From c29fa1ad5e2609575c29c217c341690205f0b5b0 Mon Sep 17 00:00:00 2001
From: Jason Gross <jasongross9@gmail.com>
Date: Thu, 4 Aug 2016 11:44:34 -0700
Subject: [PATCH] Implement Barrett Reduction following HAC 14.42 (#45)

From http://cacr.uwaterloo.ca/hac/about/chap14.pdf

This should take care of most of #43, at least the specification on Z
part of it.
---
 _CoqProject                                   |   1 +
 .../BarrettReduction/ZHandbook.v              | 158 ++++++++++++++++++
 2 files changed, 159 insertions(+)
 create mode 100644 src/ModularArithmetic/BarrettReduction/ZHandbook.v

diff --git a/_CoqProject b/_CoqProject
index 7a2de03e07..5d44d85f6c 100644
--- a/_CoqProject
+++ b/_CoqProject
@@ -52,6 +52,7 @@ src/ModularArithmetic/PseudoMersenneBaseParams.v
 src/ModularArithmetic/Tutorial.v
 src/ModularArithmetic/BarrettReduction/Z.v
 src/ModularArithmetic/BarrettReduction/ZGeneralized.v
+src/ModularArithmetic/BarrettReduction/ZHandbook.v
 src/Spec/CompleteEdwardsCurve.v
 src/Spec/EdDSA.v
 src/Spec/Encoding.v
diff --git a/src/ModularArithmetic/BarrettReduction/ZHandbook.v b/src/ModularArithmetic/BarrettReduction/ZHandbook.v
new file mode 100644
index 0000000000..3fc1214ebc
--- /dev/null
+++ b/src/ModularArithmetic/BarrettReduction/ZHandbook.v
@@ -0,0 +1,158 @@
+(*** Barrett Reduction *)
+(** This file implements a slightly-generalized version of Barrett
+    Reduction on [Z].  This version follows the Handbook of Applied
+    Cryptography (Algorithm 14.42) rather closely; the only deviations
+    are that we generalize from [k ± 1] to [k ± offset] for an
+    arbitrary offset, and we weaken the conditions on the base [b] in
+    [bᵏ] slightly.  Contrasted with some other versions, this version
+    does reduction modulo [b^(k+offset)] early (ensuring that we don't
+    have to carry around extra precision), but requires more stringint
+    conditions on the base ([b]), exponent ([k]), and the [offset]. *)
+Require Import Coq.ZArith.ZArith Coq.micromega.Psatz.
+Require Import Crypto.Util.ZUtil Crypto.Util.Tactics Crypto.Algebra.
+
+Local Open Scope Z_scope.
+
+Section barrett.
+  (** Quoting the Handbook of Applied Cryptography <http://cacr.uwaterloo.ca/hac/about/chap14.pdf>: *)
+  (** Barrett reduction (Algorithm 14.42) computes [r = x mod m] given
+      [x] and [m]. The algorithm requires the precomputation of the
+      quantity [µ = ⌊b²ᵏ/m⌋]; it is advantageous if many reductions
+      are performed with a single modulus. For example, each RSA
+      encryption for one entity requires reduction modulo that
+      entity’s public key modulus. The precomputation takes a fixed
+      amount of work, which is negligible in comparison to modular
+      exponentiation cost.  Typically, the radix [b] is chosen to be
+      close to the word-size of the processor. Hence, assume [b > 3] in
+      Algorithm 14.42 (see Note 14.44 (ii)). *)
+
+  (** * Barrett modular reduction *)
+  Section barrett_modular_reduction.
+    Context (m b x k μ offset : Z)
+            (m_pos : 0 < m)
+            (base_pos : 0 < b)
+            (k_good : m < b^k)
+            (μ_good : μ = b^(2*k) / m) (* [/] is [Z.div], which is truncated *)
+            (x_nonneg : 0 <= x)
+            (offset_nonneg : 0 <= offset)
+            (k_big_enough : offset <= k)
+            (x_small : x < b^(2*k))
+            (m_small : 3 * m <= b^(k+offset))
+            (** We also need that [m] is large enough; [m] larger than
+                [bᵏ⁻¹] works, but we ask for something more precise. *)
+            (m_large : x mod b^(k-offset) <= m).
+
+    Let q1 := x / b^(k-offset). Let q2 := q1 * μ. Let q3 := q2 / b^(k+offset).
+    Let r1 := x mod b^(k+offset). Let r2 := (q3 * m) mod b^(k+offset).
+    (** At this point, the HAC says "If [r < 0] then [r ← r + bᵏ⁺¹]".
+        This is equivalent to reduction modulo [b^(k+offset)], as we
+        prove below.  The version involving modular reduction has the
+        benefit of being cheaper to implement, and making the proofs
+        simpler, so we primarily use that version. *)
+    Let r_mod_3m      := (r1 - r2) mod b^(k+offset).
+    Let r_mod_3m_orig := let r := r1 - r2 in
+                         if r <? 0 then r + b^(k+offset) else r.
+
+    Lemma r_mod_3m_eq_orig : r_mod_3m = r_mod_3m_orig.
+    Proof.
+      assert (0 <= r1 < b^(k+offset)) by (subst r1; auto with zarith).
+      assert (0 <= r2 < b^(k+offset)) by (subst r2; auto with zarith).
+      subst r_mod_3m r_mod_3m_orig; cbv zeta.
+      break_match; Z.ltb_to_lt.
+      { symmetry; apply (Zmod_unique (r1 - r2) _ (-1)); lia. }
+      { symmetry; apply (Zmod_unique (r1 - r2) _ 0); lia. }
+    Qed.
+
+    (** 14.43 Fact By the division algorithm (Definition 2.82), there
+        exist integers [Q] and [R] such that [x = Qm + R] and [0 ≤ R <
+        m]. In step 1 of Algorithm 14.42 (Barrett modular reduction),
+        the following inequality is satisfied: [Q - 2 ≤ q₃ ≤ Q]. *)
+    (** We prove this by providing a more useful form for [q₃]. *)
+    Let Q := x / m.
+    Let R := x mod m.
+    Lemma q3_nice : { b : bool * bool | q3 = Q + (if fst b then -1 else 0) + (if snd b then -1 else 0) }.
+    Proof.
+      assert (0 < b^(k+offset)) by zero_bounds.
+      assert (0 < b^(k-offset)) by zero_bounds.
+      assert (x / b^(k-offset) <= b^(2*k) / b^(k-offset)) by auto with zarith lia.
+      assert (x / b^(k-offset) <= b^(k+offset)) by (autorewrite with pull_Zpow zsimplify in *; assumption).
+      subst q1 q2 q3 Q r_mod_3m r_mod_3m_orig r1 r2 R μ.
+      rewrite (Z.div_mul_diff_exact' (b^(2*k)) m (x/b^(k-offset))) by auto with lia zero_bounds.
+      rewrite (Z_div_mod_eq (_ * b^(2*k) / m) (b^(k+offset))) by lia.
+      autorewrite with push_Zmul push_Zopp zsimplify zstrip_div zdiv_to_mod.
+      rewrite Z.div_sub_mod_cond, !Z.div_sub_small; auto with zero_bounds zarith.
+      eexists (_, _); reflexivity.
+    Qed.
+
+    Fact q3_in_range : Q - 2 <= q3 <= Q.
+    Proof.
+      rewrite (proj2_sig q3_nice).
+      break_match; lia.
+    Qed.
+
+    (** 14.44 Note (partial justification of correctness of Barrett reduction) *)
+    (** (i) Algorithm 14.42 is based on the observation that [⌊x/m⌋]
+            can be written as [Q =
+            ⌊(x/bᵏ⁻¹)(b²ᵏ/m)(1/bᵏ⁺¹)⌋]. Moreover, [Q] can be
+            approximated by the quantity [q₃ = ⌊⌊x/bᵏ⁻¹⌋µ/bᵏ⁺¹⌋].
+            Fact 14.43 guarantees that [q₃] is never larger than the
+            true quotient [Q], and is at most 2 smaller. *)
+    Lemma x_minus_q3_m_in_range : 0 <= x - q3 * m < 3 * m.
+    Proof.
+      pose proof q3_in_range.
+      assert (0 <= R < m) by (subst R; auto with zarith).
+      assert (0 <= (Q - q3) * m + R < 3 * m) by nia.
+      subst Q R; autorewrite with push_Zmul zdiv_to_mod in *; lia.
+    Qed.
+
+    Lemma r_mod_3m_eq_alt : r_mod_3m = x - q3 * m.
+    Proof.
+      pose proof x_minus_q3_m_in_range.
+      subst r_mod_3m r_mod_3m_orig r1 r2.
+      autorewrite with pull_Zmod zsimplify; reflexivity.
+    Qed.
+
+    (** This version uses reduction modulo [b^(k+offset)]. *)
+    Theorem barrett_reduction_equivalent
+      : r_mod_3m mod m = x mod m.
+    Proof.
+      rewrite r_mod_3m_eq_alt.
+      autorewrite with zsimplify push_Zmod; reflexivity.
+    Qed.
+
+    (** This version, which matches the original in the HAC, uses
+        conditional addition of [b^(k+offset)]. *)
+    Theorem barrett_reduction_orig_equivalent
+      : r_mod_3m_orig mod m = x mod m.
+    Proof. rewrite <- r_mod_3m_eq_orig; apply barrett_reduction_equivalent. Qed.
+
+    Lemma r_small : 0 <= r_mod_3m < 3 * m.
+    Proof.
+      pose proof x_minus_q3_m_in_range.
+      subst Q R r_mod_3m r_mod_3m_orig r1 r2.
+      autorewrite with pull_Zmod zsimplify; lia.
+    Qed.
+
+
+    (** This version uses reduction modulo [b^(k+offset)]. *)
+    Theorem barrett_reduction_small (r := r_mod_3m)
+      : x mod m = let r := if r <? m then r else r-m in
+                  let r := if r <? m then r else r-m in
+                  r.
+    Proof.
+      pose proof r_small. cbv zeta.
+      destruct (r <? m) eqn:Hr, (r-m <? m) eqn:?; subst r; rewrite !r_mod_3m_eq_alt, ?Hr in *; Z.ltb_to_lt; try lia.
+      { symmetry; eapply (Zmod_unique x m q3); lia. }
+      { symmetry; eapply (Zmod_unique x m (q3 + 1)); lia. }
+      { symmetry; eapply (Zmod_unique x m (q3 + 2)); lia. }
+    Qed.
+
+    (** This version, which matches the original in the HAC, uses
+        conditional addition of [b^(k+offset)]. *)
+    Theorem barrett_reduction_small_orig (r := r_mod_3m_orig)
+      : x mod m = let r := if r <? m then r else r-m in
+                  let r := if r <? m then r else r-m in
+                  r.
+    Proof. subst r; rewrite <- r_mod_3m_eq_orig; apply barrett_reduction_small. Qed.
+  End barrett_modular_reduction.
+End barrett.