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Require Import Coq.ZArith.ZArith Coq.micromega.Psatz Coq.Classes.Morphisms Coq.Classes.RelationClasses.
Require Import Crypto.Spec.Ed25519.
Require Import Crypto.Specific.X86.Core.
Require Import Crypto.EdDSARepChange.
Require Import Crypto.BoundedArithmetic.Interface.
Require Import Crypto.BoundedArithmetic.X86ToZLike.
Require Import Crypto.BoundedArithmetic.X86ToZLikeProofs.
Require Import Crypto.BoundedArithmetic.Eta.
Require Import Crypto.ModularArithmetic.BarrettReduction.ZBounded.
Require Import Crypto.ModularArithmetic.ZBoundedZ.
Require Import Crypto.Spec.ModularArithmetic.
Require Import Crypto.Util.Tuple.
Require Import Crypto.Util.LetIn.
Require Import Crypto.Util.Tactics.
Import NPeano.
Local Notation modulusv := (2^252 + 27742317777372353535851937790883648493)%Z.
Local Coercion Z.of_nat : nat >-> Z.
Local Notation eta x := (fst x, snd x).
Local Notation eta3 x := (eta (fst x), snd x).
Local Notation eta4 x := (eta3 (fst x), snd x).
Local Notation eta4' x := (eta (fst x), eta (snd x)).
Section Z.
Definition SRep := Z. (*tuple x86.W (256/n).*)
Definition SRepEq : Relation_Definitions.relation SRep := Logic.eq.
Local Instance SRepEquiv : RelationClasses.Equivalence SRepEq := _.
Local Notation base := 2%Z (* TODO(@andres-erbsen): Is this the correct base, or are we using something else? *).
Local Notation smaller_bound_exp := 250%Z (* TODO(@andres-erbsen): Is this the correct smaller size (2^250), or are we using something else? *).
Lemma smaller_bound_smaller : (0 <= smaller_bound_exp <= 256)%Z. Proof. vm_compute; intuition congruence. Qed.
Lemma modulusv_in_range : 0 <= modulusv < 2 ^ 256. Proof. vm_compute; intuition congruence. Qed.
Lemma modulusv_pos : 0 < modulusv. Proof. vm_compute; reflexivity. Qed.
Section gen.
Lemma full_width_pos : (0 < 256)%Z. Proof. omega. Qed.
Let offset'0 := Eval compute in ((256 - smaller_bound_exp) / 2)%Z.
Let k'0 := Eval compute in ((256 - offset'0) / Z.log2 base)%Z.
Section params_gen.
Import BarrettBundled.
Let offset' := Eval compute in offset'0.
Let k' := Eval compute in k'0.
Local Instance x86_25519_Barrett : BarrettParameters
:= { m := modulusv;
b := base;
k := k';
offset := offset';
ops := _;
μ' := base ^ (2 * k') / modulusv }.
Local Instance x86_25519_BarrettProofs
: BarrettParametersCorrect x86_25519_Barrett
:= { props := _ }.
Proof.
vm_compute; reflexivity.
vm_compute; reflexivity.
vm_compute; clear; abstract intuition congruence.
vm_compute; clear; abstract intuition congruence.
vm_compute; clear; abstract intuition congruence.
vm_compute; clear; abstract intuition congruence.
vm_compute; clear; abstract intuition congruence.
vm_compute; reflexivity.
Defined.
End params_gen.
End gen.
Local Existing Instance x86_25519_Barrett.
Local Existing Instance x86_25519_BarrettProofs.
Declare Reduction srep := cbv [barrett_reduce_function_bundled barrett_reduce_function BarrettBundled.m BarrettBundled.b BarrettBundled.k BarrettBundled.offset BarrettBundled.μ' ZBounded.ConditionalSubtractModulus ZBounded.CarrySubSmall ZBounded.Mod_SmallBound ZBounded.Mod_SmallBound ZBounded.Mul ZBounded.DivBy_SmallBound ZBounded.DivBy_SmallerBound ZBounded.modulus_digits x86_25519_Barrett BarrettBundled.ops ZZLikeOps ZBounded.CarryAdd Z.pow2_mod].
Definition SRepDecModL : Word.word (256 + 256) -> SRep
:= Eval srep in
fun w => dlet w := (Z.of_N (Word.wordToN w)) in barrett_reduce_function_bundled w.
(*Local Arguments SRepDecModL' / _.
Ltac change_values v :=
match v with
| context vv[2 * ?x]
=> let x2 := (eval vm_compute in (2 * x)) in
let v' := context vv[x2] in
change_values v'
| context vv[Z.pos ?x + Z.pos ?y]
=> let x2 := (eval vm_compute in (Z.pos x + Z.pos y)) in
let v' := context vv[x2] in
change_values v'
| context vv[Z.pos ?x - Z.pos ?y]
=> let x2 := (eval vm_compute in (Z.pos x - Z.pos y)) in
let v' := context vv[x2] in
change_values v'
| context vv[Z.pos ?x / Z.pos ?y]
=> let x2 := (eval vm_compute in (Z.pos x / Z.pos y)) in
let v' := context vv[x2] in
change_values v'
| context vv[Z.ones (Z.pos ?x)]
=> let x2 := (eval vm_compute in (Z.ones (Z.pos x))) in
let v' := context vv[x2] in
change_values v'
| context vv[2^?x]
=> let x2 := (eval vm_compute in (2^x)) in
let v' := context vv[x2] in
change_values v'
| _ => v
end.
Definition SRepDecModL : Word.word (256 + 256) -> SRep.
Proof.
let v' := (eval cbv [SRepDecModL'] in SRepDecModL') in
let v' := (eval cbv beta iota delta [Let_In] in v') in
let v := (*change_values*) v' in
unify v v';
exact v.
Defined.*)
Check @sign_correct _ _ _ _ _ _ _ _ _ _ _ _ _ _ ed25519.
Check @verify_correct _ _ _ _ _ _ _ _ _ _ _ _ _ _ ed25519.
Definition S2Rep : ModularArithmetic.F.F l -> SRep := F.to_Z.
Lemma Z_of_nat_of_word_eq_Z_of_N_of_word
: forall (n : nat) (w : Word.word n), Z.of_nat (Word.wordToNat w) = Z.of_N (Word.wordToN w).
Admitted.
Lemma SRepDecModL_Correct : forall w : Word.word (b + b), SRepEq (S2Rep (ModularArithmetic.F.of_nat l (Word.wordToNat w))) (SRepDecModL w).
Proof.
intro w; change (SRepDecModL w) with (barrett_reduce_function_bundled (Z.of_N (Word.wordToN w))).
unfold SRepEq, S2Rep, b in *.
pose proof (barrett_reduce_correct_bundled (Z.of_N (Word.wordToN w))) as H.
simpl in H; destruct H as [H _].
{ pose proof (Word.wordToNat_bound w).
rewrite Word.wordToN_nat, nat_N_Z.
rewrite WordUtil.pow2_id in H.
change (Z.pow_pos 2 506) with (2^(253 + 253))%Z.
apply inj_lt in H.
rewrite Z.pow_Zpow, Nat2Z.inj_add in H; simpl @Z.of_nat in *.
split; auto with zarith.
admit. (* impossible *) }
{ simpl in *; rewrite H; clear H.
rewrite Z_of_nat_of_word_eq_Z_of_N_of_word; reflexivity. }
Admitted.
Definition SRepAdd : SRep -> SRep -> SRep
:= Eval srep in fun x y => barrett_reduce_function_bundled (snd (ZBounded.CarryAdd x y)).
Lemma SRepAdd_Correct : forall x y : ModularArithmetic.F.F l, SRepEq (S2Rep (ModularArithmetic.F.add x y)) (SRepAdd (S2Rep x) (S2Rep y)).
Proof.
intros x y; change (SRepAdd ?x ?y) with (barrett_reduce_function_bundled (snd (ZBounded.CarryAdd x y))).
unfold SRepEq, S2Rep, b in *; simpl.
match goal with
| [ |- context[barrett_reduce_function_bundled ?x] ]
=> pose proof (barrett_reduce_correct_bundled x) as H
end.
pose proof (ModularArithmeticTheorems.F.to_Z_range x).
pose proof (ModularArithmeticTheorems.F.to_Z_range y).
unfold l in *.
specialize_by auto using modulusv_pos.
assert (F.to_Z x + F.to_Z y < 2 * modulusv - 1) by omega.
assert (2 * modulusv - 1 <= 2 ^ (253 + 253)) by (vm_compute; clear; intuition congruence).
assert (2^(253 + 253) < 2^512) by (vm_compute; clear; intuition congruence).
assert (0 <= F.to_Z x + F.to_Z y < 2^512) by omega.
simpl in H; destruct H as [H _].
{ change (Z.pow_pos 2 506) with (2^(253 + 253))%Z.
rewrite Z.pow2_mod_spec by omega; Z.rewrite_mod_small.
simpl in *; omega. }
{ simpl in H |- *; rewrite H; clear H.
rewrite Z.pow2_mod_spec by omega; Z.rewrite_mod_small.
reflexivity. }
Qed.
Global Instance SRepAdd_Proper : Proper (SRepEq ==> SRepEq ==> SRepEq) SRepAdd.
Proof. unfold SRepEq; repeat intro; subst; reflexivity. Qed.
Definition SRepMul : SRep -> SRep -> SRep
:= Eval srep in fun x y => barrett_reduce_function_bundled (ZBounded.Mul x y).
Lemma SRepMul_Correct : forall x y : ModularArithmetic.F.F l, SRepEq (S2Rep (ModularArithmetic.F.mul x y)) (SRepMul (S2Rep x) (S2Rep y)).
Proof.
intros x y; change (SRepMul ?x ?y) with (barrett_reduce_function_bundled (ZBounded.Mul x y)).
unfold SRepEq, S2Rep, b in *; simpl.
match goal with
| [ |- context[barrett_reduce_function_bundled ?x] ]
=> pose proof (barrett_reduce_correct_bundled x) as H
end.
pose proof (ModularArithmeticTheorems.F.to_Z_range x).
pose proof (ModularArithmeticTheorems.F.to_Z_range y).
unfold l in *.
specialize_by auto using modulusv_pos.
assert (0 <= F.to_Z x * F.to_Z y < modulusv * modulusv) by nia.
assert (modulusv * modulusv <= 2 ^ (253 + 253)) by (vm_compute; clear; intuition congruence).
assert (2^(253 + 253) < 2^512) by (vm_compute; clear; intuition congruence).
simpl in H; destruct H as [H _].
{ change (Z.pow_pos 2 506) with (2^(253 + 253))%Z.
set (k := modulusv) in *; compute in k.
let v := (eval unfold k in k) in change v with k.
omega. }
{ simpl in H |- *; rewrite H; clear H.
reflexivity. }
Qed.
Global Instance SRepMul_Proper : Proper (SRepEq ==> SRepEq ==> SRepEq) SRepMul.
Proof. unfold SRepEq; repeat intro; subst; reflexivity. Qed.
Definition SRepDecModLShort : Word.word (n + 1) -> SRep
:= Eval srep in
fun w => dlet w := (Z.of_N (Word.wordToN w)) in barrett_reduce_function_bundled w.
Lemma SRepDecModLShort_Correct : forall w : Word.word (n + 1), SRepEq (S2Rep (ModularArithmetic.F.of_nat l (Word.wordToNat w))) (SRepDecModLShort w).
Proof.
intro w; change (SRepDecModLShort w) with (barrett_reduce_function_bundled (Z.of_N (Word.wordToN w))).
unfold SRepEq, S2Rep, n, b in *.
pose proof (barrett_reduce_correct_bundled (Z.of_N (Word.wordToN w))) as H.
simpl in H; destruct H as [H _].
{ pose proof (Word.wordToNat_bound w).
rewrite Word.wordToN_nat, nat_N_Z.
rewrite WordUtil.pow2_id in H.
change (Z.pow_pos 2 506) with (2^(253 + 253))%Z.
apply inj_lt in H.
rewrite Z.pow_Zpow, Nat2Z.inj_add in H; simpl @Z.of_nat in *.
split; auto with zarith.
assert (2^(254+1) < (2^(253+253))) by (vm_compute; reflexivity).
omega. }
{ simpl in *; rewrite H; clear H.
rewrite Z_of_nat_of_word_eq_Z_of_N_of_word; reflexivity. }
Qed.
Arguments Algebra.group : clear implicits.
Check @sign_correct _ _ _ _ _ _ _ _ _ _ _ _ _ _ ed25519.
Check @verify_correct _ _ _ _ _ _ _ _ _ _ _ _ _ _ ed25519.
End Z.
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