diff options
author | Andres Erbsen <andreser@mit.edu> | 2017-06-15 12:12:47 -0400 |
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committer | Andres Erbsen <andreser@mit.edu> | 2017-06-15 12:12:47 -0400 |
commit | a36568d1d73aff5d7accc79fd28be672882f9c17 (patch) | |
tree | f6046789fbee1be1527b4caa58c8c331db4dc89f /src/Curves/Edwards/XYZT | |
parent | b319173e40cb219ab3b9b80e967b264e699851ad (diff) |
Edwards coordinates precomputed addition formula
Diffstat (limited to 'src/Curves/Edwards/XYZT')
-rw-r--r-- | src/Curves/Edwards/XYZT/Basic.v | 132 | ||||
-rw-r--r-- | src/Curves/Edwards/XYZT/Precomputed.v | 78 |
2 files changed, 210 insertions, 0 deletions
diff --git a/src/Curves/Edwards/XYZT/Basic.v b/src/Curves/Edwards/XYZT/Basic.v new file mode 100644 index 000000000..1d327cd40 --- /dev/null +++ b/src/Curves/Edwards/XYZT/Basic.v @@ -0,0 +1,132 @@ +Require Import Coq.Classes.Morphisms. + +Require Import Crypto.Spec.CompleteEdwardsCurve Crypto.Curves.Edwards.AffineProofs. + +Require Import Crypto.Util.Notations Crypto.Util.GlobalSettings. +Require Export Crypto.Util.FixCoqMistakes. +Require Import Crypto.Util.Decidable. +Require Import Crypto.Util.Tactics.DestructHead. +Require Import Crypto.Util.Tactics.UniquePose. + +Section ExtendedCoordinates. + Context {F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv} + {field:@Algebra.Hierarchy.field F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv} + {char_ge_3 : @Ring.char_ge F Feq Fzero Fone Fopp Fadd Fsub Fmul (BinNat.N.succ_pos BinNat.N.two)} + {Feq_dec:DecidableRel Feq}. + Local Infix "=" := Feq : type_scope. Local Notation "a <> b" := (not (a = b)) : type_scope. + Local Notation "0" := Fzero. Local Notation "1" := Fone. + Local Infix "+" := Fadd. Local Infix "*" := Fmul. + Local Infix "-" := Fsub. Local Infix "/" := Fdiv. + Local Notation "x ^ 2" := (x*x). + + Context {a d: F} + {nonzero_a : a <> 0} + {square_a : exists sqrt_a, sqrt_a^2 = a} + {nonsquare_d : forall x, x^2 <> d}. + Local Notation Epoint := (@E.point F Feq Fone Fadd Fmul a d). + + Local Notation onCurve x y := (a*x^2 + y^2 = 1 + d*x^2*y^2) (only parsing). + (** [Extended.point] represents a point on an elliptic curve using extended projective + * Edwards coordinates 1 (see <https://eprint.iacr.org/2008/522.pdf>). *) + Definition point := { P | let '(X,Y,Z,T) := P in + a * X^2*Z^2 + Y^2*Z^2 = (Z^2)^2 + d * X^2 * Y^2 + /\ X * Y = Z * T + /\ Z <> 0 }. + Definition coordinates (P:point) : F*F*F*F := proj1_sig P. + Definition eq (P1 P2:point) := + let '(X1, Y1, Z1, _) := coordinates P1 in + let '(X2, Y2, Z2, _) := coordinates P2 in + Z2*X1 = Z1*X2 /\ Z2*Y1 = Z1*Y2. + + Ltac t_step := + match goal with + | |- Proper _ _ => intro + | _ => progress intros + | _ => progress destruct_head' prod + | _ => progress destruct_head' @E.point + | _ => progress destruct_head' point + | _ => progress destruct_head' and + | _ => progress cbv [eq CompleteEdwardsCurve.E.eq E.eq E.zero E.add E.opp fst snd coordinates E.coordinates proj1_sig] in * + | |- _ /\ _ => split | |- _ <-> _ => split + end. + Ltac t := repeat t_step; Field.fsatz. + + Global Instance Equivalence_eq : Equivalence eq. + Proof using Feq_dec field nonzero_a. split; repeat intro; t. Qed. + Global Instance DecidableRel_eq : Decidable.DecidableRel eq. + Proof. intros P Q; destruct P as [ [ [ [ ] ? ] ? ] ?], Q as [ [ [ [ ] ? ] ? ] ? ]; exact _. Defined. + + Program Definition from_twisted (P:Epoint) : point := + let xy := E.coordinates P in (fst xy, snd xy, 1, fst xy * snd xy). + Next Obligation. t. Qed. + Global Instance Proper_from_twisted : Proper (E.eq==>eq) from_twisted. + Proof using Type. cbv [from_twisted]; t. Qed. + + Program Definition to_twisted (P:point) : Epoint := + let XYZT := coordinates P in let T := snd XYZT in + let XYZ := fst XYZT in let Z := snd XYZ in + let XY := fst XYZ in let Y := snd XY in + let X := fst XY in + let iZ := Finv Z in ((X*iZ), (Y*iZ)). + Next Obligation. t. Qed. + Global Instance Proper_to_twisted : Proper (eq==>E.eq) to_twisted. + Proof using Type. cbv [to_twisted]; t. Qed. + + Lemma to_twisted_from_twisted P : E.eq (to_twisted (from_twisted P)) P. + Proof using Type. cbv [to_twisted from_twisted]; t. Qed. + Lemma from_twisted_to_twisted P : eq (from_twisted (to_twisted P)) P. + Proof using Type. cbv [to_twisted from_twisted]; t. Qed. + + Program Definition zero : point := (0, 1, 1, 0). + Next Obligation. t. Qed. + + Program Definition opp P : point := + match coordinates P return F*F*F*F with (X,Y,Z,T) => (Fopp X, Y, Z, Fopp T) end. + Next Obligation. t. Qed. + + Section TwistMinusOne. + Context {a_eq_minus1:a = Fopp 1} {twice_d} {k_eq_2d:twice_d = d+d}. + Program Definition m1add + (P1 P2:point) : point := + match coordinates P1, coordinates P2 return F*F*F*F with + (X1, Y1, Z1, T1), (X2, Y2, Z2, T2) => + let A := (Y1-X1)*(Y2-X2) in + let B := (Y1+X1)*(Y2+X2) in + let C := T1*twice_d*T2 in + let D := Z1*(Z2+Z2) in + let E := B-A in + let F := D-C in + let G := D+C in + let H := B+A in + let X3 := E*F in + let Y3 := G*H in + let T3 := E*H in + let Z3 := F*G in + (X3, Y3, Z3, T3) + end. + Next Obligation. + match goal with + | [ |- match (let (_, _) := coordinates ?P1 in let (_, _) := _ in let (_, _) := _ in let (_, _) := coordinates ?P2 in _) with _ => _ end ] + => pose proof (E.denominator_nonzero _ nonzero_a square_a _ nonsquare_d _ _ (proj2_sig (to_twisted P1)) _ _ (proj2_sig (to_twisted P2))) + end; t. + Qed. + + Global Instance isomorphic_commutative_group_m1 : + @Group.isomorphic_commutative_groups + Epoint E.eq + (E.add(nonzero_a:=nonzero_a)(square_a:=square_a)(nonsquare_d:=nonsquare_d)) + (E.zero(nonzero_a:=nonzero_a)) + (E.opp(nonzero_a:=nonzero_a)) + point eq m1add zero opp + from_twisted to_twisted. + Proof. + eapply Group.commutative_group_by_isomorphism; try exact _. + par: abstract + (cbv [to_twisted from_twisted zero opp m1add]; intros; + repeat match goal with + | |- context[E.add ?P ?Q] => + unique pose proof (E.denominator_nonzero _ nonzero_a square_a _ nonsquare_d _ _ (proj2_sig P) _ _ (proj2_sig Q)) end; + t). + Qed. + End TwistMinusOne. +End ExtendedCoordinates. diff --git a/src/Curves/Edwards/XYZT/Precomputed.v b/src/Curves/Edwards/XYZT/Precomputed.v new file mode 100644 index 000000000..6b925c020 --- /dev/null +++ b/src/Curves/Edwards/XYZT/Precomputed.v @@ -0,0 +1,78 @@ +Require Import Crypto.Util.Decidable Crypto.Util.Notations Crypto.Algebra.Hierarchy. + +Require Import Crypto.Spec.CompleteEdwardsCurve Crypto.Curves.Edwards.XYZT.Basic. +Require Import Coq.Classes.Morphisms. + +Require Import Crypto.Util.Tactics.DestructHead. +Require Import Crypto.Util.Tactics.BreakMatch. +Require Import Crypto.Util.Prod. +Require Import Crypto.Algebra.Field. + +Section ExtendedCoordinates. + Context {F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv} + {field:@Algebra.Hierarchy.field F Feq Fzero Fone Fopp Fadd Fsub Fmul Finv Fdiv} + {char_ge_3 : @Ring.char_ge F Feq Fzero Fone Fopp Fadd Fsub Fmul (BinNat.N.succ_pos BinNat.N.two)} + {Feq_dec:DecidableRel Feq}. + + Local Infix "=" := Feq : type_scope. Local Notation "a <> b" := (not (a = b)) : type_scope. + Local Notation "0" := Fzero. Local Notation "1" := Fone. + Local Infix "+" := Fadd. Local Infix "*" := Fmul. + Local Infix "-" := Fsub. Local Infix "/" := Fdiv. + Local Notation "x ^ 2" := (x*x). + + Context {a d: F} + {nonzero_a : a <> 0} + {square_a : exists sqrt_a, sqrt_a^2 = a} + {nonsquare_d : forall x, x^2 <> d}. + Local Notation Epoint := (@E.point F Feq Fone Fadd Fmul a d). + + Local Notation onCurve x y := (a*x^2 + y^2 = 1 + d*x^2*y^2) (only parsing). + Definition precomputed_point : Type := F*F*F. + + Definition of_twisted (P:Epoint) := + let '(x, y) := E.coordinates P in + (y+x, y-x, (let xy := x*y in xy+xy)*d). + + Section TwistMinusOne. + Context {a_eq_minus1:a = Fopp 1}. + Definition m1add_precomputed_coordinates (P:F*F*F*F) (Q:precomputed_point) : F*F*F*F := + let '(X1, Y1, Z1, T1) := P in + let '(ypx2, ymx2, xy2d2) := Q in + let YpX1 := Y1+X1 in + let YmX1 := Y1-X1 in + let A := YpX1*ypx2 in + let B := YmX1*ymx2 in + let C := xy2d2*T1 in + let D := Z1+Z1 in + let X3 := A-B in + let Y3 := A+B in + let Z3 := D+C in + let T3 := D-C in + (* X/Z, Y/T = x, y *) + let X3 := X3*T3 in + let Y3 := Y3*Z3 in + let Z3 := T3*Z3 in + let T3 := X3*Y3 in + (X3, Y3, Z3, T3). + + Create HintDb points_as_coordinates discriminated. + Hint Unfold XYZT.Basic.point XYZT.Basic.coordinates XYZT.Basic.from_twisted XYZT.Basic.m1add + E.point E.coordinates m1add_precomputed_coordinates of_twisted precomputed_point : points_as_coordinates. + Local Notation m1add := (Basic.m1add(nonzero_a:=nonzero_a)(square_a:=square_a)(a_eq_minus1:=a_eq_minus1)(nonsquare_d:=nonsquare_d)(k_eq_2d:=reflexivity _)). + Local Notation XYZT_of_twisted := (Basic.from_twisted(nonzero_a:=nonzero_a)(d:=d)). + Lemma m1add_precomputed_coordinates_correct P Q : + let '(X1, Y1, Z1, T1) := m1add_precomputed_coordinates (XYZT.Basic.coordinates P) (of_twisted Q) in + let '(X2, Y2, Z2, T2) := coordinates (m1add P (XYZT_of_twisted Q)) in + Z2*X1 = Z1*X2 /\ Z2*Y1 = Z1*Y2. + Proof. + repeat match goal with + | _ => progress autounfold with points_as_coordinates in * + | _ => progress destruct_head' @prod + | _ => progress destruct_head' @sig + | _ => progress destruct_head' @and + | _ => progress cbv [proj1_sig fst snd] + | |- _ /\ _ => split + end; fsatz. + Qed. + End TwistMinusOne. +End ExtendedCoordinates.
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