diff options
Diffstat (limited to 'src/Spec/MontgomeryCurve.v')
-rw-r--r-- | src/Spec/MontgomeryCurve.v | 48 |
1 files changed, 41 insertions, 7 deletions
diff --git a/src/Spec/MontgomeryCurve.v b/src/Spec/MontgomeryCurve.v index 2717f6bbc..cff35104c 100644 --- a/src/Spec/MontgomeryCurve.v +++ b/src/Spec/MontgomeryCurve.v @@ -60,6 +60,8 @@ Module M. end. Next Obligation. Proof. t. Qed. + Program Definition zero : point := ∞. + Program Definition opp (P:point) : point := match P return F*F+∞ with | (x, y) => (x, -y) @@ -73,23 +75,55 @@ Module M. Local Notation "27" := (3*9). Context {char_ge_28:@Ring.char_ge F Feq Fzero Fone Fopp Fadd Fsub Fmul 28}. - Let WeierstrassA := ((3-a^2)/(3*b^2)). - Let WeierstrassB := ((2*a^3-9*a)/(27*b^3)). + Local Notation WeierstrassA := ((3-a^2)/(3*b^2)). + Local Notation WeierstrassB := ((2*a^3-9*a)/(27*b^3)). Local Notation Wpoint := (@W.point F Feq Fadd Fmul WeierstrassA WeierstrassB). Local Notation Wadd := (@W.add F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv field Feq_dec char_ge_3 WeierstrassA WeierstrassB). + Program Definition to_Weierstrass (P:point) : Wpoint := + match coordinates P return F*F+∞ with + | (x, y) => ((x + a/3)/b, y/b) + | _ => ∞ + end. + Next Obligation. + Proof. clear char_ge_3; destruct P; t. Qed. + Program Definition of_Weierstrass (P:Wpoint) : point := match W.coordinates P return F*F+∞ with | (x,y) => (b*x-a/3, b*y) | _ => ∞ end. Next Obligation. - Proof. clear char_ge_3; subst WeierstrassA; subst WeierstrassB; destruct P; t. Qed. + Proof. clear char_ge_3; destruct P; t. Qed. - Lemma of_Weierstrass_add P1 P2 : - eq (of_Weierstrass (W.add P1 P2)) - (add (of_Weierstrass P1) (of_Weierstrass P2)). - Proof. cbv [WeierstrassA WeierstrassB eq of_Weierstrass W.add add coordinates W.coordinates proj1_sig] in *; clear char_ge_3; t. Qed. + (* TODO: move *) + Program Definition Wopp (P:Wpoint) : Wpoint := + match P return F*F+∞ with + | (x, y) => (x, -y) + | ∞ => ∞ + end. + Next Obligation. destruct P; t. Qed. + + Axiom Wgroup : @Algebra.group Wpoint (@W.eq F Feq Fadd Fmul WeierstrassA WeierstrassB) + Wadd (@W.zero F Feq Fadd Fmul WeierstrassA WeierstrassB) Wopp. + Program Definition _MW : _ /\ _ /\ _ := + @Group.group_from_redundant_representation + Wpoint W.eq Wadd W.zero Wopp + Wgroup + point eq add zero opp + of_Weierstrass + to_Weierstrass + _ _ _ _ _ + . + Next Obligation. cbv [W.eq eq to_Weierstrass of_Weierstrass W.add add coordinates W.coordinates proj1_sig] in *; t. Qed. + Next Obligation. cbv [W.eq eq to_Weierstrass of_Weierstrass W.add add coordinates W.coordinates proj1_sig] in *. clear char_ge_3. t. 2:intuition idtac. 2:intuition idtac. 2:intuition idtac. + { repeat split; destruct_head' and; t. } Qed. + Next Obligation. + (* addition case, same issue as in Weierstrass associativity *) + cbv [W.eq eq to_Weierstrass of_Weierstrass W.add add coordinates W.coordinates proj1_sig] in *. + clear char_ge_3. t. Qed. + Next Obligation. cbv [W.eq eq to_Weierstrass of_Weierstrass W.add add Wopp opp coordinates W.coordinates proj1_sig] in *. clear char_ge_3. t. Qed. + Next Obligation. cbv [W.eq eq to_Weierstrass of_Weierstrass W.add add Wopp opp coordinates W.coordinates proj1_sig] in *. clear char_ge_3. t. Qed. Section AddX. Lemma homogeneous_x_differential_addition_releations P1 P2 : |