diff options
Diffstat (limited to 'src/Spec')
| -rw-r--r-- | src/Spec/Ed25519.v | 6 | ||||
| -rw-r--r-- | src/Spec/Encoding.v | 56 | ||||
| -rw-r--r-- | src/Spec/ModularWordEncoding.v | 31 | ||||
| -rw-r--r-- | src/Spec/PointEncoding.v | 226 |
4 files changed, 72 insertions, 247 deletions
diff --git a/src/Spec/Ed25519.v b/src/Spec/Ed25519.v index 6ab47e8e5..30892c006 100644 --- a/src/Spec/Ed25519.v +++ b/src/Spec/Ed25519.v @@ -1,6 +1,6 @@ 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.Encoding Crypto.Spec.PointEncoding. +Require Import Crypto.Spec.PointEncoding Crypto.Spec.ModularWordEncoding. Require Import Crypto.Spec.EdDSA. Require Import Crypto.Spec.CompleteEdwardsCurve Crypto.CompleteEdwardsCurve.CompleteEdwardsCurveTheorems. Require Import Crypto.ModularArithmetic.PrimeFieldTheorems Crypto.ModularArithmetic.ModularArithmeticTheorems. @@ -114,6 +114,8 @@ Proof. compute; omega. Qed. +Require Import Crypto.Spec.Encoding. + Lemma q_pos : (0 < q)%Z. q_bound. Qed. Definition FqEncoding : encoding of (F q) as word (b-1) := @modular_word_encoding q (b - 1) q_pos b_valid. @@ -140,7 +142,7 @@ Proof. reflexivity. Qed. -Definition PointEncoding := @point_encoding curve25519params (b - 1) FqEncoding q_5mod8 sqrt_minus1_valid. +Definition PointEncoding := @point_encoding curve25519params (b - 1) FqEncoding. Definition H : forall n : nat, word n -> word (b + b). Admitted. Definition B : point. Admitted. (* TODO: B = decodePoint (y=4/5, x="positive") *) diff --git a/src/Spec/Encoding.v b/src/Spec/Encoding.v index 14cf9d9d9..9a2d5e5ed 100644 --- a/src/Spec/Encoding.v +++ b/src/Spec/Encoding.v @@ -1,61 +1,7 @@ -Require Import Coq.ZArith.ZArith Coq.ZArith.Zpower Coq.ZArith.ZArith. -Require Import Coq.Numbers.Natural.Peano.NPeano. -Require Import Crypto.ModularArithmetic.PrimeFieldTheorems Crypto.ModularArithmetic.ModularArithmeticTheorems. -Require Import Bedrock.Word. -Require Import Crypto.Tactics.VerdiTactics. -Require Import Crypto.Util.NatUtil. -Require Import Crypto.Util.WordUtil. - Class Encoding (T B:Type) := { enc : T -> B ; dec : B -> option T ; encoding_valid : forall x, dec (enc x) = Some x }. -Notation "'encoding' 'of' T 'as' B" := (Encoding T B) (at level 50). - -Local Open Scope nat_scope. - -Section ModularWordEncoding. - Context {m : Z} {sz : nat} {m_pos : (0 < m)%Z} {bound_check : Z.to_nat m < 2 ^ sz}. - - Definition Fm_enc (x : F m) : word sz := natToWord sz (Z.to_nat (FieldToZ x)). - - Definition Fm_dec (x_ : word sz) : option (F m) := - let z := Z.of_nat (wordToNat (x_)) in - if Z_lt_dec z m - then Some (ZToField z) - else None - . - - Lemma bound_check_N : forall x : F m, (N.of_nat (Z.to_nat x) < Npow2 sz)%N. - Proof. - intro. - pose proof (FieldToZ_range x m_pos) as x_range. - rewrite <- Nnat.N2Nat.id. - rewrite Npow2_nat. - apply (Nat2N_inj_lt (Z.to_nat x) (pow2 sz)). - rewrite Zpow_pow2. - destruct x_range as [x_low x_high]. - apply Z2Nat.inj_lt in x_high; try omega. - rewrite <- ZUtil.pow_Z2N_Zpow by omega. - replace (Z.to_nat 2) with 2%nat by auto. - omega. - Qed. - - Lemma Fm_encoding_valid : forall x, Fm_dec (Fm_enc x) = Some x. - Proof. - unfold Fm_dec, Fm_enc; intros. - pose proof (FieldToZ_range x m_pos). - rewrite wordToNat_natToWord_idempotent by apply bound_check_N. - rewrite Z2Nat.id by omega. - rewrite ZToField_idempotent. - break_if; auto; omega. - Qed. - - Instance modular_word_encoding : encoding of F m as word sz := { - enc := Fm_enc; - dec := Fm_dec; - encoding_valid := Fm_encoding_valid - }. -End ModularWordEncoding. +Notation "'encoding' 'of' T 'as' B" := (Encoding T B) (at level 50).
\ No newline at end of file diff --git a/src/Spec/ModularWordEncoding.v b/src/Spec/ModularWordEncoding.v new file mode 100644 index 000000000..9f7e3340b --- /dev/null +++ b/src/Spec/ModularWordEncoding.v @@ -0,0 +1,31 @@ +Require Import Coq.ZArith.ZArith. +Require Import Coq.Numbers.Natural.Peano.NPeano. +Require Import Crypto.ModularArithmetic.PrimeFieldTheorems. +Require Import Bedrock.Word. +Require Import Crypto.Tactics.VerdiTactics. +Require Import Crypto.Util.NatUtil. +Require Import Crypto.Util.WordUtil. +Require Import Crypto.Spec.Encoding. +Require Crypto.Encoding.ModularWordEncodingPre. + +Local Open Scope nat_scope. + +Section ModularWordEncoding. + Context {m : Z} {sz : nat} {m_pos : (0 < m)%Z} {bound_check : Z.to_nat m < 2 ^ sz}. + + Definition Fm_enc (x : F m) : word sz := NToWord sz (Z.to_N (FieldToZ x)). + + Definition Fm_dec (x_ : word sz) : option (F m) := + let z := Z.of_N (wordToN (x_)) in + if Z_lt_dec z m + then Some (ZToField z) + else None + . + + Instance modular_word_encoding : encoding of F m as word sz := { + enc := Fm_enc; + dec := Fm_dec; + encoding_valid := @ModularWordEncodingPre.Fm_encoding_valid m sz m_pos bound_check + }. + +End ModularWordEncoding. diff --git a/src/Spec/PointEncoding.v b/src/Spec/PointEncoding.v index 251e50414..38b4b4224 100644 --- a/src/Spec/PointEncoding.v +++ b/src/Spec/PointEncoding.v @@ -1,196 +1,42 @@ -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.Encoding Crypto.Encoding.EncodingTheorems. -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.NumTheoryUtil. -Require Import Bedrock.Word. -Require Import Crypto.Tactics.VerdiTactics. +Require Coq.ZArith.ZArith Coq.ZArith.Znumtheory. +Require Coq.Numbers.Natural.Peano.NPeano. +Require Crypto.Encoding.EncodingTheorems. +Require Crypto.CompleteEdwardsCurve.CompleteEdwardsCurveTheorems. +Require Crypto.ModularArithmetic.PrimeFieldTheorems. +Require Bedrock.Word. +Require Crypto.Tactics.VerdiTactics. +Require Crypto.Encoding.PointEncodingPre. +Obligation Tactic := eauto using PointEncodingPre.point_encoding_canonical. + +Require Import Crypto.Spec.Encoding Crypto.Spec.ModularArithmetic. +Require Import Crypto.Spec.CompleteEdwardsCurve. Local Open Scope F_scope. Section PointEncoding. - Context {prm:TwistedEdwardsParams} {sz : nat} {FqEncoding : encoding of F q as word sz} {q_5mod8 : q mod 8 = 5} {sqrt_minus1_valid : (@ZToField q 2 ^ Z.to_N (q / 4)) ^ 2 = opp 1}. - Existing Instance prime_q. - - Add Field Ffield : (@Ffield_theory q _) - (morphism (@Fring_morph q), - preprocess [Fpreprocess], - postprocess [Fpostprocess; try exact Fq_1_neq_0; try assumption], - constants [Fconstant], - div (@Fmorph_div_theory q), - power_tac (@Fpower_theory q) [Fexp_tac]). - - Definition sqrt_valid (a : F q) := ((sqrt_mod_q a) ^ 2 = a)%F. - - Lemma solve_sqrt_valid : forall (p : point), - sqrt_valid (solve_for_x2 (snd (proj1_sig p))). - Proof. - intros. - destruct p as [[x y] onCurve_xy]; simpl. - rewrite (solve_correct x y) in onCurve_xy. - rewrite <- onCurve_xy. - unfold sqrt_valid. - eapply sqrt_mod_q_valid; eauto. - unfold isSquare; eauto. - Grab Existential Variables. eauto. - Qed. - - Lemma solve_onCurve: forall (y : F q), sqrt_valid (solve_for_x2 y) -> - onCurve (sqrt_mod_q (solve_for_x2 y), y). - Proof. - intros. - unfold sqrt_valid in *. - apply solve_correct; auto. - Qed. - - Lemma solve_opp_onCurve: forall (y : F q), sqrt_valid (solve_for_x2 y) -> - onCurve (opp (sqrt_mod_q (solve_for_x2 y)), y). - Proof. - intros y sqrt_valid_x2. - unfold sqrt_valid in *. - apply solve_correct. - rewrite <- sqrt_valid_x2 at 2. - ring. - Qed. - -Definition sign_bit (x : F q) := (wordToN (enc (opp x)) <? wordToN (enc x))%N. -Definition point_enc (p : point) : word (S sz) := let '(x,y) := proj1_sig p in - WS (sign_bit x) (enc y). -Definition point_dec_coordinates (w : word (S sz)) : option (F q * F q) := - match dec (wtl w) with - | None => None - | Some y => let x2 := solve_for_x2 y in - let x := sqrt_mod_q x2 in - if F_eq_dec (x ^ 2) x2 - then - let p := (if Bool.eqb (whd w) (sign_bit x) then x else opp x, y) in - Some p - else None - end. - -Definition point_dec (w : word (S sz)) : option point := - match dec (wtl w) with - | None => None - | Some y => let x2 := solve_for_x2 y in - let x := sqrt_mod_q x2 in - match (F_eq_dec (x ^ 2) x2) with - | right _ => None - | left EQ => if Bool.eqb (whd w) (sign_bit x) - then Some (mkPoint (x, y) (solve_onCurve y EQ)) - else Some (mkPoint (opp x, y) (solve_opp_onCurve y EQ)) - end - end. - -Lemma point_dec_coordinates_correct w - : option_map (@proj1_sig _ _) (point_dec w) = point_dec_coordinates w. -Proof. - unfold point_dec, point_dec_coordinates. - edestruct dec; [ | reflexivity ]. - edestruct @F_eq_dec; [ | reflexivity ]. - edestruct @Bool.eqb; reflexivity. -Qed. - -Lemma y_decode : forall p, dec (wtl (point_enc p)) = Some (snd (proj1_sig p)). -Proof. - intros. - destruct p as [[x y] onCurve_p]; simpl. - exact (encoding_valid y). -Qed. - - -Lemma wordToN_enc_neq_opp : forall x, x <> 0 -> (wordToN (enc (opp x)) <> wordToN (enc x))%N. -Proof. - intros x x_nonzero. - intro false_eq. - apply x_nonzero. - apply F_eq_opp_zero; try apply two_lt_q. - apply wordToN_inj in false_eq. - apply encoding_inj in false_eq. - auto. -Qed. - -Lemma sign_bit_opp_negb : forall x, x <> 0 -> negb (sign_bit x) = sign_bit (opp x). -Proof. - intros x x_nonzero. - unfold sign_bit. - rewrite <- N.leb_antisym. - rewrite N.ltb_compare, N.leb_compare. - rewrite F_opp_involutive. - case_eq (wordToN (enc x) ?= wordToN (enc (opp x)))%N; auto. - intro wordToN_enc_eq. - pose proof (wordToN_enc_neq_opp x x_nonzero). - apply N.compare_eq_iff in wordToN_enc_eq. - congruence. -Qed. - -Lemma sign_bit_opp : forall x y, y <> 0 -> - (sign_bit x <> sign_bit y <-> sign_bit x = sign_bit (opp y)). -Proof. - split; intro sign_mismatch; case_eq (sign_bit x); case_eq (sign_bit y); - try congruence; intros y_sign x_sign; rewrite <- sign_bit_opp_negb in * by auto; - rewrite y_sign, x_sign in *; reflexivity || discriminate. -Qed. - -Lemma sign_bit_squares : forall x y, y <> 0 -> x ^ 2 = y ^ 2 -> - sign_bit x = sign_bit y -> x = y. -Proof. - intros ? ? y_nonzero squares_eq sign_match. - destruct (sqrt_solutions _ _ squares_eq) as [? | eq_opp]; auto. - assert (sign_bit x = sign_bit (opp y)) as sign_mismatch by (f_equal; auto). - apply sign_bit_opp in sign_mismatch; auto. - congruence. -Qed. - -Lemma sign_bit_match : forall x x' y : F q, onCurve (x, y) -> onCurve (x', y) -> - sign_bit x = sign_bit x' -> x = x'. -Proof. - intros ? ? ? onCurve_x onCurve_x' sign_match. - apply solve_correct in onCurve_x. - apply solve_correct in onCurve_x'. - destruct (F_eq_dec x' 0). - + subst. - rewrite Fq_pow_zero in onCurve_x' by congruence. - rewrite <- onCurve_x' in *. - eapply Fq_root_zero; eauto. - + apply sign_bit_squares; auto. - rewrite onCurve_x, onCurve_x'. - reflexivity. -Qed. - -Lemma point_encoding_valid : forall p, point_dec (point_enc p) = Some p. -Proof. - intros. - unfold point_dec. - rewrite y_decode. - pose proof solve_sqrt_valid p as solve_sqrt_valid_p. - unfold sqrt_valid in *. - destruct p as [[x y] onCurve_p]. - simpl in *. - destruct (F_eq_dec ((sqrt_mod_q (solve_for_x2 y)) ^ 2) (solve_for_x2 y)); intuition. - break_if; f_equal; apply point_eq. - + rewrite Bool.eqb_true_iff in Heqb. - pose proof (solve_onCurve y solve_sqrt_valid_p). - f_equal. - apply (sign_bit_match _ _ y); auto. - + rewrite Bool.eqb_false_iff in Heqb. - pose proof (solve_opp_onCurve y solve_sqrt_valid_p). - f_equal. - apply sign_bit_opp in Heqb. - apply (sign_bit_match _ _ y); auto. - intro eq_zero. - apply solve_correct in onCurve_p. - rewrite eq_zero in *. - rewrite Fq_pow_zero in solve_sqrt_valid_p by congruence. - rewrite <- solve_sqrt_valid_p in onCurve_p. - apply Fq_root_zero in onCurve_p. - rewrite onCurve_p in Heqb; auto. -Qed. - -Instance point_encoding : encoding of point as (word (S sz)) := { - enc := point_enc; - dec := point_dec; - encoding_valid := point_encoding_valid -}. + Context {prm: CompleteEdwardsCurve.TwistedEdwardsParams} {sz : nat} + {FqEncoding : encoding of ModularArithmetic.F (CompleteEdwardsCurve.q) as Word.word sz}. + + Definition sign_bit (x : F CompleteEdwardsCurve.q) := + match (enc x) with + | Word.WO => false + | Word.WS b _ w' => b + end. + + Definition point_enc (p : CompleteEdwardsCurve.point) : Word.word (S sz) := let '(x,y) := proj1_sig p in + Word.WS (sign_bit x) (enc y). + + Program Definition point_dec_with_spec : + {point_dec : Word.word (S sz) -> option CompleteEdwardsCurve.point + | forall w x, point_dec w = Some x -> (point_enc x = w) + } := PointEncodingPre.point_dec. + + Definition point_dec := Eval hnf in (proj1_sig point_dec_with_spec). + + Instance point_encoding : encoding of CompleteEdwardsCurve.point as (Word.word (S sz)) := { + enc := point_enc; + dec := point_dec; + encoding_valid := PointEncodingPre.point_encoding_valid + }. End PointEncoding. |