Ordinary genus-1 curves over binary fields

Short Weierstrass curves

y^^{2}+x*y=x^^{3}+a2*x^^{2}+a6

Lambda coordinates [database entry] represent x y as X L Z satisfying the following equations:

x=X/Z y/x=(L-X)/Z

Source: 2013 Oliveira–López–Aranha–Rodríguez-Henríquez "Lambda coordinates for binary elliptic curves".

- 11M for addition: 11M+2S.
- 11M for readdition: 11M+2S after 11M+2S.
- 3M for doubling: 3M+4S.

- 11.4M for addition: 11M+2S.
- 11.4M for readdition: 11M+2S after 11M+2S.
- 3.8M for doubling: 3M+4S.

Operation | Assumptions | Cost | Readdition cost |
---|---|---|---|

addition | 11M + 2S | 11M + 2S | |

doubling | a21=a2+1 and a226=a2^^{2}+a6 |
3M + 4S + 1*a2 + 1*a226 + 1*a21 | |

doubling | 4M + 4S + 1*a2 |

- Cost: 11M + 2S + 5add.
- Source: 2013 Oliveira–López–Aranha–Rodríguez-Henríquez "Lambda coordinates for binary elliptic curves"; database entry contributed by Chitchanok Chuengsatiansup.
- Explicit formulas:
A = L1*Z2 B = L2*Z1 C = A+B D = X1*Z2 E = X2*Z1 F = D+E G = F^

^{2}H = C*D I = C*E J = C*G K = J*Z2 X3 = H*I L3 = (I+G)^^{2}+K*(L1+Z1) Z3 = K*Z1

- Assumptions: a21=a2+1 and a226=a2^
^{2}+a6. - Cost: 3M + 4S + 1*a2 + 1*a226 + 1*a21 + 8add.
- Source: 2013 Oliveira–López–Aranha–Rodríguez-Henríquez "Lambda coordinates for binary elliptic curves"; database entry contributed by Chitchanok Chuengsatiansup.
- Explicit formulas:
A = Z1^

^{2}B = L1+Z1 C = L1*B D = a2*A E = C+D F = L1+X1 G = F^^{2}H = A^^{2}X3 = E^^{2}Z3 = E*A L3 = G*(G+E+A)+a226*H+X3+a21*Z3

- Cost: 4M + 4S + 1*a2 + 5add.
- Source: 2013 Oliveira–López–Aranha–Rodríguez-Henríquez "Lambda coordinates for binary elliptic curves"; database entry contributed by Chitchanok Chuengsatiansup.
- Explicit formulas:
A = Z1^

^{2}B = L1^^{2}C = L1*Z1 D = a2*A E = B+C+D F = X1*Z1 X3 = E^^{2}Z3 = E*A L3 = F^^{2}+X3+E*C+Z3