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Require Import Crypto.Spec.ModularArithmetic.
Require Import Crypto.Spec.CompleteEdwardsCurve.
Require Import Crypto.Spec.EdDSA.
Require ModularArithmetic.PrimeFieldTheorems. (* to know that Z mod p is a field *)
(* TODO: move this to a separate file *)
Require Crypto.Util.Decidable.
Require Crypto.Util.Tactics.
Module Pre.
Local Open Scope F_scope.
Lemma curve25519_params_ok {prime_q:Znumtheory.prime (2^255-19)} :
@E.twisted_edwards_params (F (2 ^ 255 - 19)) (@eq (F (2 ^ 255 - 19))) (@F.zero (2 ^ 255 - 19))
(@F.one (2 ^ 255 - 19)) (@F.add (2 ^ 255 - 19)) (@F.mul (2 ^ 255 - 19))
(@F.opp (2 ^ 255 - 19) 1)
(@F.opp (2 ^ 255 - 19) (F.of_Z (2 ^ 255 - 19) 121665) / F.of_Z (2 ^ 255 - 19) 121666).
Proof.
pose (@PrimeFieldTheorems.F.Decidable_square (2^255-19) _);
Tactics.specialize_by Decidable.vm_decide; split; Decidable.vm_decide_no_check.
Qed.
End Pre.
(* these 2 proofs can be generated using https://github.com/andres-erbsen/safecurves-primes *)
Axiom prime_q : Znumtheory.prime (2^255-19). Global Existing Instance prime_q.
Axiom prime_l : Znumtheory.prime (2^252 + 27742317777372353535851937790883648493). Global Existing Instance prime_l.
Section Ed25519.
Local Open Scope Z_scope.
Definition q : BinNums.Z := 2^255 - 19.
Definition Fq : Type := F q.
Definition l : BinNums.Z := 2^252 + 27742317777372353535851937790883648493.
Definition Fl : Type := F l.
Local Open Scope F_scope.
Definition a : Fq := F.opp 1.
Definition d : Fq := F.opp (F.of_Z _ 121665) / (F.of_Z _ 121666).
Local Open Scope nat_scope.
Definition b : nat := 256.
Definition n : nat := b - 2.
Definition c : nat := 3.
Context {H: forall n : nat, Word.word n -> Word.word (b + b)}.
Global Instance curve_params :
E.twisted_edwards_params
(F:=Fq) (Feq:=Logic.eq) (Fzero:=F.zero) (Fone:=F.one) (Fadd:=F.add) (Fmul:=F.mul)
(a:=a) (d:=d). Proof Pre.curve25519_params_ok.
Definition E : Type := E.point
(F:=Fq) (Feq:=Logic.eq) (Fone:=F.one) (Fadd:=F.add) (Fmul:=F.mul)
(a:=a) (d:=d).
Local Obligation Tactic := Decidable.vm_decide. (* to prove that B is on curve *)
Program Definition B : E :=
(F.of_Z q 15112221349535400772501151409588531511454012693041857206046113283949847762202,
F.of_Z q 4 / F.of_Z q 5).
Definition Fencode {b : nat} {m} : F m -> Word.word b :=
fun x : F m => (Word.NToWord _ (BinIntDef.Z.to_N (F.to_Z x))).
Definition sign (x : F q) : bool := BinIntDef.Z.testbit (F.to_Z x) 0.
Definition Eenc : E -> Word.word b := fun P =>
let '(x,y) := E.coordinates P in Word.WS (sign x) (Fencode y).
Definition Senc : Fl -> Word.word b := Fencode.
(* TODO(andreser): prove this after we have fast scalar multplication *)
Axiom B_order_l : CompleteEdwardsCurveTheorems.E.eq (BinInt.Z.to_nat l * B)%E E.zero.
Require Import Crypto.Util.Decidable.
Definition ed25519 :
EdDSA (E:=E) (Eadd:=E.add) (Ezero:=E.zero) (EscalarMult:=E.mul) (B:=B)
(Eopp:=Crypto.CompleteEdwardsCurve.CompleteEdwardsCurveTheorems.E.opp) (* TODO: move defn *)
(Eeq:=Crypto.CompleteEdwardsCurve.CompleteEdwardsCurveTheorems.E.eq) (* TODO: move defn *)
(l:=l) (b:=b) (n:=n) (c:=c)
(Eenc:=Eenc) (Senc:=Senc) (H:=H).
Proof. split; try exact _; try exact B_order_l; vm_decide. Qed.
End Ed25519.
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