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

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

Squared YZ coordinates with square d: r*y^

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

Projective coordinates: s=S/Z, c=C/Z, d=D/Z

Doubling-oriented XXYZZR coordinates: x=X/Z, y=Y/ZZ, XX=X^

Doubling-oriented XYZ coordinates: x=X/Z, y=Y/Z^

XXYZZ coordinates: x=X/Z, y=Y/ZZ, XX=X^

XXYZZR coordinates: x=X/Z, y=Y/ZZ, XX=X^

XYZ coordinates: x=X/Z, y=Y/Z^

Jacobian coordinates with a4=-3: x=X/Z^

Jacobian coordinates: x=X/Z^

Modified Jacobian coordinates: x=X/Z^

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^

XYZZ coordinates with a4=-3: x=X/ZZ, y=Y/ZZZ, ZZ^

XYZZ coordinates: x=X/ZZ, y=Y/ZZZ, ZZ^

XZ coordinates: x=X/Z

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

The EFD currently includes 451 explicit formulas for addition, doubling, tripling, and scaling of points in 33 representations on 10 shapes of genus-1 curves over large-characteristic fields.

- 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.

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

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

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 | |||||

Twisted Hessian, projective | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 8 | 12 | 12 | 12 | 12 | 14 | |||||

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

Twisted Edwards, extended | 8 | 9 | 9 | 9 | 8 | 8 | 8 | 7 | 7 | 7 | 7 | 9 | ||||||

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

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 | ||||||

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 | 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 | 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 | ||||||

Twisted Hessian, projective | 7.8 | 12.0 | 12.0 | 12.0 | 12.0 | 12.0 | 12.0 | 7.8 | 12.0 | 12.0 | 12.0 | 12.0 | 12.8 | |||||

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 | |||||

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 | ||||||

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 | ||||||

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 | ||||||

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 | 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 | 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 | 7.67 | 12.00 | 12.00 | 12.00 | 12.00 | 12.00 | 12.00 | 7.67 | 12.00 | 12.00 | 12.00 | 12.00 | 12.02 | |||||

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 | ||||||

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 | ||||||

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 | ||||||

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 |