diff options
| m--------- | etc/coq-scripts | 0 | ||||
| -rw-r--r-- | src/Curves/Montgomery/XZProofs.v | 37 |
2 files changed, 24 insertions, 13 deletions
diff --git a/etc/coq-scripts b/etc/coq-scripts -Subproject 878fffa6abdb29db903ccce680b937df7a4f04f +Subproject 70b32dd2da94b945f4e98064f2fbf78ae0755e6 diff --git a/src/Curves/Montgomery/XZProofs.v b/src/Curves/Montgomery/XZProofs.v index d24d1398c..ef708af00 100644 --- a/src/Curves/Montgomery/XZProofs.v +++ b/src/Curves/Montgomery/XZProofs.v @@ -1,8 +1,10 @@ Require Import Crypto.Algebra.Field. Require Import Crypto.Util.Sum Crypto.Util.Prod Crypto.Util.LetIn. Require Import Crypto.Util.Decidable. +Require Import Crypto.Util.Tactics.SetoidSubst. Require Import Crypto.Spec.MontgomeryCurve Crypto.Curves.Montgomery.Affine. Require Import Crypto.Curves.Montgomery.XZ BinPos. +Require Import Coq.Classes.Morphisms. Module M. Section MontgomeryCurve. @@ -22,10 +24,20 @@ Module M. Context {a b: F} {b_nonzero:b <> 0}. Context {a24:F} {a24_correct:(1+1+1+1)*a24 = a-(1+1)}. - Local Notation add := (M.add(a:=a)(b_nonzero:=b_nonzero)(char_ge_3:=char_ge_3)). - Local Notation opp := (M.opp(a:=a)(b_nonzero:=b_nonzero)). - Local Notation point := (@M.point F Feq Fadd Fmul a b). + Local Notation Madd := (M.add(a:=a)(b_nonzero:=b_nonzero)(char_ge_3:=char_ge_3)). + Local Notation Mopp := (M.opp(a:=a)(b_nonzero:=b_nonzero)). + Local Notation Mpoint := (@M.point F Feq Fadd Fmul a b). Local Notation xzladderstep := (M.xzladderstep(a24:=a24)(Fadd:=Fadd)(Fsub:=Fsub)(Fmul:=Fmul)). + Local Notation to_xz := (M.to_xz(Fzero:=Fzero)(Fone:=Fone)(Feq:=Feq)(Fadd:=Fadd)(Fmul:=Fmul)(a:=a)(b:=b)). + + Definition eq (P Q:F*F) := fst P * snd Q = fst Q * snd P. + + Context {nonsquare_a24:forall r, r*r <> a*a - (1+1+1+1)}. + Let y_nonzero (Q:Mpoint) : match M.coordinates Q with ∞ => True | (x,y) => x <> 0 -> y <> 0 end. + Proof. + destruct Q as [Q pfQ]; destruct Q as [[x y]|[]]; cbv -[not]; intros; trivial. + specialize (nonsquare_a24 (x+x+a)); fsatz. + Qed. Ltac t := repeat @@ -41,18 +53,17 @@ Module M. | _ => progress Prod.inversion_prod | _ => progress Tactics.BreakMatch.break_match_hyps | _ => progress Tactics.BreakMatch.break_match - | _ => progress cbv [fst snd M.coordinates M.add M.zero M.eq M.opp proj1_sig M.xzladderstep M.to_xz Let_In] in * + | _ => progress cbv [eq fst snd M.coordinates M.add M.zero M.eq M.opp proj1_sig xzladderstep M.to_xz Let_In Proper respectful] in * | |- _ /\ _ => split end. - Lemma xzladderstep_correct - (Q Q':point) x z x' z' x1 x2 z2 x3 z3 - (Hl:Logic.eq (pair(pair x2 z2)(pair x3 z3)) (xzladderstep x1 (pair x z) (pair x' z'))) - (H:match M.coordinates Q with∞=>z=0/\x<>0|(xQ,y)=>xQ=x/z/\z<>0 (* TODO *) /\ y <> 0 (* TODO: prove this from non-squareness of a^2 - 4 *) end) - (H':match M.coordinates Q' with∞=>z'=0/\x'<>0|(xQ',_)=>xQ'=x'/z'/\z'<>0 end) - (H1:match M.coordinates (add Q (opp Q')) with∞=>False|(x,y)=>x=x1/\x<>0 end): - match M.coordinates (add Q Q) with∞=>z2=0/\x2<>0|(xQQ,_)=>xQQ=x2/z2/\z2<>0 end /\ - match M.coordinates (add Q Q') with∞=>z3=0/\x3<>0|(xQQ',_)=>xQQ'=x3/z3/\z3<>0 end. - Proof using a24_correct char_ge_5. t; abstract fsatz. Qed. + Lemma to_xz_add (x1:F) (Q Q':Mpoint) + (difference_correct:match M.coordinates (Madd Q (Mopp Q')) with + | ∞ => False (* Q <> Q' *) + | (x,y) => x = x1 /\ x1 <> 0 (* Q-Q' <> (0, 0) *) + end) + : eq (to_xz (Madd Q Q )) (fst (xzladderstep x1 (to_xz Q) (to_xz Q ))) + /\ eq (to_xz (Madd Q Q')) (snd (xzladderstep x1 (to_xz Q) (to_xz Q'))). + Proof. specialize (y_nonzero Q); t; fsatz. Qed. End MontgomeryCurve. End M. |