here - Sites personnels de TELECOM ParisTech - Télécom ParisTech

138 Chapter 5. Complex multiplication and elliptic curvesProposition 5.1.26 ([244, Remark III.8.5]). Let K be a perfect field and E an elliptic curvedefined over K. Let m ≥ 2 be an integer co-prime to the characteristic of K. Let P, Q ∈ E(K)[m].up to an m-th power.e m (P, Q) = 〈P, Q〉 m /〈Q, P 〉 m ,Such a definition has the benefit to be more computational. The Tate pairing can indeed beefficiently computed in polynomial time using Miller’s algorithm [203] that we now describe.With the same notation as before, let P ∈ E(K)[m] and Q ∈ E(K). If we denote by f i afunction (unique up to a multiplicative constant) such that div(f i ) = i(P ) − ([i]P ) − (i − 1)(O E ),then we need to compute f m . According to the following proposition we can do it explicitly usinga standard binary exponentiation.Proposition 5.1.27 ([19, Lemma IX.17]). Let l and v be the lines going through {[i]P, [j]P }and {[i + j]P, O E }. Then f i+j = f i f j l/v.The corresponding algorithm is given in Algorithm 5.1.Algorithm 5.1: Miller’s algorithmInput: P ∈ E(K)[m] and Q ∈ E(K)Output: 〈P, Q〉 m1 S ∈ R E(K)2 Q ′ ← Q + S3 T ← P4 f ← 15 i ← ⌊log m⌋ − 16 while i ≥ 0 do7 Compute l and v to double T8 T ← [2]T9 f ← f 2 l(Q ′ )v(S)/l(S)v(Q ′ )10 if b i = 1 then11 Compute l and v to add T and P12 T ← T + P13 f ← fl(Q ′ )v(S)/l(S)v(Q ′ )14 i ← i − 115 return f5.1.4 Reduction of elliptic curvesLet K be a local field with uniformizer π and residue field k. Let E be an elliptic curve definedover K. There exist Weierstraß equations for E with integral coefficients and so it is possibleto define the reduction Ẽ modulo π of E from such equations. The resulting curve is obviouslypotentially singular. In fact, it is if and only if the discriminant of the chosen Weierstraß equationis divisible by π. Moreover, there exists a “best” Weierstraß equation to use for reduction: it iscalled the minimal Weierstraß equation.Proposition 5.1.28 ([244, Proposition VII.1.3]). Let K be a local field with uniformizer π. LetE be an elliptic curve defined over K. Then E has a minimal Weierstraß equation, i.e. anequation with integral coefficients such that the valuation at π of its discriminant is minimal.