diff options
author | 2016-09-16 13:31:40 -0400 | |
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committer | 2016-09-16 19:00:06 -0400 | |
commit | 1ea69cd53ff8472bb23c338d0e3fcac0a1f9ada5 (patch) | |
tree | 14379b1df13a789daf454f29324661ebb85c9f0c /src/Experiments | |
parent | 7d139ded819549c587b169e6ef54d411bc543cd4 (diff) |
Derive EdDSA.verify from equational specification
Experiments/SpecEd25519 will come back soon
Diffstat (limited to 'src/Experiments')
-rw-r--r-- | src/Experiments/EdDSARefinement.v | 244 | ||||
-rw-r--r-- | src/Experiments/SpecEd25519.v | 126 | ||||
-rw-r--r-- | src/Experiments/SpecificCurve25519.v | 17 |
3 files changed, 151 insertions, 236 deletions
diff --git a/src/Experiments/EdDSARefinement.v b/src/Experiments/EdDSARefinement.v index f8e93c6f3..afdf24604 100644 --- a/src/Experiments/EdDSARefinement.v +++ b/src/Experiments/EdDSARefinement.v @@ -1,9 +1,12 @@ +Require Import Crypto.Util.FixCoqMistakes. Require Import Crypto.Spec.EdDSA Bedrock.Word. -Require Import Coq.Classes.Morphisms. +Require Import Coq.Classes.Morphisms Coq.Relations.Relation_Definitions. Require Import Crypto.Algebra. Import Group ScalarMult. Require Import Crypto.Util.Decidable Crypto.Util.Option Crypto.Util.Tactics. Require Import Coq.omega.Omega. Require Import Crypto.Util.Notations. +Require Import Crypto.Util.Option Crypto.Util.Logic Crypto.Util.Relations Crypto.Util.WordUtil. +Require Import Crypto.Spec.ModularArithmetic Crypto.ModularArithmetic.PrimeFieldTheorems. Module Import NotationsFor8485. Import NPeano Nat. @@ -12,118 +15,173 @@ End NotationsFor8485. Section EdDSA. Context `{prm:EdDSA}. - Context {eq_dec:DecidableRel Eeq}. - Local Infix "==" := Eeq. - Local Notation valid := (@valid E Eeq Eadd EscalarMult b H l B Eenc Senc). - Local Infix "*" := EscalarMult. Local Infix "+" := Eadd. Local Infix "++" := combine. - Local Notation "P - Q" := (P + Eopp Q). - Local Arguments H {_} _. - - Context {Proper_Eenc : Proper (Eeq==>Logic.eq) Eenc}. - Context {Proper_Eopp : Proper (Eeq==>Eeq) Eopp}. - Context {Proper_Eadd : Proper (Eeq==>Eeq==>Eeq) Eadd}. - - Context {decE:word b-> option E}. - Context {decS:word b-> option nat}. - - Context {decE_canonical: forall x s, decE x = Some s -> x = Eenc s }. - Context {decS_canonical: forall x s, decS x = Some s -> x = Senc s }. - - Context {decS_Senc: forall x, decS (Senc x) = Some x}. - Context {decE_Eenc: forall x, decE (Eenc x) = Some x}. (* FIXME: equivalence relation *) - - Lemma solve_for_R : forall s B R n A, s * B == R + n * A <-> R == s*B - n*A. - Proof. - intros; split; - intro Heq; rewrite Heq; clear Heq. - { rewrite <-associative, right_inverse, right_identity; reflexivity. } - { rewrite <-associative, left_inverse, right_identity; reflexivity. } - Qed. + Local Infix "==" := Eeq. Local Infix "+" := Eadd. Local Infix "*" := EscalarMult. - Definition verify {mlen} (message:word mlen) (pk:word b) (sig:word (b+b)) : bool := - option_rect (fun _ => bool) (fun S : nat => - option_rect (fun _ => bool) (fun A : E => - weqb - (split1 b b sig) - (Eenc (S * B - (wordToNat (H (split1 b b sig ++ pk ++ message))) mod l * A)) - ) false (decE pk) - ) false (decS (split2 b b sig)) - . - - Lemma verify_correct mlen (message:word mlen) (pk:word b) (sig:word (b+b)) : - verify message pk sig = true <-> valid message pk sig. + Local Notation valid := (@valid E Eeq Eadd EscalarMult b H l B Eenc Senc). + Lemma sign_valid : forall A_ sk {n} (M:word n), A_ = public sk -> valid M A_ (sign A_ sk M). Proof. - cbv [verify option_rect option_map]. - split. - { + cbv [sign public Spec.EdDSA.valid]; intros; subst; repeat match goal with - | |- false = true -> _ => let H:=fresh "H" in intro H; discriminate H - | [H: _ |- _ ] => apply decS_canonical in H - | [H: _ |- _ ] => apply decE_canonical in H - | _ => rewrite weqb_true_iff - | _ => break_match + | |- exists _, _ => eexists + | |- _ /\ _ => eapply conj + | |- _ => reflexivity end. - intro. - subst. - rewrite <-combine_split. - rewrite Heqo. - rewrite H0. - constructor. - rewrite <-H0. - rewrite solve_for_R. - reflexivity. - } - { - intros [? ? ? ? Hvalid]. - rewrite solve_for_R in Hvalid. - rewrite split1_combine. - rewrite split2_combine. - rewrite decS_Senc. - rewrite decE_Eenc. - rewrite weqb_true_iff. - f_equiv. exact Hvalid. - } - Qed. - - Lemma sign_valid : forall A_ sk {n} (M:word n), A_ = public sk -> valid M A_ (sign A_ sk M). - Proof. - cbv [sign public]. intros. subst. split. - rewrite scalarmult_mod_order, scalarmult_add_l, cancel_left, scalarmult_mod_order, NPeano.Nat.mul_comm, scalarmult_assoc; + rewrite F.to_nat_of_nat, scalarmult_mod_order, scalarmult_add_l, cancel_left, scalarmult_mod_order, NPeano.Nat.mul_comm, scalarmult_assoc; try solve [ reflexivity - | pose proof EdDSA_l_odd; omega + | setoid_rewrite (*unify 0*) (Z2Nat.inj_iff _ 0); pose proof EdDSA_l_odd; omega + | pose proof EdDSA_l_odd; omega | apply EdDSA_l_order_B | rewrite scalarmult_assoc, mult_comm, <-scalarmult_assoc, EdDSA_l_order_B, scalarmult_zero_r; reflexivity ]. Qed. + Lemma solve_for_R_sig : forall s B R n A, { solution | s * B == R + n * A <-> R == solution }. + Proof. + eexists. + set_evars. + setoid_rewrite <-(symmetry_iff(R:=Eeq)) at 1. + setoid_rewrite <-eq_r_opp_r_inv. + setoid_rewrite opp_mul. + subst_evars. + reflexivity. + Defined. + Definition solve_for_R := Eval cbv [proj2_sig solve_for_R_sig] in (fun s B R n A => proj2_sig (solve_for_R_sig s B R n A)). + + Context {Proper_Eenc : Proper (Eeq==>Logic.eq) Eenc}. + Global Instance Proper_eq_Eenc ref : Proper (Eeq ==> iff) (fun P => Eenc P = ref). + Proof. intros ? ? Hx; rewrite Hx; reflexivity. Qed. + + Context {Edec:word b-> option E} {eq_enc_E_iff: forall P_ P, Eenc P = P_ <-> Edec P_ = Some P}. + Context {Sdec:word b-> option (F l)} {eq_enc_S_iff: forall n_ n, Senc n = n_ <-> Sdec n_ = Some n}. + + Local Infix "++" := combine. + Definition verify_sig : { verify | forall mlen (message:word mlen) (pk:word b) (sig:word (b+b)), + verify mlen message pk sig = true <-> valid message pk sig }. + Proof. + eexists; intros; set_evars. + unfold Spec.EdDSA.valid. + setoid_rewrite solve_for_R. + setoid_rewrite combine_eq_iff. + setoid_rewrite and_comm at 4. setoid_rewrite and_assoc. repeat setoid_rewrite exists_and_indep_l. + setoid_rewrite (and_rewrite_l Eenc (split1 b b sig) + (fun x y => x == _ * B + wordToNat (H _ (y ++ Eenc _ ++ message)) mod _ * Eopp _)); setoid_rewrite eq_enc_S_iff. + setoid_rewrite (@exists_and_equiv_r _ _ _ _ _ _). + setoid_rewrite <- (fun A => and_rewriteleft_l (fun x => x) (Eenc A) (fun pk EencA => exists a, + Sdec (split2 b b sig) = Some a /\ + Eenc (_ * B + wordToNat (H (b + (b + mlen)) (split1 b b sig ++ EencA ++ message)) mod _ * Eopp A) + = split1 b b sig)); setoid_rewrite (eq_enc_E_iff pk). + setoid_rewrite <-weqb_true_iff. + repeat setoid_rewrite <-option_rect_false_returns_true_iff. + + subst_evars. + (* TODO: generalize this higher order reflexivity *) + match goal with + |- ?f ?mlen ?msg ?pk ?sig = true <-> ?term = true + => let term := eval pattern sig in term in + let term := eval pattern pk in term in + let term := eval pattern msg in term in + let term := match term with + (fun msg => (fun pk => (fun sig => @?body msg pk sig) sig) pk) msg => + body + end in + let term := eval pattern mlen in term in + let term := match term with + (fun mlen => @?body mlen) mlen => body + end in + unify f term; reflexivity + end. + Defined. + Definition verify {mlen} (message:word mlen) (pk:word b) (sig:word (b+b)) : bool := + Eval cbv [proj1_sig verify_sig] in proj1_sig verify_sig mlen message pk sig. + Lemma verify_correct : forall {mlen} (message:word mlen) pk sig, + verify message pk sig = true <-> valid message pk sig. + Proof. exact (proj2_sig verify_sig). Qed. + Section ChangeRep. - Context {A Aeq Aadd Aid Aopp} {Agroup:@group A Aeq Aadd Aid Aopp}. - Context {EtoA} {Ahomom:@is_homomorphism E Eeq Eadd A Aeq Aadd EtoA}. + Context {Erep ErepEq ErepAdd ErepId ErepOpp} {Agroup:@group Erep ErepEq ErepAdd ErepId ErepOpp}. + Context {EToRep} {Ahomom:@is_homomorphism E Eeq Eadd Erep ErepEq ErepAdd EToRep}. + + Context {ERepEnc : Erep -> word b} + {ERepEnc_correct : forall P:E, Eenc P = ERepEnc (EToRep P)} + {Proper_ERepEnc:Proper (ErepEq==>Logic.eq) ERepEnc}. - Context {Aenc : A -> word b} {Proper_Aenc:Proper (Aeq==>Logic.eq) Aenc} - {Aenc_correct : forall P:E, Eenc P = Aenc (EtoA P)}. + Context {ERepDec : word b -> option Erep} + {ERepDec_correct : forall w, ERepDec w = option_map EToRep (Edec w) }. - Context {S Seq} `{@Equivalence S Seq} {natToS:nat->S} - {SAmul:S->A->A} {Proper_SAmul: Proper (Seq==>Aeq==>Aeq) SAmul} - {SAmul_correct: forall n P, Aeq (EtoA (n*P)) (SAmul (natToS n) (EtoA P))} - {SdecS} {SdecS_correct : forall w, (decS w) = (SdecS w)} (* FIXME: equivalence relation *) - {natToS_modl : forall n, Seq (natToS (n mod l)) (natToS n)}. + Context {SRep SRepEq} `{@Equivalence SRep SRepEq} {S2Rep:F l->SRep}. + + Context {SRepDecModL} {SRepDecModL_correct: forall (w:word (b+b)), SRepEq (S2Rep (F.of_nat _ (wordToNat w))) (SRepDecModL w)}. + + Context {SRepERepMul:SRep->Erep->Erep} + {SRepERepMul_correct:forall n P, ErepEq (EToRep (n*P)) (SRepERepMul (S2Rep (F.of_nat _ n)) (EToRep P))} + {Proper_SRepERepMul: Proper (SRepEq==>ErepEq==>ErepEq) SRepERepMul}. + + Context {SRepDec: word b -> option SRep} + {SRepDec_correct : forall w, option_eq SRepEq (option_map S2Rep (Sdec w)) (SRepDec w)}. Definition verify_using_representation {mlen} (message:word mlen) (pk:word b) (sig:word (b+b)) : { answer | answer = verify message pk sig }. Proof. + pose proof EdDSA_l_odd. + assert (l_pos:(0 < l)%Z) by omega. eexists. cbv [verify]. - repeat ( - setoid_rewrite Aenc_correct - || setoid_rewrite homomorphism - || setoid_rewrite homomorphism_id - || setoid_rewrite (homomorphism_inv(INV:=Eopp)(inv:=Aopp)(eq:=Aeq)(phi:=EtoA)) - || setoid_rewrite SAmul_correct - || setoid_rewrite SdecS_correct - || setoid_rewrite natToS_modl - ). + + etransitivity. Focus 2. { + eapply Proper_option_rect_nd_changebody; [intro|reflexivity]. + eapply Proper_option_rect_nd_changebody; [intro|reflexivity]. + repeat ( + rewrite ERepEnc_correct + || rewrite homomorphism + || rewrite homomorphism_id + || rewrite (homomorphism_inv(INV:=Eopp)(inv:=ErepOpp)(eq:=ErepEq)(phi:=EToRep)) + || rewrite SRepERepMul_correct + || rewrite SdecS_correct + || rewrite SRepDecModL_correct + || rewrite (@F.of_nat_to_nat _ l_pos _) + || rewrite (@F.of_nat_mod _ l_pos _) + ). + reflexivity. + } Unfocus. + + (* lazymatch goal with |- _ _ (option_rect _ ?some _ _) => idtac some end. *) + setoid_rewrite (option_rect_option_map EToRep + (fun s => + option_rect (fun _ : option _ => bool) + (fun s0 => + weqb + (ERepEnc + (ErepAdd (SRepERepMul (_ s0) (EToRep B)) + (SRepERepMul + (SRepDecModL + (H _ (split1 b b sig ++ pk ++ message))) + (ErepOpp (s))))) (split1 b b sig)) false + (Sdec (split2 b b sig))) + false); rewrite <-(ERepDec_correct pk). + + etransitivity. Focus 2. { + eapply Proper_option_rect_nd_changebody; [intro|reflexivity]. + set_evars. + setoid_rewrite (option_rect_option_map S2Rep + (fun s0 => + weqb + (ERepEnc + (ErepAdd (SRepERepMul (s0) (EToRep B)) + (SRepERepMul + (SRepDecModL (H _ (split1 b b sig ++ pk ++ message))) + (ErepOpp _)))) (split1 b b sig)) + + false). + subst_evars. + + eapply Proper_option_rect_nd_changevalue; + [repeat intro; repeat f_equiv; eassumption|reflexivity|..]. + + symmetry. + eapply SRepDec_correct. + } Unfocus. + reflexivity. Defined. End ChangeRep. diff --git a/src/Experiments/SpecEd25519.v b/src/Experiments/SpecEd25519.v deleted file mode 100644 index 692254512..000000000 --- a/src/Experiments/SpecEd25519.v +++ /dev/null @@ -1,126 +0,0 @@ -Require Import Coq.ZArith.ZArith Coq.ZArith.Zpower Coq.ZArith.ZArith Coq.ZArith.Znumtheory. -Require Import Coq.Numbers.Natural.Peano.NPeano Coq.NArith.NArith. -Require Import Crypto.Spec.ModularWordEncoding. -Require Import Crypto.Encoding.ModularWordEncodingTheorems. -Require Import Crypto.Spec.EdDSA. -Require Import Crypto.Spec.CompleteEdwardsCurve Crypto.CompleteEdwardsCurve.CompleteEdwardsCurveTheorems. -Require Import Crypto.ModularArithmetic.PrimeFieldTheorems Crypto.ModularArithmetic.ModularArithmeticTheorems. -Require Import Crypto.Util.NatUtil Crypto.Util.ZUtil Crypto.Util.WordUtil Crypto.Util.NumTheoryUtil. -Require Import Bedrock.Word. -Require Import Crypto.Tactics.VerdiTactics. -Require Import Coq.Logic.Decidable Crypto.Util.Decidable. -Require Import Coq.omega.Omega. - -(* TODO: move to PrimeFieldTheorems *) -Lemma minus1_is_square {q} : prime q -> (q mod 4)%Z = 1%Z -> (exists y, y*y = F.opp (F.of_Z q 1))%F. - intros; pose proof prime_ge_2 q _. - rewrite F.square_iff. - destruct (minus1_square_1mod4 q) as [b b_id]; trivial; exists b. - rewrite b_id, F.to_Z_opp, F.to_Z_of_Z, Z.mod_opp_l_nz, !Zmod_small; - (repeat (omega || rewrite Zmod_small)). -Qed. - -Definition q : Z := (2 ^ 255 - 19)%Z. -Global Instance prime_q : prime q. Admitted. -Lemma two_lt_q : (2 < q)%Z. Proof. reflexivity. Qed. -Lemma char_gt_2 : (1 + 1 <> (0:F q))%F. vm_decide_no_check. Qed. - -Definition a : F q := F.opp 1%F. -Lemma nonzero_a : a <> 0%F. Proof. vm_decide_no_check. Qed. -Lemma square_a : exists b, (b*b=a)%F. -Proof. pose (@F.Decidable_square q _ two_lt_q a); vm_decide_no_check. Qed. -Definition d : F q := (F.opp (F.of_Z _ 121665) / (F.of_Z _ 121666))%F. - -Lemma nonsquare_d : forall x, (x*x <> d)%F. -Proof. pose (@F.Decidable_square q _ two_lt_q d). vm_decide_no_check. Qed. - -Instance curve25519params : @E.twisted_edwards_params (F q) eq 0%F 1%F F.add F.mul a d := - { - nonzero_a := nonzero_a; - char_gt_2 := char_gt_2; - square_a := square_a; - nonsquare_d := nonsquare_d - }. - -Lemma two_power_nat_Z2Nat : forall n, Z.to_nat (two_power_nat n) = (2 ^ n)%nat. -Admitted. - -Definition b := 256%nat. -Lemma b_valid : (2 ^ (b - 1) > Z.to_nat q)%nat. -Proof. - unfold q, gt. - replace (2 ^ (b - 1))%nat with (Z.to_nat (2 ^ (Z.of_nat (b - 1)))) - by (rewrite <- two_power_nat_equiv; apply two_power_nat_Z2Nat). - rewrite <- Z2Nat.inj_lt; compute; congruence. -Qed. - -Definition c := 3%nat. -Lemma c_valid : (c = 2 \/ c = 3)%nat. -Proof. - right; auto. -Qed. - -Definition n := (b - 2)%nat. -Lemma n_ge_c : (n >= c)%nat. -Proof. - unfold n, c, b; omega. -Qed. -Lemma n_le_b : (n <= b)%nat. -Proof. - unfold n, b; omega. -Qed. - -Definition l : nat := Z.to_nat (252 + 27742317777372353535851937790883648493)%Z. -Lemma prime_l : prime (Z.of_nat l). Admitted. -Lemma l_odd : (l > 2)%nat. -Proof. - unfold l, proj1_sig. - rewrite Z2Nat.inj_add; try omega. - apply lt_plus_trans. - compute; omega. -Qed. - -Require Import Crypto.Spec.Encoding. - -Lemma q_pos : (0 < q)%Z. pose proof prime_ge_2 q _; omega. Qed. -Definition FqEncoding : canonical encoding of (F q) as word (b-1) := - @modular_word_encoding q (b - 1) q_pos b_valid. - -Lemma l_pos : (0 < Z.of_nat l)%Z. pose proof prime_l; prime_bound. Qed. -Lemma l_bound : (Z.to_nat (Z.of_nat l) < 2 ^ b)%nat. -Proof. - rewrite Nat2Z.id. - rewrite <- pow2_id. - rewrite Zpow_pow2. - unfold l. - apply Z2Nat.inj_lt; compute; congruence. -Qed. -Definition FlEncoding : canonical encoding of F (Z.of_nat l) as word b := - @modular_word_encoding (Z.of_nat l) b l_pos l_bound. - -Lemma q_5mod8 : (q mod 8 = 5)%Z. cbv; reflexivity. Qed. - -Lemma sqrt_minus1_valid : ((F.of_Z q 2 ^ Z.to_N (q / 4)) ^ 2 = F.opp 1)%F. -Proof. vm_decide_no_check. Qed. - -Local Notation point := (@E.point (F q) eq 1%F F.add F.mul a d). -Local Notation zero := (E.zero(field:=F.field_modulo q)). -Local Notation add := (E.add(H:=curve25519params)). -Local Infix "*" := (E.mul(H:=curve25519params)). -Axiom H : forall n : nat, word n -> word (b + b). -Axiom B : point. (* TODO: B = decodePoint (y=4/5, x="positive") *) -Axiom B_nonzero : B <> zero. -Axiom l_order_B : E.eq (l * B) zero. -Axiom Eenc : point -> word b. -Axiom Senc : nat -> word b. - -Global Instance Ed25519 : @EdDSA point E.eq add zero E.opp E.mul b H c n l B Eenc Senc := - { - EdDSA_c_valid := c_valid; - EdDSA_n_ge_c := n_ge_c; - EdDSA_n_le_b := n_le_b; - EdDSA_B_not_identity := B_nonzero; - EdDSA_l_prime := prime_l; - EdDSA_l_odd := l_odd; - EdDSA_l_order_B := l_order_B - }. diff --git a/src/Experiments/SpecificCurve25519.v b/src/Experiments/SpecificCurve25519.v deleted file mode 100644 index 15752b2a2..000000000 --- a/src/Experiments/SpecificCurve25519.v +++ /dev/null @@ -1,17 +0,0 @@ -Require Import Crypto.Util.Notations Coq.ZArith.BinInt. -Require Import Crypto.Spec.ModularArithmetic. -Require Import Crypto.Specific.GF25519. -Require Import Crypto.Experiments.SpecEd25519. -Require Import Crypto.CompleteEdwardsCurve.ExtendedCoordinates. -Local Infix "<<" := Z.shiftr. -Local Infix "&" := Z.land. - -Section Curve25519. - - Definition twice_d : fe25519 := Eval vm_compute in - @ModularBaseSystem.encode modulus params25519 (d + d)%F. - - Definition ge25519_add := - Eval cbv beta delta [Extended.add_coordinates fe25519 add mul sub ModularBaseSystemOpt.Let_In] in - @Extended.add_coordinates fe25519 add sub mul twice_d. -End Curve25519.
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