More inline equations from the spec

Ascon.cry

@@ -25,10 +25,10 @@ private
   /** Parse function.
    *
    * The parse(𝑋, 𝑟) function parses the input bitstring 𝑋 into a sequence
-   * of blocks 𝑋₀, 𝑋₁, …, 𝑋̃ℓ, where 𝓁 ← ⌊|𝑋|/𝑟⌋ (i.e., 𝑋 ← 𝑋₀ ∥ 𝑋₁ ∥ … ∥ 𝑋̃ℓ).
+   * of blocks 𝑋₀, 𝑋₁, …, 𝑋͠ℓ, where 𝓁 ← ⌊|𝑋|/𝑟⌋ (i.e., 𝑋 ← 𝑋₀ ∥ 𝑋₁ ∥ … ∥ 𝑋͠ℓ).
    * The 𝑋ᵢ blocks for 0 ≤ i ≤ 𝓁 − 1 each have a bit length 𝑟, whereas
-   * 0 ≤ |𝑋̃ℓ| ≤ 𝑟 − 1 (see Algorithm 1). When |𝑋| mod 𝑟 = 0, the final
-   * block is empty (i.e., |𝑋̃ℓ| = 0).
+   * 0 ≤ |𝑋͠ℓ| ≤ 𝑟 − 1 (see Algorithm 1). When |𝑋| mod 𝑟 = 0, the final
+   * block is empty (i.e., |𝑋͠ℓ| = 0).
    */
   parse : {r, m} (fin m, fin r, r >= 1) => [m] -> ([m / r][r], [m % r])
   parse (M # Ml) = (split M, Ml)
@@ -36,7 +36,7 @@ private
   /** Padding rule.
    *
    * The function pad(𝑋, 𝑟) appends the bit 1 to the bitstring 𝑋, followed
-   * bythe bitstring 0ʲ, where 𝑗 is equal to (−|𝑋|−1) mod 𝑟. The length of
+   * by the bitstring 0ʲ, where 𝑗 is equal to (−|𝑋|−1) mod 𝑟. The length of
    * the output bitstring is a multiple of 𝑟 (see Algorithm 2). For examples
    * of padding when representing the data as 64-bit unsigned integers, see
    * Appendix A.2.
@@ -201,9 +201,18 @@ AEAD128_encrypt K N A P = C # T
     [K0, K1] = bitsToWords K
     [N0, N1] = bitsToWords N
 
+    // 𝑆 ← 𝐼𝑉 ‖ 𝐾 ‖ 𝑁              (15)
+    // 𝑆 ← Ascon-p[12](𝑆)          (16)
+    // 𝑆 ← 𝑆 ⊕ (0¹⁹² ‖ K)          (17)
     S0 = Ascon_p`{12} [AEAD128_IV, K0, K1, N0, N1] ^ [0, 0, 0, K0, K1]
+
     SA = AddAD S0 A
 
+    // 𝑃₀, 𝑃₁, …, 𝑃ₙ₋₁, 𝑃͠ₙ ← parse(𝑃,128)       (23)
+    // For each 𝑃ᵢ, 0 ≤ 𝑖 ≤ 𝑛−1
+    //     𝑆[0:127] ← 𝑆[0:127] ⊕ 𝑃ᵢ             (24)
+    //     𝐶ᵢ ← 𝑆[0:127]                        (25)
+    //     𝑆 ← Ascon-p[8](𝑆)                    (26)
     SCs = [XorBlock s p | s <- [SA] # map Ascon_p`{8} SCs | p <- toBlocks P]
     C = take (join (map ExtractC SCs))
 
@@ -292,14 +301,28 @@ private
   /** Absorb all of the associated data into the permutation state. */
   AddAD : {a} (fin a) => State -> [a] -> State
   AddAD S A
-      | a == 0 => DomainSep S
+
+      /* If the AD is non-empty (i.e., |𝐴| > 0):
+       *    𝐴₀, 𝐴₁, …, 𝐴ₘ₋₁, 𝐴͠ₘ ← parse(𝐴,128)   (18)
+       *    𝐴ₘ ← pad(𝐴͠ₘ,128)                     (19)
+       */
       | a > 0  => DomainSep (foldl AbsorbADBlock S (toBlocks A))
 
-  /** Absorb a single block of associated data into the permutation state. */
-  AbsorbADBlock : State -> [128] -> State
-  AbsorbADBlock S X = Ascon_p`{8} (XorBlock S X)
+      /* If the AD is empty (i.e., |𝐴| = 0): Only the final step
+       * described in (22) is applied.
+       */
+      | a == 0 => DomainSep S
 
-  /** Toggle the domain separation bit indicating end of associated data. */
+  /* Absorb a block in input into the state.
+   *     𝑆[0:127] ← 𝑆[0:127] ⊕ 𝐴ᵢ      (20)
+   *     𝑆 ← Ascon-p[8](𝑆)             (21)
+   **/
+  AbsorbADBlock : State -> [128] -> State
+  AbsorbADBlock S Ai = Ascon_p`{8} (XorBlock S Ai)
+
+  /** Toggle the domain separation bit indicating end of associated data.
+   *    𝑆 ← 𝑆 ⊕ (0³¹⁹ ‖ 1)             (22)
+   */
   DomainSep : State -> State
   DomainSep [s0, s1, s2, s3, s4] = [s0, s1, s2, s3, s4 ^ 0b1#0]