From b054c75690ab33b71d4a6abf57715c573f924aec Mon Sep 17 00:00:00 2001
From: Jade Philipoom <jadep@mit.edu>
Date: Tue, 5 Jan 2016 18:22:29 -0500
Subject: Util: added util lemmas needed to instantiate EdDSA25519.

---
 src/Util/NumTheoryUtil.v | 81 ++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 81 insertions(+)
 create mode 100644 src/Util/NumTheoryUtil.v

(limited to 'src/Util/NumTheoryUtil.v')

diff --git a/src/Util/NumTheoryUtil.v b/src/Util/NumTheoryUtil.v
new file mode 100644
index 000000000..4a54e5437
--- /dev/null
+++ b/src/Util/NumTheoryUtil.v
@@ -0,0 +1,81 @@
+Require Import Zpower Znumtheory ZArith.ZArith ZArith.Zdiv.
+Require Import Omega NPeano Arith.
+Require Import Crypto.Util.NatUtil Crypto.Util.ZUtil.
+Local Open Scope Z.
+
+Lemma euler_criterion : forall (a x p : Z) (prime_p : prime p),
+  (x * 2 + 1 = p)%Z -> (a ^ x mod p = 1)%Z ->
+    exists b : Z, (0 <= b < p /\ b * b mod p = a)%Z.
+Admitted.
+
+Lemma divide2_1mod4 : forall x, x mod 4 = 1 -> 0 <= x -> (2 | x / 2).
+Proof.
+  intros.
+  assert (Z.to_nat x mod 4 = 1)%nat. {
+    replace 1%Z with (Z.of_nat 1) in H by auto.
+    replace (x mod 4)%Z with (Z.of_nat (Z.to_nat x mod 4)) in H
+      by (rewrite mod_Zmod by omega; rewrite Z2Nat.id; auto).
+    apply Nat2Z.inj in H; auto.
+  }
+  remember (Z.to_nat x / 4)%nat as c.
+  pose proof (divide2_1mod4_nat c (Z.to_nat x) Heqc H1).
+  destruct H2.
+  replace 2%nat with (Z.to_nat 2) in H2 by auto.
+  apply inj_eq in H2.
+  rewrite div_Zdiv in H2 by (replace (Z.to_nat 2) with 2%nat by auto; omega).
+  rewrite Nat2Z.inj_mul in H2.
+  do 2 rewrite Z2Nat.id in H2 by (auto || omega).
+  rewrite Z.mul_comm in H2.
+  symmetry in H2.
+  apply Zdivide_intro in H2; auto.
+Qed.
+
+Lemma minus1_even_pow : forall x y, (2 | y) -> (1 < x) -> (0 <= y) -> ((x - 1) ^ y mod x = 1).
+Proof.
+  intros.
+  apply Zdivide_Zdiv_eq in H; try omega.
+  rewrite H.
+  rewrite Z.pow_mul_r by omega.
+  assert ((x - 1)^2 mod x = 1). {
+    replace ((x - 1)^2) with (x ^ 2 - 2 * x + 1)
+      by (do 2 rewrite Z.pow_2_r; rewrite Z.mul_sub_distr_l; do 2 rewrite Z.mul_sub_distr_r; omega).
+    rewrite Zplus_mod.
+    rewrite Z.pow_2_r.
+    rewrite <- Z.mul_sub_distr_r.
+    rewrite Z_mod_mult.
+    do 2 rewrite Zmod_1_l by auto; auto.
+  }
+  rewrite <- (Z2Nat.id (y / 2)) by omega.
+  induction (Z.to_nat (y / 2)); try apply Zmod_1_l; auto.
+  rewrite Nat2Z.inj_succ.
+  rewrite Z.pow_succ_r by apply Zle_0_nat.
+  rewrite Zmult_mod.
+  rewrite IHn.
+  rewrite H2.
+  simpl.
+  apply Zmod_1_l; auto.
+Qed.
+
+Lemma minus1_square_1mod4 : forall (p : Z) (prime_p : prime p),
+  (p mod 4 = 1)%Z -> exists b : Z, (0 <= b < p /\ b * b mod p = p - 1)%Z.
+Proof.
+  intros.
+  assert (4 <> 0)%Z by omega.
+  pose proof (Z.div_mod p 4 H0).
+  pose proof (prime_ge_2 p (prime_p)).
+  assert (2 | p / 2)%Z by (apply divide2_1mod4; (auto || omega)).
+  apply (euler_criterion (p - 1) (p / 2) p prime_p);
+    [ | apply minus1_even_pow; (omega || auto); apply Z_div_pos; omega].
+  rewrite <- H.
+  rewrite H1 at 3.
+  f_equal.
+  replace 4%Z with (2 * 2)%Z by auto.
+  rewrite <- Z.div_div by omega.
+  rewrite <- Zdivide_Zdiv_eq_2 by (auto || omega).
+  replace (2 * 2 * (p / 2) / 2)%Z with (2 * (2 * (p / 2)) / 2)%Z
+    by (f_equal; apply Z.mul_assoc).
+  rewrite ZUtil.Z_div_mul' by omega.
+  rewrite Z.mul_comm.
+  auto.
+Qed.
+
-- 
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