Adds theories/algebra/ZqCentered.ec providing the centered (signed) integer representation of Z/pZ elements.

Author: mbbarbosa

theories/algebra/ZqCentered.ec

@@ -0,0 +1,300 @@
+(* ==================================================================== *)
+require import AllCore IntDiv.
+require import StdOrder.
+(*---*) import IntOrder.
+require ZModP.
+
+(* ==================================================================== *)
+(* Centered (signed) representation of elements of the ring Z/pZ.       *)
+(*                                                                      *)
+(* For p >= 2, the centered representative `crepr : zmod -> int' maps   *)
+(* every element to the unique integer in the half-open centered        *)
+(* interval [(p+1)/2 - p, (p+1)/2). For odd primes p > 2 this interval  *)
+(* is exactly {-(p-1)/2, ..., (p-1)/2}, the standard "balanced"         *)
+(* representation used in lattice-based cryptography.                   *)
+(*                                                                      *)
+(* Provides:                                                            *)
+(*   - `to_crepr', `crepr' (centered representative on int and Zq)      *)
+(*   - range, round-trip, equality lemmas                               *)
+(*   - algebraic laws under +, *                                        *)
+(*   - centered absolute value `"`|_|"' with triangle / product /       *)
+(*     monotonicity inequalities                                        *)
+(*                                                                      *)
+(* The sub-theory `ZqCenteredField' adds prime-only lemmas:             *)
+(*   - `creprN', `creprN2', `creprND' (negation symmetry)               *)
+(*   - `abs_zpN'                                                        *)
+(* ==================================================================== *)
+
+(* ==================================================================== *)
+abstract theory ZqCentered.
+
+clone include ZModP.ZModRing.
+
+(* -------------------------------------------------------------------- *)
+(* Auxiliary congruence lemmas. *)
+
+lemma zmodcgr_ceq a b :
+     (p + 1) %/ 2 - p <= a < (p + 1) %/ 2
+  => (p + 1) %/ 2 - p <= b < (p + 1) %/ 2
+  => zmodcgr a b
+  => a = b
+  by case (0 <= a); case (0 <= b); smt().
+
+lemma zmodcgr_transitive b a c :
+  zmodcgr a b => zmodcgr b c => zmodcgr a c.
+proof. smt(). qed.
+
+lemma zmodcgr_mp a : zmodcgr a (a - p) by smt().
+
+lemma zmodcgr_refl a b : zmodcgr a b <=> zmodcgr b a by smt().
+
+(* -------------------------------------------------------------------- *)
+(* Centered representative on int.                                      *)
+(*                                                                      *)
+(* "Halfway" is rounded down: when 2 %| p the unique value with         *)
+(* asint = p/2 maps to -p/2.  For odd p the choice is irrelevant.       *)
+
+op to_crepr (x : int) : int =
+  let y = x %% p in
+  if (p + 1) %/ 2 <= y then y - p else y.
+
+lemma to_crepr_idempotent : to_crepr \o to_crepr = to_crepr.
+proof. by rewrite fun_ext /(\o) => x; smt(ge2_p). qed.
+
+lemma rg_to_crepr x :
+  (p + 1) %/ 2 - p <= to_crepr x < (p + 1) %/ 2.
+proof. smt(ge2_p rg_asint). qed.
+
+lemma to_crepr_zmodcgr x : zmodcgr x (to_crepr x).
+proof. smt(ge2_p). qed.
+
+lemma to_crepr_id x :
+  (p + 1) %/ 2 - p <= x < (p + 1) %/ 2 => to_crepr x = x.
+proof. smt(). qed.
+
+lemma to_crepr_eq x y : to_crepr x = to_crepr y <=> zmodcgr x y.
+proof. rewrite /to_crepr /= /#. qed.
+
+(* -------------------------------------------------------------------- *)
+(* Centered representative on Zq.                                       *)
+
+op crepr (x : zmod) : int =
+  let y = asint x in
+  if (p + 1) %/ 2 <= y then y - p else y.
+
+lemma rg_crepr x :
+  (p + 1) %/ 2 - p <= crepr x < (p + 1) %/ 2.
+proof. smt(ge2_p rg_asint). qed.
+
+lemma crepr_eq x y : crepr x = crepr y <=> x = y.
+proof. smt(rg_asint asint_eq). qed.
+
+lemma asint_crepr x : asint x = (crepr x) %% p.
+proof.
+rewrite /crepr /=.
+case ((p + 1) %/ 2 <= asint x) => _.
+- by rewrite -modzDr; smt(modz_small rg_asint).
+- by smt(modz_small rg_asint).
+qed.
+
+lemma creprK (x : zmod) : inzmod (crepr x) = x.
+proof.
+rewrite /crepr /=.
+case ((p + 1) %/ 2 <= asint x) => ?; last by exact asintK.
+rewrite inzmod_mod.
+suff : (asint x - p) %% p = asint x by smt(asintK).
+smt(rg_asint).
+qed.
+
+lemma inzmodK_centered x :
+     (p + 1) %/ 2 - p <= x
+  => x < (p + 1) %/ 2
+  => crepr (inzmod x) = x.
+proof. by rewrite /crepr inzmodK /#. qed.
+
+(* -------------------------------------------------------------------- *)
+(* Algebraic laws under multiplication and addition.                    *)
+
+lemma crepr_mulE x y :
+  crepr (x * y) = to_crepr (crepr x * crepr y).
+proof.
+suff : zmodcgr (crepr (x * y)) (crepr x * crepr y).
++ move => *.
+  have ? := to_crepr_zmodcgr (crepr x * crepr y).
+  by have : zmodcgr (crepr (x * y)) (to_crepr (crepr x * crepr y));
+     smt(to_crepr_zmodcgr rg_to_crepr rg_crepr zmodcgr_ceq).
+rewrite /crepr /=.
+case ((p + 1) %/ 2 <= asint (x * y)) => ?.
+- case ((p + 1) %/ 2 <= asint x) => ?.
+  + case ((p + 1) %/ 2 <= asint y) => ?.
+    + rewrite -modzMm -(zmodcgr_mp (asint x)) -(zmodcgr_mp (asint y)).
+      by rewrite -(zmodcgr_mp (asint (x * y))) mulE -modzMm modz_mod.
+    + rewrite -modzMm -(zmodcgr_mp (asint x)) -(zmodcgr_mp (asint (x * y))).
+      by rewrite mulE -modzMm modz_mod.
+  + case ((p + 1) %/ 2 <= asint y) => ?.
+    + rewrite -modzMm -(zmodcgr_mp (asint y)).
+      by rewrite -(zmodcgr_mp (asint (x * y))) mulE -modzMm modz_mod.
+    + rewrite -modzMm -(zmodcgr_mp (asint (x * y))).
+      by rewrite mulE -modzMm modz_mod.
+- case ((p + 1) %/ 2 <= asint x) => ?.
+  + case ((p + 1) %/ 2 <= asint y) => ?.
+    + rewrite -modzMm -(zmodcgr_mp (asint x)) -(zmodcgr_mp (asint y)).
+      by rewrite mulE -modzMm modz_mod.
+    + rewrite -modzMm -(zmodcgr_mp (asint x)).
+      by rewrite mulE -modzMm modz_mod.
+  + case ((p + 1) %/ 2 <= asint y) => ?.
+    + rewrite -modzMm -(zmodcgr_mp (asint y)).
+      by rewrite mulE -modzMm modz_mod.
+    + rewrite -modzMm.
+      by rewrite mulE -modzMm modz_mod.
+qed.
+
+lemma creprD (x y : zmod) :
+  crepr (x + y) = to_crepr (crepr x + crepr y).
+proof.
+suff : zmodcgr (crepr (x + y)) (crepr x + crepr y).
++ move => *.
+  have ? := to_crepr_zmodcgr (crepr x + crepr y).
+  by have : zmodcgr (crepr (x + y)) (to_crepr (crepr x + crepr y));
+     smt(to_crepr_zmodcgr rg_to_crepr rg_crepr zmodcgr_ceq).
+rewrite /crepr addE /=.
+apply (zmodcgr_transitive (asint x + asint y)).
+- case ((p + 1) %/ 2 <= (asint x + asint y) %% p) => _.
+  + have -> : (asint x + asint y) %% p - p
+            = (asint x + asint y) %% p + (-1) * p by ring.
+    by rewrite modzMDr modz_mod.
+  + by rewrite modz_mod.
+case ((p + 1) %/ 2 <= asint x) => _.
+- case ((p + 1) %/ 2 <= asint y) => _.
+  + have -> : asint x - p + (asint y - p)
+            = asint x + asint y + (-2) * p by ring.
+    by rewrite modzMDr.
+  + have -> : asint x - p + asint y
+            = asint x + asint y + (-1) * p by ring.
+    by rewrite modzMDr.
+- case ((p + 1) %/ 2 <= asint y) => _ //.
+  have -> : asint x + (asint y - p)
+          = asint x + asint y + (-1) * p by ring.
+  by rewrite modzMDr.
+qed.
+
+(* -------------------------------------------------------------------- *)
+(* Centered absolute value.                                             *)
+
+op "`|_|" (x : zmod) : int = `| crepr x |.
+
+lemma to_crepr_abs x : `|to_crepr x| <= `|x|.
+proof. smt(rg_to_crepr). qed.
+
+lemma abs_zp_small (x : int) :
+  `|x| < p %/ 2 => `|inzmod x| = `|x|.
+proof. by move=> ?; rewrite /"`|_|"; smt(ltr_norml inzmodK). qed.
+
+lemma abs_zp_triangle (x y : zmod) :
+  `|x + y| <= `|x| + `|y|.
+proof.
+rewrite /"`|_|" creprD.
+have := to_crepr_abs (crepr x + crepr y).
+smt().
+qed.
+
+lemma abs_zp_zero : `|zero| = 0.
+proof. by rewrite /"`|_|" /crepr zeroE /=; smt(ge2_p). qed.
+
+lemma abs_zp_one : `|one| = 1.
+proof. by rewrite /"`|_|" /crepr /=; rewrite oneE; smt(ge2_p). qed.
+
+lemma ge0_abs_zp (x : zmod) : 0 <= `|x|.
+proof. smt(). qed.
+
+lemma abs_zp_ub (x : zmod) : `|x| <= p %/ 2.
+proof. smt(rg_asint). qed.
+
+lemma abs_zp_prod (x y : zmod) :
+  `|x * y| <= `|x| * `|y|.
+proof. by rewrite /"`|_|" crepr_mulE; smt(to_crepr_abs). qed.
+
+lemma abs_zpN (x : zmod) : `|x| = `|-x|.
+proof.
+rewrite /"`|_|" /crepr /=.
+case (x = zero) => ?; first by smt(ZModpRing.oppr0).
+have ? : asint x <> 0 by have := asint_inj x zero; smt(zeroE).
+suff : asint (-x) = p - asint x by smt().
+rewrite oppE -modzDl.
+suff : (p - asint x) %% p = ((p - asint x) + p) %% p by smt(rg_asint).
+by smt(modzDr).
+qed.
+
+end ZqCentered.
+
+(* ==================================================================== *)
+(* Field version: when p is prime, additionally provide negation        *)
+(* symmetry for the centered representative.                            *)
+(* ==================================================================== *)
+abstract theory ZqCenteredField.
+
+const p : { int | prime p } as prime_p.
+
+clone include ZqCentered with
+  op p <- p
+  proof ge2_p by smt(gt1_prime prime_p).
+
+(* prime_p is now in scope; ZqCentered's content is available. *)
+
+lemma modz_small2 a :
+  p <= a < 2 * p => a %% p = a - p
+  by smt().
+
+lemma zmodcgrN a :
+  a <> zero =>
+  (- asint a) %% p = - asint a %% p + p.
+proof.
+move => ?; have ? := rg_asint a.
+have ? : asint a <> 0 by have := asint_inj a zero; smt(zeroE).
+rewrite modNz; 1,2: by smt(prime_p).
+rewrite -modzDm modNz //=; 1: smt(prime_p).
+suff : (asint a %% p + (p - 1)) %% p = (asint a %% p + (p - 1)) - p by smt().
+rewrite (modz_small (asint a) p) 1:/# /=.
+by rewrite modz_small2 /#.
+qed.
+
+lemma creprN (a : zmod) :
+  p <> 2 => crepr (-a) = -crepr a.
+proof.
+have ? := rg_asint a.
+rewrite /crepr !oppE; case (a = zero).
++ by move => -> /=; rewrite zeroE /=; smt(prime_p).
+move => ? /=; rewrite !zmodcgrN //.
+have -> : (- asint a %% p) + p = p - asint a
+  by smt(rg_asint modz_small edivzP).
+have ? : asint a <> 0 by have := asint_inj a zero; smt(zeroE).
+case ((p + 1) %/ 2 < asint a); 1: by smt().
+case (asint a < (p + 1) %/ 2); 1: by smt().
+move => *; have -> : asint a = (p + 1) %/ 2 by smt(rg_asint).
+have ? : p + 1 = (p + 1) %/ 2 + (p + 1) %/ 2; last by smt().
+have ? : p %% 2 <> 0.
++ have := prime_p.
+  rewrite /prime => [#? Hmod].
+  by have := Hmod 2 => /= /#.
+  smt().
+qed.
+
+lemma creprN2 (a : zmod) :
+  p = 2 => crepr (-a) = crepr a.
+proof.
+move => Hp.
+have ? := rg_asint a.
+rewrite /crepr !oppE; case (a = zero).
++ by move => -> /=; rewrite zeroE /=; smt(prime_p).
+move => ? /=; rewrite !zmodcgrN /#.
+qed.
+
+lemma creprND (a b : zmod) :
+  crepr (a - b) = to_crepr (crepr a - crepr b).
+proof.
+have ? := rg_asint a; have ? := rg_asint b.
+case (p = 2); 2: by smt(creprD creprN).
+by smt(creprD creprN2).
+qed.
+
+end ZqCenteredField.