02	14.69	14.02	14.69	11.35	11.35	11.35	5.02	11.35	11.35	11.35
Montgomery, XZ	3.34							2.34						5.34	4.34	8.68	7.68	101.00
Edwards, squared YZ with square d	2.68							2.01						5.34	4.34	8.02	7.02	101.00
Edwards, YZ with square d	2.68							1.34						6.68	5.68	8.02	7.02	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

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

Jacobi quartics: y^2=x^4+2*a*x^2+1

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

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

Twisted Edwards curves: a*x^2+y^2=1+d*x^2*y^2

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