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anza-ed25519-verified/verification/Proofs/ScalarPackSpec.lean
mrwulf 0ae92c235b PHASE-2 HALF-LIFT on the anza fork: verify_accepts_iff_point, button-enforced 
THE HALF-LIFT IS NOW PROVEN ON ALL FOUR PYRAMIDS. anza's shape: the hash
oracle is one sha512_hash3 bind, -A is the STORED minus_A field (no
negation call), and the scalar arrives already parsed - so the recompute
inversion is five flat bind_ok_inv steps (two anonymous slice reads, the
oracle, the reduction, the dsm) and the assembly takes ExtValid/OnCurveExt
of self.minus_A directly. Files 1-4 are dalek's verbatim modulo the
curve25519 namespace; every proof compiled FIRST TRY.

verify_accepts_iff_point: accept IFF the signature's R bytes are the
canonical encoding of the recomputed [k]*minus_A + [s]*B (valid,
on-curve, certified model). Five new standard certificates; Phase 3b
enforces anza's tight boundary (SHA-512 oracle + foreign Signature type
+ its two accessors) on BOTH apex and half-lift. Full button green fresh.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
2026-07-05 17:25:25 +02:00

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/- ──────────────────────────────────────────────────────────────────────────────
Proofs/ScalarPackSpec.lean — phase 2, half-lift prerequisite: the scalar
PACK step (`Scalar52::to_bytes` / `Scalar52::pack`) and the composed
`Scalar::from_bytes_mod_order_wide` — the hash-to-scalar entry the
verifier's recompute path calls.
`Scalar52::to_bytes` is pure bit-packing (radix 2⁵² → little-endian bytes,
NO reduction — its callers guarantee canonical limbs): 30 shift-extracts
plus 2 limb-boundary bytes (6, 19 — the only offsets where 52j is not
byte-aligned). `scalar_pack` (the ℕ identity) mirrors ToBytesMath's
bytes_pack at the 52-bit offsets; the boundary ORs become additions via
the same disjointness idiom.
THE COMPOSED SPEC (`from_bytes_mod_order_wide_spec`): for a 64-byte input
with value T, the returned Scalar's 32 bytes have value V with
V < ℓ and (V : ZMod ℓ) = T
— i.e. the verifier's k = SHA-512 output reduced mod ℓ, in EXACTLY the
byte-value form the dsm certificate (`vartime_double_base_mul_spec`)
takes as its scalar premises (V < ℓ < 2²⁵³).
────────────────────────────────────────────────────────────────────────────── -/
import Proofs.ToBytesSpec
import Proofs.ScalarFromBytesSpec
open Aeneas Aeneas.Std Result
open curve25519
set_option maxHeartbeats 8000000
set_option linter.unusedSimpArgs false
set_option maxRecDepth 8000
namespace ScalarProofs
open Aeneas.Std.WP
open CurveFieldProofs (bytesVal nat_shr byte_split_0 xor_top_bit)
/-- Offset-4 byte split for 52-bit limbs (the pack's odd-offset limbs 1, 3):
low 4 bits close the boundary byte, six whole bytes follow. -/
theorem byte_split_52_4 (f : ℕ) :
f % 2^4 + (f / 2^4 % 2^8) * 2^4 + (f / 2^12 % 2^8) * 2^12
+ (f / 2^20 % 2^8) * 2^20 + (f / 2^28 % 2^8) * 2^28
+ (f / 2^36 % 2^8) * 2^36 + (f / 2^44) * 2^44 = f := by
have e1 : f / 2^4 / 2^8 = f / 2^12 := by rw [Nat.div_div_eq_div_mul]; norm_num
have e2 : f / 2^12 / 2^8 = f / 2^20 := by rw [Nat.div_div_eq_div_mul]; norm_num
have e3 : f / 2^20 / 2^8 = f / 2^28 := by rw [Nat.div_div_eq_div_mul]; norm_num
have e4 : f / 2^28 / 2^8 = f / 2^36 := by rw [Nat.div_div_eq_div_mul]; norm_num
have e5 : f / 2^36 / 2^8 = f / 2^44 := by rw [Nat.div_div_eq_div_mul]; norm_num
have d0 := Nat.div_add_mod f (2^4)
have d1 := Nat.div_add_mod (f / 2^4) (2^8)
have d2 := Nat.div_add_mod (f / 2^12) (2^8)
have d3 := Nat.div_add_mod (f / 2^20) (2^8)
have d4 := Nat.div_add_mod (f / 2^28) (2^8)
have d5 := Nat.div_add_mod (f / 2^36) (2^8)
omega
/-- THE 52-BIT PACKING IDENTITY: the 32 bytes `Scalar52::to_bytes` emits
assemble to the limb value (boundary bytes 6 and 19 in additive form). -/
theorem scalar_pack (l0 l1 l2 l3 l4 : ℕ)
(h0 : l0 < 2^52) (h1 : l1 < 2^52) (h2 : l2 < 2^52)
(h3 : l3 < 2^52) (h4 : l4 < 2^48) :
(l0 % 2^8)
+ (l0 / 2^8 % 2^8) * 2^8
+ (l0 / 2^16 % 2^8) * 2^16
+ (l0 / 2^24 % 2^8) * 2^24
+ (l0 / 2^32 % 2^8) * 2^32
+ (l0 / 2^40 % 2^8) * 2^40
+ (l0 / 2^48 + (l1 % 2^4) * 2^4) * 2^48
+ (l1 / 2^4 % 2^8) * 2^56
+ (l1 / 2^12 % 2^8) * 2^64
+ (l1 / 2^20 % 2^8) * 2^72
+ (l1 / 2^28 % 2^8) * 2^80
+ (l1 / 2^36 % 2^8) * 2^88
+ (l1 / 2^44 % 2^8) * 2^96
+ (l2 % 2^8) * 2^104
+ (l2 / 2^8 % 2^8) * 2^112
+ (l2 / 2^16 % 2^8) * 2^120
+ (l2 / 2^24 % 2^8) * 2^128
+ (l2 / 2^32 % 2^8) * 2^136
+ (l2 / 2^40 % 2^8) * 2^144
+ (l2 / 2^48 + (l3 % 2^4) * 2^4) * 2^152
+ (l3 / 2^4 % 2^8) * 2^160
+ (l3 / 2^12 % 2^8) * 2^168
+ (l3 / 2^20 % 2^8) * 2^176
+ (l3 / 2^28 % 2^8) * 2^184
+ (l3 / 2^36 % 2^8) * 2^192
+ (l3 / 2^44 % 2^8) * 2^200
+ (l4 % 2^8) * 2^208
+ (l4 / 2^8 % 2^8) * 2^216
+ (l4 / 2^16 % 2^8) * 2^224
+ (l4 / 2^24 % 2^8) * 2^232
+ (l4 / 2^32 % 2^8) * 2^240
+ (l4 / 2^40) * 2^248
= l0 + l1 * 2^52 + l2 * 2^104 + l3 * 2^156 + l4 * 2^208 := by
have s0 := byte_split_0 l0
have s1 := byte_split_52_4 l1
have s2 := byte_split_0 l2
have s3 := byte_split_52_4 l3
-- limb 4 has only 6 bytes: split_0's seventh chunk vanishes (l4 < 2^48)
have s4 := byte_split_0 l4
have h4z : l4 / 2^48 = 0 := Nat.div_eq_of_lt h4
-- l1/2^44, l3/2^44, l4/2^40 all fit a byte: the mods are the identity
have h1m : l1 / 2^44 % 2^8 = l1 / 2^44 :=
Nat.mod_eq_of_lt (Nat.div_lt_of_lt_mul (show l1 < 2^44 * 2^8 by omega))
have h3m : l3 / 2^44 % 2^8 = l3 / 2^44 :=
Nat.mod_eq_of_lt (Nat.div_lt_of_lt_mul (show l3 < 2^44 * 2^8 by omega))
have h4m : l4 / 2^40 % 2^8 = l4 / 2^40 :=
Nat.mod_eq_of_lt (Nat.div_lt_of_lt_mul (show l4 < 2^40 * 2^8 by omega))
rw [h1m, h3m]
rw [h4m, h4z] at s4
zify at s0 s1 s2 s3 s4 ⊢
linear_combination s0 + 2^52 * s1 + 2^104 * s2 + 2^156 * s3 + 2^208 * s4
/-- **`Scalar52::to_bytes` is the canonical serializer of canonical limbs**:
for limbs below 2⁵² with value below 2²⁵³ (every reduced scalar), the 32
output bytes denote exactly the limb value. -/
theorem scalar52_to_bytes_spec (a : Sc) (l0 l1 l2 l3 l4 : U64)
(hl : (↑a : List U64) = [l0, l1, l2, l3, l4])
(hb0 : l0.val < 2^52) (hb1 : l1.val < 2^52) (hb2 : l2.val < 2^52)
(hb3 : l3.val < 2^52) (hb4 : l4.val < 2^48) :
backend.serial.u64.scalar.Scalar52.to_bytes a
⦃ s => bytesVal s = scVal a ⦄ := by
unfold backend.serial.u64.scalar.Scalar52.to_bytes
-- limb 0 read
step as ⟨v0, hv0⟩
simp [hl] at hv0
-- byte 0 (limb 0 >> 0)
step as ⟨x0, hx0⟩
rw [hv0] at hx0
step as ⟨b0, hb0⟩
have hb0v : b0.val = l0.val % 2^8 := by
rw [hb0, UScalar.cast_val_eq, hx0, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s1, hs1⟩
have hsl0 : (↑s1 : List Std.U8) = [b0, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs1, Array.set_val_eq, Array.repeat_val]
rfl
clear hx0 hb0 hs1
-- byte 1 (limb 0 >> 8)
step as ⟨x1, hx1⟩
rw [hv0] at hx1
step as ⟨b1, hb1⟩
have hb1v : b1.val = l0.val / 2^8 % 2^8 := by
rw [hb1, UScalar.cast_val_eq, hx1, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s2, hs2⟩
have hsl1 : (↑s2 : List Std.U8) = [b0, b1, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs2, Array.set_val_eq, hsl0]
rfl
clear hx1 hb1 hs2 hsl0
-- byte 2 (limb 0 >> 16)
step as ⟨x2, hx2⟩
rw [hv0] at hx2
step as ⟨b2, hb2⟩
have hb2v : b2.val = l0.val / 2^16 % 2^8 := by
rw [hb2, UScalar.cast_val_eq, hx2, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s3, hs3⟩
have hsl2 : (↑s3 : List Std.U8) = [b0, b1, b2, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs3, Array.set_val_eq, hsl1]
rfl
clear hx2 hb2 hs3 hsl1
-- byte 3 (limb 0 >> 24)
step as ⟨x3, hx3⟩
rw [hv0] at hx3
step as ⟨b3, hb3⟩
have hb3v : b3.val = l0.val / 2^24 % 2^8 := by
rw [hb3, UScalar.cast_val_eq, hx3, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s4, hs4⟩
have hsl3 : (↑s4 : List Std.U8) = [b0, b1, b2, b3, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs4, Array.set_val_eq, hsl2]
rfl
clear hx3 hb3 hs4 hsl2
-- byte 4 (limb 0 >> 32)
step as ⟨x4, hx4⟩
rw [hv0] at hx4
step as ⟨b4, hb4⟩
have hb4v : b4.val = l0.val / 2^32 % 2^8 := by
rw [hb4, UScalar.cast_val_eq, hx4, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s5, hs5⟩
have hsl4 : (↑s5 : List Std.U8) = [b0, b1, b2, b3, b4, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs5, Array.set_val_eq, hsl3]
rfl
clear hx4 hb4 hs5 hsl3
-- byte 5 (limb 0 >> 40)
step as ⟨x5, hx5⟩
rw [hv0] at hx5
step as ⟨b5, hb5⟩
have hb5v : b5.val = l0.val / 2^40 % 2^8 := by
rw [hb5, UScalar.cast_val_eq, hx5, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s6, hs6⟩
have hsl5 : (↑s6 : List Std.U8) = [b0, b1, b2, b3, b4, b5, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs6, Array.set_val_eq, hsl4]
rfl
clear hx5 hb5 hs6 hsl4
-- byte 6 (boundary: limb 0 >> 48 | limb 1 << 4)
step as ⟨x6, hx6⟩
rw [hv0] at hx6
step as ⟨v1, hv1⟩
simp [hl] at hv1
step as ⟨y6, hy6⟩
rw [hv1] at hy6
have hy6v : y6.val = l1.val * 2^4 := by
rw [hy6]
simp only [Nat.shiftLeft_eq]
rw [Nat.mod_eq_of_lt (show l1.val * 2^4 < U64.size by scalar_tac)]
step as ⟨z6, hz6⟩
step as ⟨b6, hb6⟩
have hb6v : b6.val = l0.val / 2^48 + (l1.val % 2^4) * 2^4 := by
rw [hb6, UScalar.cast_val_eq]
norm_num [UScalarTy.numBits]
rw [hz6, UScalar.val_or, hx6, nat_shr, hy6v]
have hxlt : l0.val / 2^48 < 2^4 := by omega
have hor := Nat.two_pow_add_eq_or_of_lt (b := l0.val / 2^48) (i := 4) hxlt (l1.val)
have hadd : l0.val / 2^48 ||| l1.val * 2^4 = l0.val / 2^48 + l1.val * 2^4 := by
calc l0.val / 2^48 ||| l1.val * 2^4
= l0.val / 2^48 ||| 2^4 * l1.val := by rw [Nat.mul_comm]
_ = 2^4 * l1.val ||| l0.val / 2^48 := Nat.lor_comm _ _
_ = 2^4 * l1.val + l0.val / 2^48 := hor.symm
_ = l0.val / 2^48 + l1.val * 2^4 := by ring
rw [hadd]
omega
step as ⟨s7, hs7⟩
have hsl6 : (↑s7 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs7, Array.set_val_eq, hsl5]
rfl
clear hx6 hy6 hy6v hz6 hb6 hs7 hsl5
-- byte 7 (limb 1 >> 4)
step as ⟨x7, hx7⟩
rw [hv1] at hx7
step as ⟨b7, hb7⟩
have hb7v : b7.val = l1.val / 2^4 % 2^8 := by
rw [hb7, UScalar.cast_val_eq, hx7, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s8, hs8⟩
have hsl7 : (↑s8 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs8, Array.set_val_eq, hsl6]
rfl
clear hx7 hb7 hs8 hsl6
-- byte 8 (limb 1 >> 12)
step as ⟨x8, hx8⟩
rw [hv1] at hx8
step as ⟨b8, hb8⟩
have hb8v : b8.val = l1.val / 2^12 % 2^8 := by
rw [hb8, UScalar.cast_val_eq, hx8, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s9, hs9⟩
have hsl8 : (↑s9 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs9, Array.set_val_eq, hsl7]
rfl
clear hx8 hb8 hs9 hsl7
-- byte 9 (limb 1 >> 20)
step as ⟨x9, hx9⟩
rw [hv1] at hx9
step as ⟨b9, hb9⟩
have hb9v : b9.val = l1.val / 2^20 % 2^8 := by
rw [hb9, UScalar.cast_val_eq, hx9, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s10, hs10⟩
have hsl9 : (↑s10 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs10, Array.set_val_eq, hsl8]
rfl
clear hx9 hb9 hs10 hsl8
-- byte 10 (limb 1 >> 28)
step as ⟨x10, hx10⟩
rw [hv1] at hx10
step as ⟨b10, hb10⟩
have hb10v : b10.val = l1.val / 2^28 % 2^8 := by
rw [hb10, UScalar.cast_val_eq, hx10, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s11, hs11⟩
have hsl10 : (↑s11 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs11, Array.set_val_eq, hsl9]
rfl
clear hx10 hb10 hs11 hsl9
-- byte 11 (limb 1 >> 36)
step as ⟨x11, hx11⟩
rw [hv1] at hx11
step as ⟨b11, hb11⟩
have hb11v : b11.val = l1.val / 2^36 % 2^8 := by
rw [hb11, UScalar.cast_val_eq, hx11, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s12, hs12⟩
have hsl11 : (↑s12 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs12, Array.set_val_eq, hsl10]
rfl
clear hx11 hb11 hs12 hsl10
-- byte 12 (limb 1 >> 44)
step as ⟨x12, hx12⟩
rw [hv1] at hx12
step as ⟨b12, hb12⟩
have hb12v : b12.val = l1.val / 2^44 % 2^8 := by
rw [hb12, UScalar.cast_val_eq, hx12, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s13, hs13⟩
have hsl12 : (↑s13 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs13, Array.set_val_eq, hsl11]
rfl
clear hx12 hb12 hs13 hsl11
-- limb 2 read
step as ⟨v2, hv2⟩
simp [hl] at hv2
-- byte 13 (limb 2 >> 0)
step as ⟨x13, hx13⟩
rw [hv2] at hx13
step as ⟨b13, hb13⟩
have hb13v : b13.val = l2.val % 2^8 := by
rw [hb13, UScalar.cast_val_eq, hx13, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s14, hs14⟩
have hsl13 : (↑s14 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs14, Array.set_val_eq, hsl12]
rfl
clear hx13 hb13 hs14 hsl12
-- byte 14 (limb 2 >> 8)
step as ⟨x14, hx14⟩
rw [hv2] at hx14
step as ⟨b14, hb14⟩
have hb14v : b14.val = l2.val / 2^8 % 2^8 := by
rw [hb14, UScalar.cast_val_eq, hx14, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s15, hs15⟩
have hsl14 : (↑s15 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs15, Array.set_val_eq, hsl13]
rfl
clear hx14 hb14 hs15 hsl13
-- byte 15 (limb 2 >> 16)
step as ⟨x15, hx15⟩
rw [hv2] at hx15
step as ⟨b15, hb15⟩
have hb15v : b15.val = l2.val / 2^16 % 2^8 := by
rw [hb15, UScalar.cast_val_eq, hx15, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s16, hs16⟩
have hsl15 : (↑s16 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs16, Array.set_val_eq, hsl14]
rfl
clear hx15 hb15 hs16 hsl14
-- byte 16 (limb 2 >> 24)
step as ⟨x16, hx16⟩
rw [hv2] at hx16
step as ⟨b16, hb16⟩
have hb16v : b16.val = l2.val / 2^24 % 2^8 := by
rw [hb16, UScalar.cast_val_eq, hx16, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s17, hs17⟩
have hsl16 : (↑s17 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs17, Array.set_val_eq, hsl15]
rfl
clear hx16 hb16 hs17 hsl15
-- byte 17 (limb 2 >> 32)
step as ⟨x17, hx17⟩
rw [hv2] at hx17
step as ⟨b17, hb17⟩
have hb17v : b17.val = l2.val / 2^32 % 2^8 := by
rw [hb17, UScalar.cast_val_eq, hx17, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s18, hs18⟩
have hsl17 : (↑s18 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs18, Array.set_val_eq, hsl16]
rfl
clear hx17 hb17 hs18 hsl16
-- byte 18 (limb 2 >> 40)
step as ⟨x18, hx18⟩
rw [hv2] at hx18
step as ⟨b18, hb18⟩
have hb18v : b18.val = l2.val / 2^40 % 2^8 := by
rw [hb18, UScalar.cast_val_eq, hx18, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s19, hs19⟩
have hsl18 : (↑s19 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs19, Array.set_val_eq, hsl17]
rfl
clear hx18 hb18 hs19 hsl17
-- byte 19 (boundary: limb 2 >> 48 | limb 3 << 4)
step as ⟨x19, hx19⟩
rw [hv2] at hx19
step as ⟨v3, hv3⟩
simp [hl] at hv3
step as ⟨y19, hy19⟩
rw [hv3] at hy19
have hy19v : y19.val = l3.val * 2^4 := by
rw [hy19]
simp only [Nat.shiftLeft_eq]
rw [Nat.mod_eq_of_lt (show l3.val * 2^4 < U64.size by scalar_tac)]
step as ⟨z19, hz19⟩
step as ⟨b19, hb19⟩
have hb19v : b19.val = l2.val / 2^48 + (l3.val % 2^4) * 2^4 := by
rw [hb19, UScalar.cast_val_eq]
norm_num [UScalarTy.numBits]
rw [hz19, UScalar.val_or, hx19, nat_shr, hy19v]
have hxlt : l2.val / 2^48 < 2^4 := by omega
have hor := Nat.two_pow_add_eq_or_of_lt (b := l2.val / 2^48) (i := 4) hxlt (l3.val)
have hadd : l2.val / 2^48 ||| l3.val * 2^4 = l2.val / 2^48 + l3.val * 2^4 := by
calc l2.val / 2^48 ||| l3.val * 2^4
= l2.val / 2^48 ||| 2^4 * l3.val := by rw [Nat.mul_comm]
_ = 2^4 * l3.val ||| l2.val / 2^48 := Nat.lor_comm _ _
_ = 2^4 * l3.val + l2.val / 2^48 := hor.symm
_ = l2.val / 2^48 + l3.val * 2^4 := by ring
rw [hadd]
omega
step as ⟨s20, hs20⟩
have hsl19 : (↑s20 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs20, Array.set_val_eq, hsl18]
rfl
clear hx19 hy19 hy19v hz19 hb19 hs20 hsl18
-- byte 20 (limb 3 >> 4)
step as ⟨x20, hx20⟩
rw [hv3] at hx20
step as ⟨b20, hb20⟩
have hb20v : b20.val = l3.val / 2^4 % 2^8 := by
rw [hb20, UScalar.cast_val_eq, hx20, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s21, hs21⟩
have hsl20 : (↑s21 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs21, Array.set_val_eq, hsl19]
rfl
clear hx20 hb20 hs21 hsl19
-- byte 21 (limb 3 >> 12)
step as ⟨x21, hx21⟩
rw [hv3] at hx21
step as ⟨b21, hb21⟩
have hb21v : b21.val = l3.val / 2^12 % 2^8 := by
rw [hb21, UScalar.cast_val_eq, hx21, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s22, hs22⟩
have hsl21 : (↑s22 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs22, Array.set_val_eq, hsl20]
rfl
clear hx21 hb21 hs22 hsl20
-- byte 22 (limb 3 >> 20)
step as ⟨x22, hx22⟩
rw [hv3] at hx22
step as ⟨b22, hb22⟩
have hb22v : b22.val = l3.val / 2^20 % 2^8 := by
rw [hb22, UScalar.cast_val_eq, hx22, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s23, hs23⟩
have hsl22 : (↑s23 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs23, Array.set_val_eq, hsl21]
rfl
clear hx22 hb22 hs23 hsl21
-- byte 23 (limb 3 >> 28)
step as ⟨x23, hx23⟩
rw [hv3] at hx23
step as ⟨b23, hb23⟩
have hb23v : b23.val = l3.val / 2^28 % 2^8 := by
rw [hb23, UScalar.cast_val_eq, hx23, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s24, hs24⟩
have hsl23 : (↑s24 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs24, Array.set_val_eq, hsl22]
rfl
clear hx23 hb23 hs24 hsl22
-- byte 24 (limb 3 >> 36)
step as ⟨x24, hx24⟩
rw [hv3] at hx24
step as ⟨b24, hb24⟩
have hb24v : b24.val = l3.val / 2^36 % 2^8 := by
rw [hb24, UScalar.cast_val_eq, hx24, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s25, hs25⟩
have hsl24 : (↑s25 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs25, Array.set_val_eq, hsl23]
rfl
clear hx24 hb24 hs25 hsl23
-- byte 25 (limb 3 >> 44)
step as ⟨x25, hx25⟩
rw [hv3] at hx25
step as ⟨b25, hb25⟩
have hb25v : b25.val = l3.val / 2^44 % 2^8 := by
rw [hb25, UScalar.cast_val_eq, hx25, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s26, hs26⟩
have hsl25 : (↑s26 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs26, Array.set_val_eq, hsl24]
rfl
clear hx25 hb25 hs26 hsl24
-- limb 4 read
step as ⟨v4, hv4⟩
simp [hl] at hv4
-- byte 26 (limb 4 >> 0)
step as ⟨x26, hx26⟩
rw [hv4] at hx26
step as ⟨b26, hb26⟩
have hb26v : b26.val = l4.val % 2^8 := by
rw [hb26, UScalar.cast_val_eq, hx26, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s27, hs27⟩
have hsl26 : (↑s27 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, b26, 0#u8, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs27, Array.set_val_eq, hsl25]
rfl
clear hx26 hb26 hs27 hsl25
-- byte 27 (limb 4 >> 8)
step as ⟨x27, hx27⟩
rw [hv4] at hx27
step as ⟨b27, hb27⟩
have hb27v : b27.val = l4.val / 2^8 % 2^8 := by
rw [hb27, UScalar.cast_val_eq, hx27, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s28, hs28⟩
have hsl27 : (↑s28 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, b26, b27, 0#u8, 0#u8, 0#u8, 0#u8] := by
simp only [hs28, Array.set_val_eq, hsl26]
rfl
clear hx27 hb27 hs28 hsl26
-- byte 28 (limb 4 >> 16)
step as ⟨x28, hx28⟩
rw [hv4] at hx28
step as ⟨b28, hb28⟩
have hb28v : b28.val = l4.val / 2^16 % 2^8 := by
rw [hb28, UScalar.cast_val_eq, hx28, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s29, hs29⟩
have hsl28 : (↑s29 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, b26, b27, b28, 0#u8, 0#u8, 0#u8] := by
simp only [hs29, Array.set_val_eq, hsl27]
rfl
clear hx28 hb28 hs29 hsl27
-- byte 29 (limb 4 >> 24)
step as ⟨x29, hx29⟩
rw [hv4] at hx29
step as ⟨b29, hb29⟩
have hb29v : b29.val = l4.val / 2^24 % 2^8 := by
rw [hb29, UScalar.cast_val_eq, hx29, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s30, hs30⟩
have hsl29 : (↑s30 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, b26, b27, b28, b29, 0#u8, 0#u8] := by
simp only [hs30, Array.set_val_eq, hsl28]
rfl
clear hx29 hb29 hs30 hsl28
-- byte 30 (limb 4 >> 32)
step as ⟨x30, hx30⟩
rw [hv4] at hx30
step as ⟨b30, hb30⟩
have hb30v : b30.val = l4.val / 2^32 % 2^8 := by
rw [hb30, UScalar.cast_val_eq, hx30, nat_shr]
norm_num [UScalarTy.numBits]
step as ⟨s31, hs31⟩
have hsl30 : (↑s31 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, b26, b27, b28, b29, b30, 0#u8] := by
simp only [hs31, Array.set_val_eq, hsl29]
rfl
clear hx30 hb30 hs31 hsl29
-- byte 31 (limb 4 >> 40)
step as ⟨x31, hx31⟩
rw [hv4] at hx31
step as ⟨b31, hb31⟩
have hb31v : b31.val = l4.val / 2^40 := by
rw [hb31, UScalar.cast_val_eq, hx31, nat_shr]
norm_num [UScalarTy.numBits]
omega
step as ⟨s32, hs32⟩
have hsl31 : (↑s32 : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, b26, b27, b28, b29, b30, b31] := by
simp only [hs32, Array.set_val_eq, hsl30]
rfl
clear hx31 hb31 hs32 hsl30
try simp only [spec_ok]
-- assembly via the 52-bit packing identity
have hsum : bytesVal s32 = l0.val + l1.val * 2^52 + l2.val * 2^104
+ l3.val * 2^156 + l4.val * 2^208 := by
simp only [bytesVal, hsl31]
rw [hb0v, hb1v, hb2v, hb3v, hb4v, hb5v, hb6v, hb7v, hb8v, hb9v, hb10v,
hb11v, hb12v, hb13v, hb14v, hb15v, hb16v, hb17v, hb18v, hb19v, hb20v,
hb21v, hb22v, hb23v, hb24v, hb25v, hb26v, hb27v, hb28v, hb29v, hb30v,
hb31v]
exact scalar_pack l0.val l1.val l2.val l3.val l4.val hb0 hb1 hb2 hb3 hb4
rw [hsum, scVal_eq a l0 l1 l2 l3 l4 hl]
unfold scLimbs
ring
/-- **The hash-to-scalar entry is total and canonical**: for a 64-byte input
of value T, `from_bytes_mod_order_wide` returns a Scalar whose 32 bytes
denote V with V < ℓ and V ≡ T (mod ℓ) — exactly the scalar premises the
dsm certificate consumes. -/
theorem from_bytes_mod_order_wide_spec (input : Std.Array Std.U8 64#usize)
(b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 b14 b15 b16 b17 b18 b19 b20 b21 b22 b23 b24 b25 b26 b27 b28 b29 b30 b31 b32 b33 b34 b35 b36 b37 b38 b39 b40 b41 b42 b43 b44 b45 b46 b47 b48 b49 b50 b51 b52 b53 b54 b55 b56 b57 b58 b59 b60 b61 b62 b63 : Std.U8)
(hb : (↑input : List Std.U8) = [b0, b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18, b19, b20, b21, b22, b23, b24, b25, b26, b27, b28, b29, b30, b31, b32, b33, b34, b35, b36, b37, b38, b39, b40, b41, b42, b43, b44, b45, b46, b47, b48, b49, b50, b51, b52, b53, b54, b55, b56, b57, b58, b59, b60, b61, b62, b63])
(T : ℕ) (hT : T = b0.val + b1.val * 2^8 + b2.val * 2^16 + b3.val * 2^24 + b4.val * 2^32 + b5.val * 2^40 + b6.val * 2^48 + b7.val * 2^56 + b8.val * 2^64 + b9.val * 2^72 + b10.val * 2^80 + b11.val * 2^88 + b12.val * 2^96 + b13.val * 2^104 + b14.val * 2^112 + b15.val * 2^120 + b16.val * 2^128 + b17.val * 2^136 + b18.val * 2^144 + b19.val * 2^152 + b20.val * 2^160 + b21.val * 2^168 + b22.val * 2^176 + b23.val * 2^184 + b24.val * 2^192 + b25.val * 2^200 + b26.val * 2^208 + b27.val * 2^216 + b28.val * 2^224 + b29.val * 2^232 + b30.val * 2^240 + b31.val * 2^248 + b32.val * 2^256 + b33.val * 2^264 + b34.val * 2^272 + b35.val * 2^280 + b36.val * 2^288 + b37.val * 2^296 + b38.val * 2^304 + b39.val * 2^312 + b40.val * 2^320 + b41.val * 2^328 + b42.val * 2^336 + b43.val * 2^344 + b44.val * 2^352 + b45.val * 2^360 + b46.val * 2^368 + b47.val * 2^376 + b48.val * 2^384 + b49.val * 2^392 + b50.val * 2^400 + b51.val * 2^408 + b52.val * 2^416 + b53.val * 2^424 + b54.val * 2^432 + b55.val * 2^440 + b56.val * 2^448 + b57.val * 2^456 + b58.val * 2^464 + b59.val * 2^472 + b60.val * 2^480 + b61.val * 2^488 + b62.val * 2^496 + b63.val * 2^504) :
scalar.Scalar.from_bytes_mod_order_wide input
⦃ k => bytesVal k.bytes < Ell ∧ (bytesVal k.bytes : ZMod Ell) = (T : ZMod Ell) ⦄ := by
unfold scalar.Scalar.from_bytes_mod_order_wide scalar.Scalar52.pack
step with (from_bytes_wide_spec input b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 b14 b15 b16 b17 b18 b19 b20 b21 b22 b23 b24 b25 b26 b27 b28 b29 b30 b31 b32 b33 b34 b35 b36 b37 b38 b39 b40 b41 b42 b43 b44 b45 b46 b47 b48 b49 b50 b51 b52 b53 b54 b55 b56 b57 b58 b59 b60 b61 b62 b63 hb T hT)
as ⟨r, hex, hlt, hden⟩
obtain ⟨s0, s1, s2, s3, s4, hrl, hs0, hs1, hs2, hs3, hs4⟩ := hex
-- top-limb tightening: scVal r < ℓ < 2²⁵³ forces s4 < 2⁴⁵ < 2⁴⁸
have hs4' : s4.val < 2^48 := by
have hv := scVal_eq r s0 s1 s2 s3 s4 hrl
have : scVal r < 2^253 := by
have : Ell < 2^253 := by unfold Ell; norm_num
omega
unfold scLimbs at hv
omega
step with (scalar52_to_bytes_spec r s0 s1 s2 s3 s4 hrl hs0 hs1 hs2 hs3 hs4')
as ⟨bytes, hbv⟩
try simp only [spec_ok]
constructor
· simpa [hbv] using hlt
· rw [hbv]
exact hden
end ScalarProofs
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