Counting interface for the indifferentiability arc

CompElliptic/CurveForms/ShortWeierstrass.lean

@@ -60,6 +60,18 @@ theorem not_onCurve_zero {a b : F} (hb : b ≠ 0) : ¬ OnCurve a b (0, 0) := by
   have h' : (0 : F) ^ 2 = (0 : F) ^ 3 + a * 0 + b := h
   simpa using h'.symm
 
+omit [DecidableEq F] in
+/-- Two points on the curve sharing an `x`-coordinate have `y`-coordinates equal up to sign,
+because their curve equations subtract to `(y₁ - y₂)·(y₁ + y₂) = 0`. -/
+theorem y_eq_pm_of_onCurve_x_eq {a b x y₁ y₂ : F}
+    (h₁ : OnCurve a b (x, y₁)) (h₂ : OnCurve a b (x, y₂)) : y₁ = y₂ ∨ y₁ = -y₂ := by
+  have h : (y₁ - y₂) * (y₁ + y₂) = 0 := by
+    simp only [OnCurve] at h₁ h₂
+    linear_combination h₁ - h₂
+  rcases mul_eq_zero.mp h with h | h
+  · exact Or.inl (sub_eq_zero.mp h)
+  · exact Or.inr (by linear_combination h)
+
 /-- Negation `(x, y) ↦ (x, -y)`; fixes the `(0, 0)` sentinel. -/
 def neg (p : F × F) : F × F := (p.1, -p.2)
 
@@ -426,6 +438,36 @@ omit [DecidableEq F] in
 /-- Negation negates the `y`-coordinate — the one fact that makes `2 • P = 0` say `P.y = -P.y`. -/
 @[simp] theorem SWPoint.neg_y {E : SWCurve F} (P : SWPoint E) : (-P).y = -P.y := rfl
 
+/-- Addition computes its `x`-coordinate by the raw `add` on the coordinate pairs. -/
+lemma SWPoint.add_x {E : SWCurve F} (P Q : SWPoint E) :
+    (P + Q).x = (add E.A (P.x, P.y) (Q.x, Q.y)).1 := rfl
+
+/-- Addition computes its `y`-coordinate by the raw `add` on the coordinate pairs. -/
+lemma SWPoint.add_y {E : SWCurve F} (P Q : SWPoint E) :
+    (P + Q).y = (add E.A (P.x, P.y) (Q.x, Q.y)).2 := rfl
+
+omit [DecidableEq F] in
+/-- A nonzero representable point is on the curve: its `Valid` disjunction cannot be the `(0, 0)`
+sentinel. -/
+theorem SWPoint.onCurve_of_ne_zero {E : SWCurve F} {P : SWPoint E} (h : P ≠ 0) :
+    OnCurve E.A E.B (P.x, P.y) := by
+  rcases P.onCurve with hc | h0
+  · exact hc
+  · exact absurd (SWPoint.ext_pair (by rw [h0]; rfl)) h
+
+omit [DecidableEq F] in
+/-- Nonzero representable points sharing an `x`-coordinate are equal up to sign: both are on the
+curve (`onCurve_of_ne_zero`), so their `y`-coordinates agree up to sign
+(`y_eq_pm_of_onCurve_x_eq`). -/
+theorem SWPoint.eq_pm_of_x_eq {E : SWCurve F} {P Q : SWPoint E}
+    (hP : P ≠ 0) (hQ : Q ≠ 0) (hx : P.x = Q.x) : P = Q ∨ P = -Q := by
+  have hPC := onCurve_of_ne_zero hP
+  have hQC := onCurve_of_ne_zero hQ
+  rw [hx] at hPC
+  rcases y_eq_pm_of_onCurve_x_eq hPC hQC with hy | hy
+  · exact Or.inl (ext_pair (by rw [hx, hy]))
+  · exact Or.inr (ext_pair (by rw [hx, hy]; rfl))
+
 /-! ### Fast (logarithmic) scalar multiplication
 
 The spec-level `smul` is linear (`n` additions), so it cannot be evaluated by `decide` or

CompElliptic/Fields/Pasta.lean

@@ -166,6 +166,11 @@ abbrev VestaBaseField := PallasScalarField
 /-- Vesta scalar field = Pallas base field. -/
 abbrev VestaScalarField := PallasBaseField
 
+/-- The Pallas base field (= the Vesta scalar field), under its `pasta_curves` letter. -/
+abbrev Fp := PallasBaseField
+/-- The Pallas scalar field (= the Vesta base field), under its `pasta_curves` letter. -/
+abbrev Fq := PallasScalarField
+
 /-- Tonelli–Shanks data for the Pallas base field `𝔽ₚ`: `p-1 = 2^32 · T`, with `rootOfUnity = 5ᵀ`
 (`pallas.py`). -/
 def pallasBase : TonelliShanks PallasBaseField where

CompElliptic/Hashing/FibreBound.lean

@@ -0,0 +1,221 @@
+/-
+Copyright (c) 2026 CompElliptic Contributors.
+Released under the Apache License, Version 2.0, or the MIT license, at your option,
+as described in the files LICENSE-APACHE and LICENSE-MIT.
+Authors: Daira-Emma Hopwood
+-/
+import CompElliptic.Hashing.SimplifiedSWU
+import Mathlib.Algebra.Polynomial.Roots
+import Mathlib.Tactic.ComputeDegree
+
+/-!
+# Fibre bounds for the simplified SWU mapping
+
+Away from the input `u = 0`, every point of the target curve has at most 10
+preimages under `SSWUParams.map`. This is the counting fact the
+indifferentiability arc's rejection sampler consumes (zcash/ironwood#198,
+CompElliptic#25): to sample a uniform preimage of a point under the deployed
+two-term construction, the sampler needs each single-term fibre to be
+computable and small; smallness is what makes its acceptance probability an
+explicit constant. The input `u = 0` is excluded rather than counted: the
+probability analysis already carries a negligible correction for pairs
+containing `0` (the deployed-vs-zero-repaired transport), and the sampler
+rides the same correction.
+
+The bound is by abscissa, and the argument is elementary root counting. On a
+field where `-1` is a square (both Pasta base fields, `q ≡ 1 (mod 4)`), a
+nonzero input is never exceptional: `ta = t² + t = 0` forces `t = 0` (which
+is `u = 0`) or `t = -1` (which would make `-1/Z = u²` a square against `Z`
+nonsquare). So a nonzero preimage `u` of a point with abscissa `x` satisfies
+`xnum/xdiv = x`, where `xnum` is one of the two branch numerators and the
+denominator is `xdiv = A·(-ta)`. Clearing the denominator turns membership
+in *either* branch into the vanishing of the product polynomial
+
+`Φ_x(u) = (x·(-A·ta) - x1num) · (x·(-A·ta) - x2num)`.
+
+This is an explicit polynomial in `u` through `t = Z·u²`. Its second factor has
+degree exactly 6 with leading coefficient `-B·Z³ ≠ 0`, so `Φ_x` is a nonzero
+polynomial of degree at most 10, and has at most 10 roots.
+
+The constant is deliberately crude; the proof stays at the level of one
+product polynomial. The optimal constant is 4: each branch equation is a
+quadratic in `t`, and oddness pairs `±u` across `P` and `-P` (which are
+distinct — the target curves have no 2-torsion), so each `t` contributes one
+preimage. Tightening to that constant is a planned follow-up.
+-/
+
+namespace CompElliptic.Hashing
+
+open Finset Polynomial CompElliptic.CurveForms.ShortWeierstrass
+
+namespace SSWUParams
+
+variable {F : Type*} [Field F] [Fintype F] [DecidableEq F]
+
+/-- The polynomial `t = Z·u²`, as a polynomial in `u`. -/
+noncomputable def tPoly (G : SSWUParams F) : Polynomial F := C G.Z * X^2
+
+/-- The polynomial `ta = t² + t`, as a polynomial in `u`. -/
+noncomputable def taPoly (G : SSWUParams F) : Polynomial F :=
+  G.tPoly^2 + G.tPoly
+
+/-- The abscissa-fibre polynomial `Φ_x`: away from `u = 0`, a preimage of
+abscissa `x` under either branch is a root. The first factor is the branch-1
+equation `x·(-A·ta) = x1num`, the second the branch-2 equation
+`x·(-A·ta) = x2num`, both cleared of the denominator `xdiv = A·(-ta)`. -/
+noncomputable def fibrePoly (G : SSWUParams F) (x : F) : Polynomial F :=
+  (C x * (-(C G.E.A) * G.taPoly) - C G.E.B * (G.taPoly + 1)) *
+    (C x * (-(C G.E.A) * G.taPoly) - G.tPoly * (C G.E.B * (G.taPoly + 1)))
+
+/-- The second factor of `Φ_x` has degree exactly 6, because its leading
+coefficient `-B·Z³` does not vanish for any `x`. -/
+theorem fibrePoly_snd_natDegree (G : SSWUParams F) (x : F) :
+    (C x * (-(C G.E.A) * G.taPoly)
+      - G.tPoly * (C G.E.B * (G.taPoly + 1))).natDegree = 6 := by
+  rw [taPoly, tPoly]
+  compute_degree!
+  exact ⟨G.Z_nonzero, G.E.B_nonzero, G.Z_nonzero⟩
+
+/-- The second factor of `Φ_x` is nonzero, because it has degree 6. -/
+theorem fibrePoly_snd_ne_zero (G : SSWUParams F) (x : F) :
+    C x * (-(C G.E.A) * G.taPoly)
+      - G.tPoly * (C G.E.B * (G.taPoly + 1)) ≠ 0 := by
+  intro h
+  have hdeg := G.fibrePoly_snd_natDegree x
+  rw [h, natDegree_zero] at hdeg
+  exact absurd hdeg (by norm_num)
+
+/-- The first factor of `Φ_x` is nonzero, because it either has degree 4
+(when `A·x + B ≠ 0`) or is the nonzero constant `-B`. -/
+theorem fibrePoly_fst_ne_zero (G : SSWUParams F) (x : F) :
+    C x * (-(C G.E.A) * G.taPoly) - C G.E.B * (G.taPoly + 1) ≠ 0 := by
+  have hexpand : C x * (-(C G.E.A) * G.taPoly) - C G.E.B * (G.taPoly + 1)
+      = -(C (G.E.A * x + G.E.B)) * G.taPoly - C G.E.B := by
+    rw [map_add, map_mul]
+    ring
+  rw [hexpand]
+  intro h
+  by_cases hAB : G.E.A * x + G.E.B = 0
+  · rw [hAB, map_zero, neg_zero, zero_mul, zero_sub, neg_eq_zero,
+      C_eq_zero] at h
+    exact G.E.B_nonzero h
+  · have hdeg : (-(C (G.E.A * x + G.E.B)) * G.taPoly - C G.E.B).natDegree
+        = 4 := by
+      rw [taPoly, tPoly]
+      compute_degree!
+      refine ⟨fun hc => hAB ?_, G.Z_nonzero⟩
+      linear_combination -hc
+    rw [h, natDegree_zero] at hdeg
+    exact absurd hdeg (by norm_num)
+
+/-- `Φ_x` is nonzero, because both its factors are. -/
+theorem fibrePoly_ne_zero (G : SSWUParams F) (x : F) : G.fibrePoly x ≠ 0 :=
+  mul_ne_zero (G.fibrePoly_fst_ne_zero x) (G.fibrePoly_snd_ne_zero x)
+
+theorem fibrePoly_natDegree_le (G : SSWUParams F) (x : F) :
+    (G.fibrePoly x).natDegree ≤ 10 := by
+  refine natDegree_mul_le.trans ?_
+  have h1 : (C x * (-(C G.E.A) * G.taPoly)
+      - C G.E.B * (G.taPoly + 1)).natDegree ≤ 4 := by
+    rw [taPoly, tPoly]
+    compute_degree
+  have h2 := (G.fibrePoly_snd_natDegree x).le
+  omega
+
+/-- Away from `u = 0`, a preimage of abscissa `x` is a root of `Φ_x`. The
+denominator is `A·(-ta)` with `ta ≠ 0`, and whichever branch the square-root
+split took, the corresponding factor of `Φ_x` vanishes. -/
+theorem eval_fibrePoly_eq_zero (G : SSWUParams F) {x u : F}
+    (hta : (G.Z * u^2)^2 + G.Z * u^2 ≠ 0)
+    (hx : (G.mapXYUpToSign u).1 = x) :
+    (G.fibrePoly x).eval u = 0 := by
+  have heval_t : G.tPoly.eval u = G.Z * u^2 := by
+    simp [tPoly]
+  have heval_ta : G.taPoly.eval u = (G.Z * u^2)^2 + G.Z * u^2 := by
+    simp [taPoly, tPoly]
+  simp only [mapXYUpToSign] at hx
+  rw [if_neg hta] at hx
+  have hxdiv : G.E.A * -((G.Z * u^2)^2 + G.Z * u^2) ≠ 0 :=
+    mul_ne_zero G.A_nonzero (neg_ne_zero.mpr hta)
+  rw [div_eq_iff hxdiv] at hx
+  rw [fibrePoly, eval_mul]
+  split_ifs at hx with hsr
+  · apply mul_eq_zero_of_left
+    simp only [eval_sub, eval_mul, eval_neg, eval_C, eval_add, eval_one,
+      heval_ta]
+    linear_combination -hx
+  · apply mul_eq_zero_of_right
+    simp only [eval_sub, eval_mul, eval_neg, eval_C, eval_add, eval_one,
+      heval_ta, heval_t]
+    linear_combination -hx
+
+/-- A nonzero input is never exceptional when `-1` is a square, because
+`ta = 0` forces `t = 0` (that is, `u = 0`) or `t = -1`, and the latter would
+exhibit the nonsquare `Z` as `-1` times a square of an inverse. -/
+theorem ta_ne_zero_of_u_ne_zero (G : SSWUParams F) (hsq : IsSquare (-1 : F)) {u : F}
+    (hu : u ≠ 0) : (G.Z * u^2)^2 + G.Z * u^2 ≠ 0 := by
+  intro h
+  have hfac : G.Z * u^2 * (G.Z * u^2 + 1) = 0 := by linear_combination h
+  rcases mul_eq_zero.mp hfac with h0 | h1
+  · rcases mul_eq_zero.mp h0 with hZ | hu2
+    · exact G.Z_nonzero hZ
+    · exact hu (sq_eq_zero_iff.mp hu2)
+  · obtain ⟨s, hs⟩ := hsq
+    have hZu : G.Z * u^2 = -1 := by linear_combination h1
+    refine G.Z_nonsquare ⟨s/u, ?_⟩
+    rw [div_mul_div_comm, show u*u = u^2 from (pow_two u).symm]
+    exact (eq_div_iff (pow_ne_zero 2 hu)).mpr (hZu.trans hs)
+
+/-- **At most 10 nonzero preimages per abscissa.** On a field where `-1` is
+a square, every nonzero preimage is a root of the degree-≤10 polynomial
+`Φ_x`. -/
+theorem card_abscissaFibre_le (G : SSWUParams F) (hsq : IsSquare (-1 : F))
+    (x : F) :
+    (univ.filter fun u => u ≠ 0 ∧ (G.mapXYUpToSign u).1 = x).card ≤ 10 := by
+  have hsub : univ.filter (fun u => u ≠ 0 ∧ (G.mapXYUpToSign u).1 = x)
+      ⊆ (G.fibrePoly x).roots.toFinset := by
+    intro u hu
+    rw [mem_filter] at hu
+    rw [Multiset.mem_toFinset, mem_roots (G.fibrePoly_ne_zero x)]
+    exact G.eval_fibrePoly_eq_zero (G.ta_ne_zero_of_u_ne_zero hsq hu.2.1) hu.2.2
+  calc (univ.filter fun u => u ≠ 0 ∧ (G.mapXYUpToSign u).1 = x).card
+      ≤ (G.fibrePoly x).roots.toFinset.card := Finset.card_le_card hsub
+    _ ≤ (G.fibrePoly x).roots.card := Multiset.toFinset_card_le _
+    _ ≤ (G.fibrePoly x).natDegree := (G.fibrePoly x).card_roots'
+    _ ≤ 10 := G.fibrePoly_natDegree_le x
+
+/-- **At most 10 nonzero preimages per point** under the simplified SWU
+mapping. A preimage of `P` is in particular a preimage of its abscissa. The
+optimal constant is 4 (a planned follow-up); the input `u = 0` is excluded,
+priced by the probability analysis's existing zero-pair correction. -/
+theorem card_map_fibre_le (G : SSWUParams F) (hsq : IsSquare (-1 : F))
+    (P : SWPoint G.E) :
+    (univ.filter fun u => u ≠ 0 ∧ G.map u = P).card ≤ 10 := by
+  refine (Finset.card_le_card ?_).trans (G.card_abscissaFibre_le hsq P.x)
+  intro u hu
+  rw [mem_filter] at hu ⊢
+  exact ⟨mem_univ u, hu.2.1,
+    by rw [show (G.mapXYUpToSign u).1 = (G.map u).x from rfl, hu.2.2]⟩
+
+end SSWUParams
+
+/-- Composing with an injective map does not grow fibres: a fibre bound for
+`f` is a fibre bound for `g ∘ f`. Stated with the auxiliary predicate that
+the fibre-bound statements carry. -/
+theorem card_fibre_comp_le {α β γ : Type*} [Fintype α] [DecidableEq β]
+    [DecidableEq γ] {f : α → β} {g : β → γ} (hg : Function.Injective g)
+    {pred : α → Prop} [DecidablePred pred] {n : ℕ}
+    (h : ∀ Q : β, (univ.filter fun u => pred u ∧ f u = Q).card ≤ n)
+    (P : γ) :
+    (univ.filter fun u => pred u ∧ g (f u) = P).card ≤ n := by
+  rcases (univ.filter fun u => pred u ∧ g (f u) = P).eq_empty_or_nonempty
+    with he | ⟨u₀, hu₀⟩
+  · rw [he, Finset.card_empty]
+    exact Nat.zero_le n
+  · rw [mem_filter] at hu₀
+    refine (Finset.card_le_card ?_).trans (h (f u₀))
+    intro u hu
+    rw [mem_filter] at hu ⊢
+    exact ⟨mem_univ u, hu.2.1, hg (hu.2.2.trans hu₀.2.2.symm)⟩
+
+end CompElliptic.Hashing

CompElliptic/Hashing/PastaSSWU.lean

@@ -9,6 +9,7 @@ import CompElliptic.Isogenies.Homomorphism
 import CompElliptic.Curves.PastaOrder
 import CompElliptic.Hashing.SimplifiedSWU
 import CompElliptic.Hashing.SignedLift
+import CompElliptic.Hashing.FibreBound
 import Mathlib.Tactic.ReduceModChar
 
 /-!
@@ -71,6 +72,12 @@ theorem neg_thirteen_not_isSquare : ¬ IsSquare (-13 : PallasBaseField) := by
   reduce_mod_char
   decide
 
+/-- `-1` is a square in the Pallas base field (`q ≡ 1 (mod 4)`), by Euler's
+criterion with the power evaluated by fast modular exponentiation. -/
+theorem isSquare_neg_one : IsSquare (-1 : PallasBaseField) := by
+  rw [ZMod.euler_criterion PALLAS_BASE_CARD (by decide : (-1 : PallasBaseField) ≠ 0)]
+  reduce_mod_char
+
 /-- The precomputed resolvent root for RFC 9380's criterion 3 on iso-Pallas is not
 a cube, by `not_exists_pow_eq_of_pow_ne_one` with the power evaluated by fast
 modular exponentiation, as for `neg_five_not_isCube`. -/
@@ -153,6 +160,15 @@ character-sum analysis consumes. -/
 theorem isOdd_zeroRepaired_mapToCurve : IsOdd (zeroRepaired mapToCurve) :=
   isOdd_zeroRepaired fun _ hu => mapToCurve_neg hu
 
+/-- **At most 10 nonzero preimages per point** under the deployed
+`mapToCurve`. The isogeny is injective on rational points, so the SSWU fibre
+bound carries over. This is the counting interface for the
+indifferentiability sampler (zcash/ironwood#198). -/
+theorem card_mapToCurve_fibre_le (P : SWPoint curve) :
+    (Finset.univ.filter fun u => u ≠ 0 ∧ mapToCurve u = P).card ≤ 10 :=
+  card_fibre_comp_le iso_map_bijective.injective
+    (fun Q => sswu.card_map_fibre_le isSquare_neg_one Q) P
+
 /-- The zero-repair transport, composed at the deployed mapping: for every
 character, the deployed and repaired character sums differ by exactly
 `ψ (mapToCurve 0) - 1`. Conclusions about the literally-odd
@@ -248,6 +264,12 @@ theorem neg_thirteen_not_isSquare : ¬ IsSquare (-13 : VestaBaseField) := by
   reduce_mod_char
   decide
 
+/-- `-1` is a square in the Vesta base field (`q ≡ 1 (mod 4)`), by Euler's
+criterion with the power evaluated by fast modular exponentiation. -/
+theorem isSquare_neg_one : IsSquare (-1 : VestaBaseField) := by
+  rw [ZMod.euler_criterion PALLAS_SCALAR_CARD (by decide : (-1 : VestaBaseField) ≠ 0)]
+  reduce_mod_char
+
 /-- The precomputed resolvent root for RFC 9380's criterion 3 on iso-Vesta is not
 a cube, by `not_exists_pow_eq_of_pow_ne_one` with the power evaluated by fast
 modular exponentiation, as for `neg_five_not_isCube`. -/
@@ -325,6 +347,15 @@ character-sum analysis consumes. -/
 theorem isOdd_zeroRepaired_mapToCurve : IsOdd (zeroRepaired mapToCurve) :=
   isOdd_zeroRepaired fun _ hu => mapToCurve_neg hu
 
+/-- **At most 10 nonzero preimages per point** under the deployed
+`mapToCurve`. The isogeny is injective on rational points, so the SSWU fibre
+bound carries over. This is the counting interface for the
+indifferentiability sampler (zcash/ironwood#198). -/
+theorem card_mapToCurve_fibre_le (P : SWPoint curve) :
+    (Finset.univ.filter fun u => u ≠ 0 ∧ mapToCurve u = P).card ≤ 10 :=
+  card_fibre_comp_le iso_map_bijective.injective
+    (fun Q => sswu.card_map_fibre_le isSquare_neg_one Q) P
+
 /-- The zero-repair transport, composed at the deployed mapping: for every
 character, the deployed and repaired character sums differ by exactly
 `ψ (mapToCurve 0) - 1`. Conclusions about the literally-odd