    -----------------------------------------------------------------------------
    -- |
    -- Module    : Documentation.SBV.Examples.Crypto.AES
    -- Copyright : (c) Levent Erkok
    -- License   : BSD3
    -- Maintainer: erkokl@gmail.com
    -- Stability : experimental
    --
    -- An implementation of AES (Advanced Encryption Standard), using SBV.
    -- For details on AES, see <http://en.wikipedia.org/wiki/Advanced_Encryption_Standard>.
    --
    -- We do a T-box implementation, which leads to good C code as we can take
    -- advantage of look-up tables. Note that we make virtually no attempt to
    -- optimize our Haskell code. The concern here is not with getting Haskell running
    -- fast at all. The idea is to program the T-Box implementation as naturally and clearly
    -- as possible in Haskell, and have SBV's code-generator generate fast C code automatically.
    -- Therefore, we merely use ordinary Haskell lists as our data-structures, and do not
    -- bother with any unboxing or strictness annotations. Thus, we achieve the separation
    -- of concerns: Correctness via clairty and simplicity and proofs on the Haskell side,
    -- performance by relying on SBV's code generator. If necessary, the generated code
    -- can be FFI'd back into Haskell to complete the loop.
    --
    -- All 3 valid key sizes (128, 192, and 256) as required by the FIPS-197 standard
    -- are supported.
    -----------------------------------------------------------------------------

    {-# LANGUAGE DataKinds        #-}
    {-# LANGUAGE ParallelListComp #-}
    {-# LANGUAGE TypeApplications #-}

    {-# OPTIONS_GHC -Wall -Werror #-}

    module Documentation.SBV.Examples.Crypto.AES where

    import Control.Monad (void)

    import Data.SBV
    import Data.SBV.Tools.CodeGen
    import Data.SBV.Tools.Polynomial

    import Data.List (transpose)
    import Data.Maybe (fromJust)

    import Numeric (showHex)

    -----------------------------------------------------------------------------
    -- * Formalizing GF(2^8)
    -----------------------------------------------------------------------------

    -- | An element of the Galois Field 2^8, which are essentially polynomials with
    -- maximum degree 7. They are conveniently represented as values between 0 and 255.
    type GF28 = SWord 8

    -- | Multiplication in GF(2^8). This is simple polynomial multipliation, followed
    -- by the irreducible polynomial @x^8+x^4+x^3+x^1+1@. We simply use the 'pMult'
    -- function exported by SBV to do the operation.
    gf28Mult :: GF28 -> GF28 -> GF28
    gf28Mult :: GF28 -> GF28 -> GF28
    gf28Mult x :: GF28
    x y :: GF28
    y = (GF28, GF28, [Int]) -> GF28
    forall a. Polynomial a => (a, a, [Int]) -> a
    pMult (GF28
    x, GF28
    y, [8, 4, 3, 1, 0])

    -- | Exponentiation by a constant in GF(2^8). The implementation uses the usual
    -- square-and-multiply trick to speed up the computation.
    gf28Pow :: GF28 -> Int -> GF28
    gf28Pow :: GF28 -> Int -> GF28
    gf28Pow n :: GF28
    n = Int -> GF28
    forall a. (Integral a, Bits a) => a -> GF28
    pow
      where sq :: GF28 -> GF28
    sq x :: GF28
    x  = GF28
    x GF28 -> GF28 -> GF28
    `gf28Mult` GF28
    x
            pow :: a -> GF28
    pow 0    = 1
            pow i :: a
    i
             | a -> Bool
    forall a. Integral a => a -> Bool
    odd a
    i = GF28
    n GF28 -> GF28 -> GF28
    `gf28Mult` GF28 -> GF28
    sq (a -> GF28
    pow (a
    i a -> Int -> a
    forall a. Bits a => a -> Int -> a
    `shiftR` 1))
             | Bool
    True  = GF28 -> GF28
    sq (a -> GF28
    pow (a
    i a -> Int -> a
    forall a. Bits a => a -> Int -> a
    `shiftR` 1))

    -- | Computing inverses in GF(2^8). By the mathematical properties of GF(2^8)
    -- and the particular irreducible polynomial used @x^8+x^5+x^3+x^1+1@, it
    -- turns out that raising to the 254 power gives us the multiplicative inverse.
    -- Of course, we can prove this using SBV:
    --
    -- >>> prove $ \x -> x ./= 0 .=> x `gf28Mult` gf28Inverse x .== 1
    -- Q.E.D.
    --
    -- Note that we exclude @0@ in our theorem, as it does not have a
    -- multiplicative inverse.
    gf28Inverse :: GF28 -> GF28
    gf28Inverse :: GF28 -> GF28
    gf28Inverse x :: GF28
    x = GF28
    x GF28 -> Int -> GF28
    `gf28Pow` 254

    -----------------------------------------------------------------------------
    -- * Implementing AES
    -----------------------------------------------------------------------------

    -----------------------------------------------------------------------------
    -- ** Types and basic operations
    -----------------------------------------------------------------------------
    -- | AES state. The state consists of four 32-bit words, each of which is in turn treated
    -- as four GF28's, i.e., 4 bytes. The T-Box implementation keeps the four-bytes together
    -- for efficient representation.
    type State = [SWord 32]

    -- | The key, which can be 128, 192, or 256 bits. Represented as a sequence of 32-bit words.
    type Key = [SWord 32]

    -- | The key schedule. AES executes in rounds, and it treats first and last round keys slightly
    -- differently than the middle ones. We reflect that choice by being explicit about it in our type.
    -- The length of the middle list of keys depends on the key-size, which in turn determines
    -- the number of rounds.
    type KS = (Key, [Key], Key)

    -- | Rotating a state row by a fixed amount to the right.
    rotR :: [GF28] -> Int -> [GF28]
    rotR :: [GF28] -> Int -> [GF28]
    rotR [a :: GF28
    a, b :: GF28
    b, c :: GF28
    c, d :: GF28
    d] 1 = [GF28
    d, GF28
    a, GF28
    b, GF28
    c]
    rotR [a :: GF28
    a, b :: GF28
    b, c :: GF28
    c, d :: GF28
    d] 2 = [GF28
    c, GF28
    d, GF28
    a, GF28
    b]
    rotR [a :: GF28
    a, b :: GF28
    b, c :: GF28
    c, d :: GF28
    d] 3 = [GF28
    b, GF28
    c, GF28
    d, GF28
    a]
    rotR xs :: [GF28]
    xs           i :: Int
    i = [Char] -> [GF28]
    forall a. HasCallStack => [Char] -> a
    error ([Char] -> [GF28]) -> [Char] -> [GF28]
    forall a b. (a -> b) -> a -> b
    $ "rotR: Unexpected input: " [Char] -> [Char] -> [Char]
    forall a. [a] -> [a] -> [a]
    ++ ([GF28], Int) -> [Char]
    forall a. Show a => a -> [Char]
    show ([GF28]
    xs, Int
    i)

    -----------------------------------------------------------------------------
    -- ** The key schedule
    -----------------------------------------------------------------------------

    -- | Definition of round-constants, as specified in Section 5.2 of the AES standard.
    roundConstants :: [GF28]
    roundConstants :: [GF28]
    roundConstants = 0 GF28 -> [GF28] -> [GF28]
    forall a. a -> [a] -> [a]
    : [ GF28 -> Int -> GF28
    gf28Pow 2 (Int
    kInt -> Int -> Int
    forall a. Num a => a -> a -> a
    -1) | Int
    k <- [1 .. ] ]

    -- | The @InvMixColumns@ transformation, as described in Section 5.3.3 of the standard. Note
    -- that this transformation is only used explicitly during key-expansion in the T-Box implementation
    -- of AES.
    invMixColumns :: State -> State
    invMixColumns :: State -> State
    invMixColumns state :: State
    state = ([GF28] -> SWord 32) -> [[GF28]] -> State
    forall a b. (a -> b) -> [a] -> [b]
    map [GF28] -> SWord 32
    forall a. ByteConverter a => [GF28] -> a
    fromBytes ([[GF28]] -> State) -> [[GF28]] -> State
    forall a b. (a -> b) -> a -> b
    $ [[GF28]] -> [[GF28]]
    forall a. [[a]] -> [[a]]
    transpose ([[GF28]] -> [[GF28]]) -> [[GF28]] -> [[GF28]]
    forall a b. (a -> b) -> a -> b
    $ [[GF28]] -> [[GF28]]
    mmult ((SWord 32 -> [GF28]) -> State -> [[GF28]]
    forall a b. (a -> b) -> [a] -> [b]
    map SWord 32 -> [GF28]
    forall a. ByteConverter a => a -> [GF28]
    toBytes State
    state)
     where dot :: [a -> c] -> [a] -> c
    dot f :: [a -> c]
    f   = (c -> c -> c) -> [c] -> c
    forall (t :: * -> *) a. Foldable t => (a -> a -> a) -> t a -> a
    foldr1 c -> c -> c
    forall a. Bits a => a -> a -> a
    xor ([c] -> c) -> ([a] -> [c]) -> [a] -> c
    forall b c a. (b -> c) -> (a -> b) -> a -> c
    . ((a -> c) -> a -> c) -> [a -> c] -> [a] -> [c]
    forall a b c. (a -> b -> c) -> [a] -> [b] -> [c]
    zipWith (a -> c) -> a -> c
    forall a b. (a -> b) -> a -> b
    ($) [a -> c]
    f
           mmult :: [[GF28]] -> [[GF28]]
    mmult n :: [[GF28]]
    n = [([GF28] -> GF28) -> [[GF28]] -> [GF28]
    forall a b. (a -> b) -> [a] -> [b]
    map ([GF28 -> GF28] -> [GF28] -> GF28
    forall c a. Bits c => [a -> c] -> [a] -> c
    dot [GF28 -> GF28]
    r) [[GF28]]
    n | [GF28 -> GF28]
    r <- [ [GF28 -> GF28
    mE, GF28 -> GF28
    mB, GF28 -> GF28
    mD, GF28 -> GF28
    m9]
                                           , [GF28 -> GF28
    m9, GF28 -> GF28
    mE, GF28 -> GF28
    mB, GF28 -> GF28
    mD]
                                           , [GF28 -> GF28
    mD, GF28 -> GF28
    m9, GF28 -> GF28
    mE, GF28 -> GF28
    mB]
                                           , [GF28 -> GF28
    mB, GF28 -> GF28
    mD, GF28 -> GF28
    m9, GF28 -> GF28
    mE]
                                           ]]
           -- table-lookup versions of gf28Mult with the constants used in invMixColumns
           mE :: GF28 -> GF28
    mE = [GF28] -> GF28 -> GF28 -> GF28
    forall a b.
    (Mergeable a, Ord b, SymVal b, Num b) =>
    [a] -> a -> SBV b -> a
    select [GF28]
    mETable 0
           mB :: GF28 -> GF28
    mB = [GF28] -> GF28 -> GF28 -> GF28
    forall a b.
    (Mergeable a, Ord b, SymVal b, Num b) =>
    [a] -> a -> SBV b -> a
    select [GF28]
    mBTable 0
           mD :: GF28 -> GF28
    mD = [GF28] -> GF28 -> GF28 -> GF28
    forall a b.
    (Mergeable a, Ord b, SymVal b, Num b) =>
    [a] -> a -> SBV b -> a
    select [GF28]
    mDTable 0
           m9 :: GF28 -> GF28
    m9 = [GF28] -> GF28 -> GF28 -> GF28
    forall a b.
    (Mergeable a, Ord b, SymVal b, Num b) =>
    [a] -> a -> SBV b -> a
    select [GF28]
    m9Table 0
           mETable :: [GF28]
    mETable = (GF28 -> GF28) -> [GF28] -> [GF28]
    forall a b. (a -> b) -> [a] -> [b]
    map (GF28 -> GF28 -> GF28
    gf28Mult 0xE) [0..255]
           mBTable :: [GF28]
    mBTable = (GF28 -> GF28) -> [GF28] -> [GF28]
    forall a b. (a -> b) -> [a] -> [b]
    map (GF28 -> GF28 -> GF28
    gf28Mult 0xB) [0..255]
           mDTable :: [GF28]
    mDTable = (GF28 -> GF28) -> [GF28] -> [GF28]
    forall a b. (a -> b) -> [a] -> [b]
    map (GF28 -> GF28 -> GF28
    gf28Mult 0xD) [0..255]
           m9Table :: [GF28]
    m9Table = (GF28 -> GF28) -> [GF28] -> [GF28]
    forall a b. (a -> b) -> [a] -> [b]
    map (GF28 -> GF28 -> GF28
    gf28Mult 0x9) [0..255]

    -- | Key expansion. Starting with the given key, returns an infinite sequence of
    -- words, as described by the AES standard, Section 5.2, Figure 11.
    keyExpansion :: Int -> Key -> [Key]
    keyExpansion :: Int -> State -> [State]
    keyExpansion nk :: Int
    nk key :: State
    key = State -> [State]
    forall a. [a] -> [[a]]
    chop4 State
    keys
       where keys :: [SWord 32]
             keys :: State
    keys = State
    key State -> State -> State
    forall a. [a] -> [a] -> [a]
    ++ [Int -> SWord 32 -> SWord 32 -> SWord 32
    nextWord Int
    i SWord 32
    prev SWord 32
    old | Int
    i <- [Int
    nk ..] | SWord 32
    prev <- Int -> State -> State
    forall a. Int -> [a] -> [a]
    drop (Int
    nkInt -> Int -> Int
    forall a. Num a => a -> a -> a
    -1) State
    keys | SWord 32
    old <- State
    keys]
             chop4 :: [a] -> [[a]]
             chop4 :: [a] -> [[a]]
    chop4 xs :: [a]
    xs = let (f :: [a]
    f, r :: [a]
    r) = Int -> [a] -> ([a], [a])
    forall a. Int -> [a] -> ([a], [a])
    splitAt 4 [a]
    xs in [a]
    f [a] -> [[a]] -> [[a]]
    forall a. a -> [a] -> [a]
    : [a] -> [[a]]
    forall a. [a] -> [[a]]
    chop4 [a]
    r
             nextWord :: Int -> SWord 32 -> SWord 32 -> SWord 32
             nextWord :: Int -> SWord 32 -> SWord 32 -> SWord 32
    nextWord i :: Int
    i prev :: SWord 32
    prev old :: SWord 32
    old
               | Int
    i Int -> Int -> Int
    forall a. Integral a => a -> a -> a
    `mod` Int
    nk Int -> Int -> Bool
    forall a. Eq a => a -> a -> Bool
    == 0           = SWord 32
    old SWord 32 -> SWord 32 -> SWord 32
    forall a. Bits a => a -> a -> a
    `xor` SWord 32 -> GF28 -> SWord 32
    subWordRcon (SWord 32
    prev SWord 32 -> Int -> SWord 32
    forall a. Bits a => a -> Int -> a
    `rotateL` 8) ([GF28]
    roundConstants [GF28] -> Int -> GF28
    forall a. [a] -> Int -> a
    !! (Int
    i Int -> Int -> Int
    forall a. Integral a => a -> a -> a
    `div` Int
    nk))
               | Int
    i Int -> Int -> Int
    forall a. Integral a => a -> a -> a
    `mod` Int
    nk Int -> Int -> Bool
    forall a. Eq a => a -> a -> Bool
    == 4 Bool -> Bool -> Bool
    && Int
    nk Int -> Int -> Bool
    forall a. Ord a => a -> a -> Bool
    > 6 = SWord 32
    old SWord 32 -> SWord 32 -> SWord 32
    forall a. Bits a => a -> a -> a
    `xor` SWord 32 -> GF28 -> SWord 32
    subWordRcon SWord 32
    prev 0
               | Bool
    True                      = SWord 32
    old SWord 32 -> SWord 32 -> SWord 32
    forall a. Bits a => a -> a -> a
    `xor` SWord 32
    prev
             subWordRcon :: SWord 32 -> GF28 -> SWord 32
             subWordRcon :: SWord 32 -> GF28 -> SWord 32
    subWordRcon w :: SWord 32
    w rc :: GF28
    rc = [GF28] -> SWord 32
    forall a. ByteConverter a => [GF28] -> a
    fromBytes [GF28
    a GF28 -> GF28 -> GF28
    forall a. Bits a => a -> a -> a
    `xor` GF28
    rc, GF28
    b, GF28
    c, GF28
    d]
                where [a :: GF28
    a, b :: GF28
    b, c :: GF28
    c, d :: GF28
    d] = (GF28 -> GF28) -> [GF28] -> [GF28]
    forall a b. (a -> b) -> [a] -> [b]
    map GF28 -> GF28
    sbox ([GF28] -> [GF28]) -> [GF28] -> [GF28]
    forall a b. (a -> b) -> a -> b
    $ SWord 32 -> [GF28]
    forall a. ByteConverter a => a -> [GF28]
    toBytes SWord 32
    w

    -----------------------------------------------------------------------------
    -- ** The S-box transformation
    -----------------------------------------------------------------------------

    -- | The values of the AES S-box table. Note that we describe the S-box programmatically
    -- using the mathematical construction given in Section 5.1.1 of the standard. However,
    -- the code-generation will turn this into a mere look-up table, as it is just a
    -- constant table, all computation being done at \"compile-time\".
    sboxTable :: [GF28]
    sboxTable :: [GF28]
    sboxTable = [GF28 -> GF28
    xformByte (GF28 -> GF28
    gf28Inverse GF28
    b) | GF28
    b <- [0 .. 255]]
      where xformByte :: GF28 -> GF28
            xformByte :: GF28 -> GF28
    xformByte b :: GF28
    b = (GF28 -> GF28 -> GF28) -> GF28 -> [GF28] -> GF28
    forall (t :: * -> *) a b.
    Foldable t =>
    (a -> b -> b) -> b -> t a -> b
    foldr GF28 -> GF28 -> GF28
    forall a. Bits a => a -> a -> a
    xor 0x63 [GF28
    b GF28 -> Int -> GF28
    forall a. Bits a => a -> Int -> a
    `rotateR` Int
    i | Int
    i <- [0, 4, 5, 6, 7]]

    -- | The sbox transformation. We simply select from the sbox table. Note that we
    -- are obliged to give a default value (here @0@) to be used if the index is out-of-bounds
    -- as required by SBV's 'select' function. However, that will never happen since
    -- the table has all 256 elements in it.
    sbox :: GF28 -> GF28
    sbox :: GF28 -> GF28
    sbox = [GF28] -> GF28 -> GF28 -> GF28
    forall a b.
    (Mergeable a, Ord b, SymVal b, Num b) =>
    [a] -> a -> SBV b -> a
    select [GF28]
    sboxTable 0

    -----------------------------------------------------------------------------
    -- ** The inverse S-box transformation
    -----------------------------------------------------------------------------

    -- | The values of the inverse S-box table. Again, the construction is programmatic.
    unSBoxTable :: [GF28]
    unSBoxTable :: [GF28]
    unSBoxTable = [GF28 -> GF28
    gf28Inverse (GF28 -> GF28
    xformByte GF28
    b) | GF28
    b <- [0 .. 255]]
      where xformByte :: GF28 -> GF28
            xformByte :: GF28 -> GF28
    xformByte b :: GF28
    b = (GF28 -> GF28 -> GF28) -> GF28 -> [GF28] -> GF28
    forall (t :: * -> *) a b.
    Foldable t =>
    (a -> b -> b) -> b -> t a -> b
    foldr GF28 -> GF28 -> GF28
    forall a. Bits a => a -> a -> a
    xor 0x05 [GF28
    b GF28 -> Int -> GF28
    forall a. Bits a => a -> Int -> a
    `rotateR` Int
    i | Int
    i <- [2, 5, 7]]

    -- | The inverse s-box transformation.
    unSBox :: GF28 -> GF28
    unSBox :: GF28 -> GF28
    unSBox = [GF28] -> GF28 -> GF28 -> GF28
    forall a b.
    (Mergeable a, Ord b, SymVal b, Num b) =>
    [a] -> a -> SBV b -> a
    select [GF28]
    unSBoxTable 0

    -- | Prove that the 'sbox' and 'unSBox' are inverses. We have:
    --
    -- >>> prove sboxInverseCorrect
    -- Q.E.D.
    --
    sboxInverseCorrect :: GF28 -> SBool
    sboxInverseCorrect :: GF28 -> SBool
    sboxInverseCorrect x :: GF28
    x = GF28 -> GF28
    unSBox (GF28 -> GF28
    sbox GF28
    x) GF28 -> GF28 -> SBool
    forall a. EqSymbolic a => a -> a -> SBool
    .== GF28
    x SBool -> SBool -> SBool
    .&& GF28 -> GF28
    sbox (GF28 -> GF28
    unSBox GF28
    x) GF28 -> GF28 -> SBool
    forall a. EqSymbolic a => a -> a -> SBool
    .== GF28
    x

    -----------------------------------------------------------------------------
    -- ** AddRoundKey transformation
    -----------------------------------------------------------------------------

    -- | Adding the round-key to the current state. We simply exploit the fact
    -- that addition is just xor in implementing this transformation.
    addRoundKey :: Key -> State -> State
    addRoundKey :: State -> State -> State
    addRoundKey = (SWord 32 -> SWord 32 -> SWord 32) -> State -> State -> State
    forall a b c. (a -> b -> c) -> [a] -> [b] -> [c]
    zipWith SWord 32 -> SWord 32 -> SWord 32
    forall a. Bits a => a -> a -> a
    xor

    -----------------------------------------------------------------------------
    -- ** Tables for T-Box encryption
    -----------------------------------------------------------------------------

    -- | T-box table generation function for encryption
    t0Func :: GF28 -> [GF28]
    t0Func :: GF28 -> [GF28]
    t0Func a :: GF28
    a = [GF28
    s GF28 -> GF28 -> GF28
    `gf28Mult` 2, GF28
    s, GF28
    s, GF28
    s GF28 -> GF28 -> GF28
    `gf28Mult` 3] where s :: GF28
    s = GF28 -> GF28
    sbox GF28
    a

    -- | First look-up table used in encryption
    t0 :: GF28 -> SWord 32
    t0 :: GF28 -> SWord 32
    t0 = State -> SWord 32 -> GF28 -> SWord 32
    forall a b.
    (Mergeable a, Ord b, SymVal b, Num b) =>
    [a] -> a -> SBV b -> a
    select State
    t0Table 0 where t0Table :: State
    t0Table = [[GF28] -> SWord 32
    forall a. ByteConverter a => [GF28] -> a
    fromBytes (GF28 -> [GF28]
    t0Func GF28
    a)          | GF28
    a <- [0..255]]

    -- | Second look-up table used in encryption
    t1 :: GF28 -> SWord 32
    t1 :: GF28 -> SWord 32
    t1 = State -> SWord 32 -> GF28 -> SWord 32
    forall a b.
    (Mergeable a, Ord b, SymVal b, Num b) =>
    [a] -> a -> SBV b -> a
    select State
    t1Table 0 where t1Table :: State
    t1Table = [[GF28] -> SWord 32
    forall a. ByteConverter a => [GF28] -> a
    fromBytes (GF28 -> [GF28]
    t0Func GF28
    a [GF28] -> Int -> [GF28]
    `rotR` 1) | GF28
    a <- [0..255]]

    -- | Third look-up table used in encryption
    t2 :: GF28 -> SWord 32
    t2 :: GF28 -> SWord 32
    t2 = State -> SWord 32 -> GF28 -> SWord 32
    forall a b.
    (Mergeable a, Ord b, SymVal b, Num b) =>
    [a] -> a -> SBV b -> a
    select State
    t2Table 0 where t2Table :: State
    t2Table = [[GF28] -> SWord 32
    forall a. ByteConverter a => [GF28] -> a
    fromBytes (GF28 -> [GF28]
    t0Func GF28
    a [GF28] -> Int -> [GF28]
    `rotR` 2) | GF28
    a <- [0..255]]

    -- | Fourth look-up table used in encryption
    t3 :: GF28 -> SWord 32
    t3 :: GF28 -> SWord 32
    t3 = State -> SWord 32 -> GF28 -> SWord 32
    forall a b.
    (Mergeable a, Ord b, SymVal b, Num b) =>
    [a] -> a -> SBV b -> a
    select State
    t3Table 0 where t3Table :: State
    t3Table = [[GF28] -> SWord 32
    forall a. ByteConverter a => [GF28] -> a
    fromBytes (GF28 -> [GF28]
    t0Func GF28
    a [GF28] -> Int -> [GF28]
    `rotR` 3) | GF28
    a <- [0..255]]

    -----------------------------------------------------------------------------
    -- ** Tables for T-Box decryption
    -----------------------------------------------------------------------------

    -- | T-box table generating function for decryption
    u0Func :: GF28 -> [GF28]
    u0Func :: GF28 -> [GF28]
    u0Func a :: GF28
    a = [GF28
    s GF28 -> GF28 -> GF28
    `gf28Mult` 0xE, GF28
    s GF28 -> GF28 -> GF28
    `gf28Mult` 0x9, GF28
    s GF28 -> GF28 -> GF28
    `gf28Mult` 0xD, GF28
    s GF28 -> GF28 -> GF28
    `gf28Mult` 0xB] where s :: GF28
    s = GF28 -> GF28
    unSBox GF28
    a

    -- | First look-up table used in decryption
    u0 :: GF28 -> SWord 32
    u0 :: GF28 -> SWord 32
    u0 = State -> SWord 32 -> GF28 -> SWord 32
    forall a b.
    (Mergeable a, Ord b, SymVal b, Num b) =>
    [a] -> a -> SBV b -> a
    select State
    t0Table 0 where t0Table :: State
    t0Table = [[GF28] -> SWord 32
    forall a. ByteConverter a => [GF28] -> a
    fromBytes (GF28 -> [GF28]
    u0Func GF28
    a)          | GF28
    a <- [0..255]]

    -- | Second look-up table used in decryption
    u1 :: GF28 -> SWord 32
    u1 :: GF28 -> SWord 32
    u1 = State -> SWord 32 -> GF28 -> SWord 32
    forall a b.
    (Mergeable a, Ord b, SymVal b, Num b) =>
    [a] -> a -> SBV b -> a
    select State
    t1Table 0 where t1Table :: State
    t1Table = [[GF28] -> SWord 32
    forall a. ByteConverter a => [GF28] -> a
    fromBytes (GF28 -> [GF28]
    u0Func GF28
    a [GF28] -> Int -> [GF28]
    `rotR` 1) | GF28
    a <- [0..255]]

    -- | Third look-up table used in decryption
    u2 :: GF28 -> SWord 32
    u2 :: GF28 -> SWord 32
    u2 = State -> SWord 32 -> GF28 -> SWord 32
    forall a b.
    (Mergeable a, Ord b, SymVal b, Num b) =>
    [a] -> a -> SBV b -> a
    select State
    t2Table 0 where t2Table :: State
    t2Table = [[GF28] -> SWord 32
    forall a. ByteConverter a => [GF28] -> a
    fromBytes (GF28 -> [GF28]
    u0Func GF28
    a [GF28] -> Int -> [GF28]
    `rotR` 2) | GF28
    a <- [0..255]]

    -- | Fourth look-up table used in decryption
    u3 :: GF28 -> SWord 32
    u3 :: GF28 -> SWord 32
    u3 = State -> SWord 32 -> GF28 -> SWord 32
    forall a b.
    (Mergeable a, Ord b, SymVal b, Num b) =>
    [a] -> a -> SBV b -> a
    select State
    t3Table 0 where t3Table :: State
    t3Table = [[GF28] -> SWord 32
    forall a. ByteConverter a => [GF28] -> a
    fromBytes (GF28 -> [GF28]
    u0Func GF28
    a [GF28] -> Int -> [GF28]
    `rotR` 3) | GF28
    a <- [0..255]]

    -----------------------------------------------------------------------------
    -- ** AES rounds
    -----------------------------------------------------------------------------

    -- | Generic round function. Given the function to perform one round, a key-schedule,
    -- and a starting state, it performs the AES rounds.
    doRounds :: (Bool -> State -> Key -> State) -> KS -> State -> State
    doRounds :: (Bool -> State -> State -> State) -> KS -> State -> State
    doRounds rnd :: Bool -> State -> State -> State
    rnd (ikey :: State
    ikey, rkeys :: [State]
    rkeys, fkey :: State
    fkey) sIn :: State
    sIn = Bool -> State -> State -> State
    rnd Bool
    True ([State] -> State
    forall a. [a] -> a
    last [State]
    rs) State
    fkey
      where s0 :: State
    s0 = State
    ikey State -> State -> State
    `addRoundKey` State
    sIn
            rs :: [State]
    rs = State
    s0 State -> [State] -> [State]
    forall a. a -> [a] -> [a]
    : [Bool -> State -> State -> State
    rnd Bool
    False State
    s State
    k | State
    s <- [State]
    rs | State
    k <- [State]
    rkeys ]

    -- | One encryption round. The first argument indicates whether this is the final round
    -- or not, in which case the construction is slightly different.
    aesRound :: Bool -> State -> Key -> State
    aesRound :: Bool -> State -> State -> State
    aesRound isFinal :: Bool
    isFinal s :: State
    s key :: State
    key = State
    d State -> State -> State
    `addRoundKey` State
    key
      where d :: State
    d = (Int -> SWord 32) -> [Int] -> State
    forall a b. (a -> b) -> [a] -> [b]
    map (Bool -> Int -> SWord 32
    f Bool
    isFinal) [0..3]
            a :: [[GF28]]
    a = (SWord 32 -> [GF28]) -> State -> [[GF28]]
    forall a b. (a -> b) -> [a] -> [b]
    map SWord 32 -> [GF28]
    forall a. ByteConverter a => a -> [GF28]
    toBytes State
    s
            f :: Bool -> Int -> SWord 32
    f True j :: Int
    j = [GF28] -> SWord 32
    forall a. ByteConverter a => [GF28] -> a
    fromBytes [ GF28 -> GF28
    sbox ([[GF28]]
    a [[GF28]] -> Int -> [GF28]
    forall a. [a] -> Int -> a
    !! ((Int
    jInt -> Int -> Int
    forall a. Num a => a -> a -> a
    +0) Int -> Int -> Int
    forall a. Integral a => a -> a -> a
    `mod` 4) [GF28] -> Int -> GF28
    forall a. [a] -> Int -> a
    !! 0)
                                 , GF28 -> GF28
    sbox ([[GF28]]
    a [[GF28]] -> Int -> [GF28]
    forall a. [a] -> Int -> a
    !! ((Int
    jInt -> Int -> Int
    forall a. Num a => a -> a -> a
    +1) Int -> Int -> Int
    forall a. Integral a => a -> a -> a
    `mod` 4) [GF28] -> Int -> GF28
    forall a. [a] -> Int -> a
    !! 1)
                                 , GF28 -> GF28
    sbox ([[GF28]]
    a [[GF28]] -> Int -> [GF28]
    forall a. [a] -> Int -> a
    !! ((Int
    jInt -> Int -> Int
    forall a. Num a => a -> a -> a
    +2) Int -> Int -> Int
    forall a. Integral a => a -> a -> a
    `mod` 4) [GF28] -> Int -> GF28
    forall a. [a] -> Int -> a
    !! 2)
                                 , GF28 -> GF28
    sbox ([[GF28]]
    a [[GF28]] -> Int -> [GF28]
    forall a. [a] -> Int -> a
    !! ((Int
    jInt -> Int -> Int
    forall a. Num a => a -> a -> a
    +3) Int -> Int -> Int
    forall a. Integral a => a -> a -> a
    `mod` 4) [GF28] -> Int -> GF28
    forall a. [a] -> Int -> a
    !! 3)
                                 ]
            f False j :: Int
    j = SWord 32
    e0 SWord 32 -> SWord 32 -> SWord 32
    forall a. Bits a => a -> a -> a
    `xor` SWord 32
    e1 SWord 32 -> SWord 32 -> SWord 32
    forall a. Bits a => a -> a -> a
    `xor` SWord 32
    e2 SWord 32 -> SWord 32 -> SWord 32
    forall a. Bits a => a -> a -> a
    `xor` SWord 32
    e3
                  where e0 :: SWord 32
    e0 = GF28 -> SWord 32
    t0 ([[GF28]]
    a [[GF28]] -> Int -> [GF28]
    forall a. [a] -> Int -> a
    !! ((Int
    jInt -> Int -> Int
    forall a. Num a => a -> a -> a
    +0) Int -> Int -> Int
    forall a. Integral a => a -> a -> a
    `mod` 4) [GF28] -> Int -> GF28
    forall a. [a] -> Int -> a
    !! 0)
                        e1 :: SWord 32
    e1 = GF28 -> SWord 32
    t1 ([[GF28]]
    a [[GF28]] -> Int -> [GF28]
    forall a. [a] -> Int -> a
    !! ((Int
    jInt -> Int -> Int
    forall a. Num a => a -> a -> a
    +1) Int -> Int -> Int
    forall a. Integral a => a -> a -> a
    `mod` 4) [GF28] -> Int -> GF28
    forall a. [a] -> Int -> a
    !! 1)
                        e2 :: SWord 32
    e2 = GF28 -> SWord 32
    t2 ([[GF28]]
    a [[GF28]] -> Int -> [GF28]
    forall a. [a] -> Int -> a
    !! ((Int
    jInt -> Int -> Int
    forall a. Num a => a -> a -> a
    +2) Int -> Int -> Int
    forall a. Integral a => a -> a -> a
    `mod` 4) [GF28] -> Int -> GF28
    forall a. [a] -> Int -> a
    !! 2)
                        e3 :: SWord 32
    e3 = GF28 -> SWord 32
    t3 ([[GF28]]
    a [[GF28]] -> Int -> [GF28]
    forall a. [a] -> Int -> a
    !! ((Int
    jInt -> Int -> Int
    forall a. Num a => a -> a -> a
    +3) Int -> Int -> Int
    forall a. Integral a => a -> a -> a
    `mod` 4) [GF28] -> Int -> GF28
    forall a. [a] -> Int -> a
    !! 3)

    -- | One decryption round. Similar to the encryption round, the first argument
    -- indicates whether this is the final round or not.
    aesInvRound :: Bool -> State -> Key -> State
    aesInvRound :: Bool -> State -> State -> State
    aesInvRound isFinal :: Bool
    isFinal s :: State
    s key :: State
    key = State
    d State -> State -> State
    `addRoundKey` State
    key
      where d :: State
    d = (Int -> SWord 32) -> [Int] -> State
    forall a b. (a -> b) -> [a] -> [b]
    map (Bool -> Int -> SWord 32
    f Bool
    isFinal) [0..3]
            a :: [[GF28]]
    a = (SWord 32 -> [GF28]) -> State -> [[GF28]]
    forall a b. (a -> b) -> [a] -> [b]
    map SWord 32 -> [GF28]
    forall a. ByteConverter a => a -> [GF28]
    toBytes State
    s
            f :: Bool -> Int -> SWord 32
    f True j :: Int
    j = [GF28] -> SWord 32
    forall a. ByteConverter a => [GF28] -> a
    fromBytes [ GF28 -> GF28
    unSBox ([[GF28]]
    a [[GF28]] -> Int -> [GF28]
    forall a. [a] -> Int -> a
    !! ((Int
    jInt -> Int -> Int
    forall a. Num a => a -> a -> a
    +0) Int -> Int -> Int
    forall a. Integral a => a -> a -> a
    `mod` 4) [GF28] -> Int -> GF28
    forall a. [a] -> Int -> a
    !! 0)
                                 , GF28 -> GF28
    unSBox ([[GF28]]
    a [[GF28]] -> Int -> [GF28]
    forall a. [a] -> Int -> a
    !! ((Int
    jInt -> Int -> Int
    forall a. Num a => a -> a -> a
    +3) Int -> Int -> Int
    forall a. Integral a => a -> a -> a
    `mod` 4) [GF28] -> Int -> GF28
    forall a. [a] -> Int -> a
    !! 1)
                                 , GF28 -> GF28
    unSBox ([[GF28]]
    a [[GF28]] -> Int -> [GF28]
    forall a. [a] -> Int -> a
    !! ((Int
    jInt -> Int -> Int
    forall a. Num a => a -> a -> a
    +2) Int -> Int -> Int
    forall a. Integral a => a -> a -> a
    `mod` 4) [GF28] -> Int -> GF28
    forall a. [a] -> Int -> a
    !! 2)
                                 , GF28 -> GF28
    unSBox ([[GF28]]
    a [[GF28]] -> Int -> [GF28]
    forall a. [a] -> Int -> a
    !! ((Int
    jInt -> Int -> Int
    forall a. Num a => a -> a -> a
    +1) Int -> Int -> Int
    forall a. Integral a => a -> a -> a
    `mod` 4) [GF28] -> Int -> GF28
    forall a. [a] -> Int -> a
    !! 3)
                                 ]
            f False j :: Int
    j = SWord 32
    e0 SWord 32 -> SWord 32 -> SWord 32
    forall a. Bits a => a -> a -> a
    `xor` SWord 32
    e1 SWord 32 -> SWord 32 -> SWord 32
    forall a. Bits a => a -> a -> a
    `xor` SWord 32
    e2 SWord 32 -> SWord 32 -> SWord 32
    forall a. Bits a => a -> a -> a
    `xor` SWord 32
    e3
                  where e0 :: SWord 32
    e0 = GF28 -> SWord 32
    u0 ([[GF28]]
    a [[GF28]] -> Int -> [GF28]
    forall a. [a] -> Int -> a
    !! ((Int
    jInt -> Int -> Int
    forall a. Num a => a -> a -> a
    +0) Int -> Int -> Int
    forall a. Integral a => a -> a -> a
    `mod` 4) [GF28] -> Int -> GF28
    forall a. [a] -> Int -> a
    !! 0)
                        e1 :: SWord 32
    e1 = GF28 -> SWord 32
    u1 ([[GF28]]
    a [[GF28]] -> Int -> [GF28]
    forall a. [a] -> Int -> a
    !! ((Int
    jInt -> Int -> Int
    forall a. Num a => a -> a -> a
    +3) Int -> Int -> Int
    forall a. Integral a => a -> a -> a
    `mod` 4) [GF28] -> Int -> GF28
    forall a. [a] -> Int -> a
    !! 1)
                        e2 :: SWord 32
    e2 = GF28 -> SWord 32
    u2 ([[GF28]]
    a [[GF28]] -> Int -> [GF28]
    forall a. [a] -> Int -> a
    !! ((Int
    jInt -> Int -> Int
    forall a. Num a => a -> a -> a
    +2) Int -> Int -> Int
    forall a. Integral a => a -> a -> a
    `mod` 4) [GF28] -> Int -> GF28
    forall a. [a] -> Int -> a
    !! 2)
                        e3 :: SWord 32
    e3 = GF28 -> SWord 32
    u3 ([[GF28]]
    a [[GF28]] -> Int -> [GF28]
    forall a. [a] -> Int -> a
    !! ((Int
    jInt -> Int -> Int
    forall a. Num a => a -> a -> a
    +1) Int -> Int -> Int
    forall a. Integral a => a -> a -> a
    `mod` 4) [GF28] -> Int -> GF28
    forall a. [a] -> Int -> a
    !! 3)

    -----------------------------------------------------------------------------
    -- * AES API
    -----------------------------------------------------------------------------

    -- | Key schedule. Given a 128, 192, or 256 bit key, expand it to get key-schedules
    -- for encryption and decryption. The key is given as a sequence of 32-bit words.
    -- (4 elements for 128-bits, 6 for 192, and 8 for 256.)
    aesKeySchedule :: Key -> (KS, KS)
    aesKeySchedule :: State -> (KS, KS)
    aesKeySchedule key :: State
    key
      | Int
    nk Int -> [Int] -> Bool
    forall (t :: * -> *) a. (Foldable t, Eq a) => a -> t a -> Bool
    `elem` [4, 6, 8]
      = (KS
    encKS, KS
    decKS)
      | Bool
    True
      = [Char] -> (KS, KS)
    forall a. HasCallStack => [Char] -> a
    error "aesKeySchedule: Invalid key size"
      where nk :: Int
    nk = State -> Int
    forall (t :: * -> *) a. Foldable t => t a -> Int
    length State
    key
            nr :: Int
    nr = Int
    nk Int -> Int -> Int
    forall a. Num a => a -> a -> a
    + 6
            encKS :: KS
    encKS@(f :: State
    f, m :: [State]
    m, l :: State
    l) = ([State] -> State
    forall a. [a] -> a
    head [State]
    rKeys, Int -> [State] -> [State]
    forall a. Int -> [a] -> [a]
    take (Int
    nrInt -> Int -> Int
    forall a. Num a => a -> a -> a
    -1) ([State] -> [State]
    forall a. [a] -> [a]
    tail [State]
    rKeys), [State]
    rKeys [State] -> Int -> State
    forall a. [a] -> Int -> a
    !! Int
    nr)
            decKS :: KS
    decKS = (State
    l, (State -> State) -> [State] -> [State]
    forall a b. (a -> b) -> [a] -> [b]
    map State -> State
    invMixColumns ([State] -> [State]
    forall a. [a] -> [a]
    reverse [State]
    m), State
    f)
            rKeys :: [State]
    rKeys = Int -> State -> [State]
    keyExpansion Int
    nk State
    key

    -- | Block encryption. The first argument is the plain-text, which must have
    -- precisely 4 elements, for a total of 128-bits of input. The second
    -- argument is the key-schedule to be used, obtained by a call to 'aesKeySchedule'.
    -- The output will always have 4 32-bit words, which is the cipher-text.
    aesEncrypt :: [SWord 32] -> KS -> [SWord 32]
    aesEncrypt :: State -> KS -> State
    aesEncrypt pt :: State
    pt encKS :: KS
    encKS
      | State -> Int
    forall (t :: * -> *) a. Foldable t => t a -> Int
    length State
    pt Int -> Int -> Bool
    forall a. Eq a => a -> a -> Bool
    == 4
      = (Bool -> State -> State -> State) -> KS -> State -> State
    doRounds Bool -> State -> State -> State
    aesRound KS
    encKS State
    pt
      | Bool
    True
      = [Char] -> State
    forall a. HasCallStack => [Char] -> a
    error "aesEncrypt: Invalid plain-text size"

    -- | Block decryption. The arguments are the same as in 'aesEncrypt', except
    -- the first argument is the cipher-text and the output is the corresponding
    -- plain-text.
    aesDecrypt :: [SWord 32] -> KS -> [SWord 32]
    aesDecrypt :: State -> KS -> State
    aesDecrypt ct :: State
    ct decKS :: KS
    decKS
      | State -> Int
    forall (t :: * -> *) a. Foldable t => t a -> Int
    length State
    ct Int -> Int -> Bool
    forall a. Eq a => a -> a -> Bool
    == 4
      = (Bool -> State -> State -> State) -> KS -> State -> State
    doRounds Bool -> State -> State -> State
    aesInvRound KS
    decKS State
    ct
      | Bool
    True
      = [Char] -> State
    forall a. HasCallStack => [Char] -> a
    error "aesDecrypt: Invalid cipher-text size"

    -----------------------------------------------------------------------------
    -- * Test vectors
    -----------------------------------------------------------------------------

    -----------------------------------------------------------------------------
    -- ** 128-bit enc/dec test
    -----------------------------------------------------------------------------

    -- | 128-bit encryption test, from Appendix C.1 of the AES standard:
    --
    -- >>> map hex8 t128Enc
    -- ["69c4e0d8","6a7b0430","d8cdb780","70b4c55a"]
    --
    t128Enc :: [SWord 32]
    t128Enc :: State
    t128Enc = State -> KS -> State
    aesEncrypt State
    pt KS
    ks
      where pt :: State
    pt  = [0x00112233, 0x44556677, 0x8899aabb, 0xccddeeff]
            key :: State
    key = [0x00010203, 0x04050607, 0x08090a0b, 0x0c0d0e0f]
            (ks :: KS
    ks, _) = State -> (KS, KS)
    aesKeySchedule State
    key

    -- | 128-bit decryption test, from Appendix C.1 of the AES standard:
    --
    -- >>> map hex8 t128Dec
    -- ["00112233","44556677","8899aabb","ccddeeff"]
    --
    t128Dec :: [SWord 32]
    t128Dec :: State
    t128Dec = State -> KS -> State
    aesDecrypt State
    ct KS
    ks
      where ct :: State
    ct  = [0x69c4e0d8, 0x6a7b0430, 0xd8cdb780, 0x70b4c55a]
            key :: State
    key = [0x00010203, 0x04050607, 0x08090a0b, 0x0c0d0e0f]
            (_, ks :: KS
    ks) = State -> (KS, KS)
    aesKeySchedule State
    key

    -----------------------------------------------------------------------------
    -- ** 192-bit enc/dec test
    -----------------------------------------------------------------------------

    -- | 192-bit encryption test, from Appendix C.2 of the AES standard:
    --
    -- >>> map hex8 t192Enc
    -- ["dda97ca4","864cdfe0","6eaf70a0","ec0d7191"]
    --
    t192Enc :: [SWord 32]
    t192Enc :: State
    t192Enc = State -> KS -> State
    aesEncrypt State
    pt KS
    ks
      where pt :: State
    pt  = [0x00112233, 0x44556677, 0x8899aabb, 0xccddeeff]
            key :: State
    key = [0x00010203, 0x04050607, 0x08090a0b, 0x0c0d0e0f, 0x10111213, 0x14151617]
            (ks :: KS
    ks, _) = State -> (KS, KS)
    aesKeySchedule State
    key

    -- | 192-bit decryption test, from Appendix C.2 of the AES standard:
    --
    -- >>> map hex8 t192Dec
    -- ["00112233","44556677","8899aabb","ccddeeff"]
    --
    t192Dec :: [SWord 32]
    t192Dec :: State
    t192Dec = State -> KS -> State
    aesDecrypt State
    ct KS
    ks
      where ct :: State
    ct  = [0xdda97ca4, 0x864cdfe0, 0x6eaf70a0, 0xec0d7191]
            key :: State
    key = [0x00010203, 0x04050607, 0x08090a0b, 0x0c0d0e0f, 0x10111213, 0x14151617]
            (_, ks :: KS
    ks) = State -> (KS, KS)
    aesKeySchedule State
    key

    -----------------------------------------------------------------------------
    -- ** 256-bit enc/dec test
    -----------------------------------------------------------------------------

    -- | 256-bit encryption, from Appendix C.3 of the AES standard:
    --
    -- >>> map hex8 t256Enc
    -- ["8ea2b7ca","516745bf","eafc4990","4b496089"]
    --
    t256Enc :: [SWord 32]
    t256Enc :: State
    t256Enc = State -> KS -> State
    aesEncrypt State
    pt KS
    ks
      where pt :: State
    pt  = [0x00112233, 0x44556677, 0x8899aabb, 0xccddeeff]
            key :: State
    key = [0x00010203, 0x04050607, 0x08090a0b, 0x0c0d0e0f, 0x10111213, 0x14151617, 0x18191a1b, 0x1c1d1e1f]
            (ks :: KS
    ks, _) = State -> (KS, KS)
    aesKeySchedule State
    key

    -- | 256-bit decryption, from Appendix C.3 of the AES standard:
    --
    -- >>> map hex8 t256Dec
    -- ["00112233","44556677","8899aabb","ccddeeff"]
    --
    t256Dec :: [SWord 32]
    t256Dec :: State
    t256Dec = State -> KS -> State
    aesDecrypt State
    ct KS
    ks
      where ct :: State
    ct  = [0x8ea2b7ca, 0x516745bf, 0xeafc4990, 0x4b496089]
            key :: State
    key = [0x00010203, 0x04050607, 0x08090a0b, 0x0c0d0e0f, 0x10111213, 0x14151617, 0x18191a1b, 0x1c1d1e1f]
            (_, ks :: KS
    ks) = State -> (KS, KS)
    aesKeySchedule State
    key

    -----------------------------------------------------------------------------
    -- * Verification
    -- ${verifIntro}
    -----------------------------------------------------------------------------
    {- $verifIntro
      While SMT based technologies can prove correct many small properties fairly quickly, it would
      be naive for them to automatically verify that our AES implementation is correct. (By correct,
      we mean decryption follewed by encryption yielding the same result.) However, we can state
      this property precisely using SBV, and use quick-check to gain some confidence.
    -}

    -- | Correctness theorem for 128-bit AES. Ideally, we would run:
    --
    -- @
    --   prove aes128IsCorrect
    -- @
    --
    -- to get a proof automatically. Unfortunately, while SBV will successfully generate the proof
    -- obligation for this theorem and ship it to the SMT solver, it would be naive to expect the SMT-solver
    -- to finish that proof in any reasonable time with the currently available SMT solving technologies.
    -- Instead, we can issue:
    --
    -- @
    --   quickCheck aes128IsCorrect
    -- @
    --
    -- and get some degree of confidence in our code. Similar predicates can be easily constructed for 192, and
    -- 256 bit cases as well.
    aes128IsCorrect :: (SWord 32, SWord 32, SWord 32, SWord 32)  -- ^ plain-text words
                    -> (SWord 32, SWord 32, SWord 32, SWord 32)  -- ^ key-words
                    -> SBool                                 -- ^ True if round-trip gives us plain-text back
    aes128IsCorrect :: (SWord 32, SWord 32, SWord 32, SWord 32)
    -> (SWord 32, SWord 32, SWord 32, SWord 32) -> SBool
    aes128IsCorrect (i0 :: SWord 32
    i0, i1 :: SWord 32
    i1, i2 :: SWord 32
    i2, i3 :: SWord 32
    i3) (k0 :: SWord 32
    k0, k1 :: SWord 32
    k1, k2 :: SWord 32
    k2, k3 :: SWord 32
    k3) = State
    pt State -> State -> SBool
    forall a. EqSymbolic a => a -> a -> SBool
    .== State
    pt'
       where pt :: State
    pt  = [SWord 32
    i0, SWord 32
    i1, SWord 32
    i2, SWord 32
    i3]
             key :: State
    key = [SWord 32
    k0, SWord 32
    k1, SWord 32
    k2, SWord 32
    k3]
             (encKS :: KS
    encKS, decKS :: KS
    decKS) = State -> (KS, KS)
    aesKeySchedule State
    key
             ct :: State
    ct  = State -> KS -> State
    aesEncrypt State
    pt KS
    encKS
             pt' :: State
    pt' = State -> KS -> State
    aesDecrypt State
    ct KS
    decKS

    -----------------------------------------------------------------------------
    -- * Code generation
    -- ${codeGenIntro}
    -----------------------------------------------------------------------------
    {- $codeGenIntro
       We have emphasized that our T-Box implementation in Haskell was guided by clarity and correctness, not
       performance. Indeed, our implementation is hardly the fastest AES implementation in Haskell. However,
       we can use it to automatically generate straight-line C-code that can run fairly fast.

       For the purposes of illustration, we only show here how to generate code for a 128-bit AES block-encrypt
       function, that takes 8 32-bit words as an argument. The first 4 are the 128-bit input, and the final
       four are the 128-bit key. The impact of this is that the generated function would expand the key for
       each block of encryption, a needless task unless we change the key in every block. In a more serios application,
       we would instead generate code for both the 'aesKeySchedule' and the 'aesEncrypt' functions, thus reusing the
       key-schedule over many applications of the encryption call. (Unfortunately doing this is rather cumbersome right
       now, since Haskell does not support fixed-size lists.)
    -}

    -- | Code generation for 128-bit AES encryption.
    --
    -- The following sample from the generated code-lines show how T-Boxes are rendered as C arrays:
    --
    -- @
    --   static const SWord32 table1[] = {
    --       0xc66363a5UL, 0xf87c7c84UL, 0xee777799UL, 0xf67b7b8dUL,
    --       0xfff2f20dUL, 0xd66b6bbdUL, 0xde6f6fb1UL, 0x91c5c554UL,
    --       0x60303050UL, 0x02010103UL, 0xce6767a9UL, 0x562b2b7dUL,
    --       0xe7fefe19UL, 0xb5d7d762UL, 0x4dababe6UL, 0xec76769aUL,
    --       ...
    --       }
    -- @
    --
    -- The generated program has 5 tables (one sbox table, and 4-Tboxes), all converted to fast C arrays. Here
    -- is a sample of the generated straightline C-code:
    --
    -- @
    --   const SWord8  s1915 = (SWord8) s1912;
    --   const SWord8  s1916 = table0[s1915];
    --   const SWord16 s1917 = (((SWord16) s1914) << 8) | ((SWord16) s1916);
    --   const SWord32 s1918 = (((SWord32) s1911) << 16) | ((SWord32) s1917);
    --   const SWord32 s1919 = s1844 ^ s1918;
    --   const SWord32 s1920 = s1903 ^ s1919;
    -- @
    --
    -- The GNU C-compiler does a fine job of optimizing this straightline code to generate a fairly efficient C implementation.
    cgAES128BlockEncrypt :: IO ()
    cgAES128BlockEncrypt :: IO ()
    cgAES128BlockEncrypt = Maybe [Char] -> [Char] -> SBVCodeGen () -> IO ()
    forall a. Maybe [Char] -> [Char] -> SBVCodeGen a -> IO a
    compileToC Maybe [Char]
    forall a. Maybe a
    Nothing "aes128BlockEncrypt" (SBVCodeGen () -> IO ()) -> SBVCodeGen () -> IO ()
    forall a b. (a -> b) -> a -> b
    $ do
            State
    pt  <- Int -> [Char] -> SBVCodeGen State
    forall a. SymVal a => Int -> [Char] -> SBVCodeGen [SBV a]
    cgInputArr 4 "pt"        -- plain-text as an array of 4 Word32's
            State
    key <- Int -> [Char] -> SBVCodeGen State
    forall a. SymVal a => Int -> [Char] -> SBVCodeGen [SBV a]
    cgInputArr 4 "key"       -- key as an array of 4 Word32s
            -- Use the test values from Appendix C.1 of the AES standard as the driver values
            [Integer] -> SBVCodeGen ()
    cgSetDriverValues ([Integer] -> SBVCodeGen ()) -> [Integer] -> SBVCodeGen ()
    forall a b. (a -> b) -> a -> b
    $    [0x00112233, 0x44556677, 0x8899aabb, 0xccddeeff]
                                [Integer] -> [Integer] -> [Integer]
    forall a. [a] -> [a] -> [a]
    ++ [0x00010203, 0x04050607, 0x08090a0b, 0x0c0d0e0f]
            let (encKs :: KS
    encKs, _) = State -> (KS, KS)
    aesKeySchedule State
    key
            [Char] -> State -> SBVCodeGen ()
    forall a. SymVal a => [Char] -> [SBV a] -> SBVCodeGen ()
    cgOutputArr "ct" (State -> SBVCodeGen ()) -> State -> SBVCodeGen ()
    forall a b. (a -> b) -> a -> b
    $ State -> KS -> State
    aesEncrypt State
    pt KS
    encKs

    -----------------------------------------------------------------------------
    -- * C-library generation
    -- ${libraryIntro}
    -----------------------------------------------------------------------------
    {- $libraryIntro
       The 'cgAES128BlockEncrypt' example shows how to generate code for 128-bit AES encryption. As the generated
       function performs encryption on a given block, it performs key expansion as necessary. However, this is
       not quite practical: We would like to expand the key only once, and encrypt the stream of plain-text blocks using
       the same expanded key (potentially using some crypto-mode), until we decide to change the key. In this
       section, we show how to use SBV to instead generate a library of functions that can be used in such a scenario.
       The generated library is a typical @.a@ archive, that can be linked using the C-compiler as usual.
    -}

    -- | Components of the AES implementation that the library is generated from
    aesLibComponents :: Int -> [(String, SBVCodeGen ())]
    aesLibComponents :: Int -> [([Char], SBVCodeGen ())]
    aesLibComponents sz :: Int
    sz = [ ("aes" [Char] -> [Char] -> [Char]
    forall a. [a] -> [a] -> [a]
    ++ Int -> [Char]
    forall a. Show a => a -> [Char]
    show Int
    sz [Char] -> [Char] -> [Char]
    forall a. [a] -> [a] -> [a]
    ++ "KeySchedule",  SBVCodeGen ()
    keySchedule)
                          , ("aes" [Char] -> [Char] -> [Char]
    forall a. [a] -> [a] -> [a]
    ++ Int -> [Char]
    forall a. Show a => a -> [Char]
    show Int
    sz [Char] -> [Char] -> [Char]
    forall a. [a] -> [a] -> [a]
    ++ "BlockEncrypt", SBVCodeGen ()
    enc)
                          , ("aes" [Char] -> [Char] -> [Char]
    forall a. [a] -> [a] -> [a]
    ++ Int -> [Char]
    forall a. Show a => a -> [Char]
    show Int
    sz [Char] -> [Char] -> [Char]
    forall a. [a] -> [a] -> [a]
    ++ "BlockDecrypt", SBVCodeGen ()
    dec)
                          ]
      where -- key-schedule
            nk :: Int
    nk
             | Int
    sz Int -> Int -> Bool
    forall a. Eq a => a -> a -> Bool
    == 128 = 4
             | Int
    sz Int -> Int -> Bool
    forall a. Eq a => a -> a -> Bool
    == 192 = 6
             | Int
    sz Int -> Int -> Bool
    forall a. Eq a => a -> a -> Bool
    == 256 = 8
             | Bool
    True      = [Char] -> Int
    forall a. HasCallStack => [Char] -> a
    error ([Char] -> Int) -> [Char] -> Int
    forall a b. (a -> b) -> a -> b
    $ "aesLibComponents: Size must be one of 128, 192, or 256; received: " [Char] -> [Char] -> [Char]
    forall a. [a] -> [a] -> [a]
    ++ Int -> [Char]
    forall a. Show a => a -> [Char]
    show Int
    sz
            -- We get 4*(nr+1) keys, where nr = nk + 6
            nr :: Int
    nr = Int
    nk Int -> Int -> Int
    forall a. Num a => a -> a -> a
    + 6
            xk :: Int
    xk = 4 Int -> Int -> Int
    forall a. Num a => a -> a -> a
    * (Int
    nr Int -> Int -> Int
    forall a. Num a => a -> a -> a
    + 1)
            keySchedule :: SBVCodeGen ()
    keySchedule = do State
    key <- Int -> [Char] -> SBVCodeGen State
    forall a. SymVal a => Int -> [Char] -> SBVCodeGen [SBV a]
    cgInputArr Int
    nk "key"     -- key
                             let (encKS :: KS
    encKS, decKS :: KS
    decKS) = State -> (KS, KS)
    aesKeySchedule State
    key
                             [Char] -> State -> SBVCodeGen ()
    forall a. SymVal a => [Char] -> [SBV a] -> SBVCodeGen ()
    cgOutputArr "encKS" (KS -> State
    ksToXKey KS
    encKS)
                             [Char] -> State -> SBVCodeGen ()
    forall a. SymVal a => [Char] -> [SBV a] -> SBVCodeGen ()
    cgOutputArr "decKS" (KS -> State
    ksToXKey KS
    decKS)
            -- encryption
            enc :: SBVCodeGen ()
    enc = do State
    pt   <- Int -> [Char] -> SBVCodeGen State
    forall a. SymVal a => Int -> [Char] -> SBVCodeGen [SBV a]
    cgInputArr 4  "pt"    -- plain-text
                     State
    xkey <- Int -> [Char] -> SBVCodeGen State
    forall a. SymVal a => Int -> [Char] -> SBVCodeGen [SBV a]
    cgInputArr Int
    xk "xkey"  -- expanded key
                     [Char] -> State -> SBVCodeGen ()
    forall a. SymVal a => [Char] -> [SBV a] -> SBVCodeGen ()
    cgOutputArr "ct" (State -> SBVCodeGen ()) -> State -> SBVCodeGen ()
    forall a b. (a -> b) -> a -> b
    $ State -> KS -> State
    aesEncrypt State
    pt (State -> KS
    xkeyToKS State
    xkey)
            -- decryption
            dec :: SBVCodeGen ()
    dec = do State
    pt   <- Int -> [Char] -> SBVCodeGen State
    forall a. SymVal a => Int -> [Char] -> SBVCodeGen [SBV a]
    cgInputArr 4  "ct"    -- cipher-text
                     State
    xkey <- Int -> [Char] -> SBVCodeGen State
    forall a. SymVal a => Int -> [Char] -> SBVCodeGen [SBV a]
    cgInputArr Int
    xk "xkey"  -- expanded key
                     [Char] -> State -> SBVCodeGen ()
    forall a. SymVal a => [Char] -> [SBV a] -> SBVCodeGen ()
    cgOutputArr "pt" (State -> SBVCodeGen ()) -> State -> SBVCodeGen ()
    forall a b. (a -> b) -> a -> b
    $ State -> KS -> State
    aesDecrypt State
    pt (State -> KS
    xkeyToKS State
    xkey)
            -- Transforming back and forth from our KS type to a flat array used by the generated C code
            -- Turn a series of expanded keys to our internal KS type
            xkeyToKS :: [SWord 32] -> KS
            xkeyToKS :: State -> KS
    xkeyToKS xs :: State
    xs = (State
    f, [State]
    m, State
    l)
               where f :: State
    f  = Int -> State -> State
    forall a. Int -> [a] -> [a]
    take 4 State
    xs                             -- first round key
                     m :: [State]
    m  = State -> [State]
    forall a. [a] -> [[a]]
    chop4 (Int -> State -> State
    forall a. Int -> [a] -> [a]
    take (Int
    xk Int -> Int -> Int
    forall a. Num a => a -> a -> a
    - 8) (Int -> State -> State
    forall a. Int -> [a] -> [a]
    drop 4 State
    xs))     -- middle rounds
                     l :: State
    l  = Int -> State -> State
    forall a. Int -> [a] -> [a]
    drop (Int
    xk Int -> Int -> Int
    forall a. Num a => a -> a -> a
    - 4) State
    xs                      -- last round key
            -- Turn a KS to a series of expanded key words
            ksToXKey :: KS -> [SWord 32]
            ksToXKey :: KS -> State
    ksToXKey (f :: State
    f, m :: [State]
    m, l :: State
    l) = State
    f State -> State -> State
    forall a. [a] -> [a] -> [a]
    ++ [State] -> State
    forall (t :: * -> *) a. Foldable t => t [a] -> [a]
    concat [State]
    m State -> State -> State
    forall a. [a] -> [a] -> [a]
    ++ State
    l
            -- chunk in fours. (This function must be in some standard library, where?)
            chop4 :: [a] -> [[a]]
            chop4 :: [a] -> [[a]]
    chop4 [] = []
            chop4 xs :: [a]
    xs = let (f :: [a]
    f, r :: [a]
    r) = Int -> [a] -> ([a], [a])
    forall a. Int -> [a] -> ([a], [a])
    splitAt 4 [a]
    xs in [a]
    f [a] -> [[a]] -> [[a]]
    forall a. a -> [a] -> [a]
    : [a] -> [[a]]
    forall a. [a] -> [[a]]
    chop4 [a]
    r

    -- | Generate code for AES functionality; given the key size.
    cgAESLibrary :: Int -> Maybe FilePath -> IO ()
    cgAESLibrary :: Int -> Maybe [Char] -> IO ()
    cgAESLibrary sz :: Int
    sz mbd :: Maybe [Char]
    mbd
      | Int
    sz Int -> [Int] -> Bool
    forall (t :: * -> *) a. (Foldable t, Eq a) => a -> t a -> Bool
    `elem` [128, 192, 256] = IO [()] -> IO ()
    forall (f :: * -> *) a. Functor f => f a -> f ()
    void (IO [()] -> IO ()) -> IO [()] -> IO ()
    forall a b. (a -> b) -> a -> b
    $ Maybe [Char] -> [Char] -> [([Char], SBVCodeGen ())] -> IO [()]
    forall a.
    Maybe [Char] -> [Char] -> [([Char], SBVCodeGen a)] -> IO [a]
    compileToCLib Maybe [Char]
    mbd [Char]
    nm (Int -> [([Char], SBVCodeGen ())]
    aesLibComponents Int
    sz)
      | Bool
    True                      = [Char] -> IO ()
    forall a. HasCallStack => [Char] -> a
    error ([Char] -> IO ()) -> [Char] -> IO ()
    forall a b. (a -> b) -> a -> b
    $ "cgAESLibrary: Size must be one of 128, 192, or 256, received: " [Char] -> [Char] -> [Char]
    forall a. [a] -> [a] -> [a]
    ++ Int -> [Char]
    forall a. Show a => a -> [Char]
    show Int
    sz
      where nm :: [Char]
    nm = "aes" [Char] -> [Char] -> [Char]
    forall a. [a] -> [a] -> [a]
    ++ Int -> [Char]
    forall a. Show a => a -> [Char]
    show Int
    sz [Char] -> [Char] -> [Char]
    forall a. [a] -> [a] -> [a]
    ++ "Lib"

    -- | Generate a C library, containing functions for performing 128-bit enc/dec/key-expansion.
    -- A note on performance: In a very rough speed test, the generated code was able to do
    -- 6.3 million block encryptions per second on a decent MacBook Pro. On the same machine, OpenSSL
    -- reports 8.2 million block encryptions per second. So, the generated code is about 25% slower
    -- as compared to the highly optimized OpenSSL implementation. (Note that the speed test was done
    -- somewhat simplistically, so these numbers should be considered very rough estimates.)
    cgAES128Library :: IO ()
    cgAES128Library :: IO ()
    cgAES128Library = Int -> Maybe [Char] -> IO ()
    cgAESLibrary 128 Maybe [Char]
    forall a. Maybe a
    Nothing

    --------------------------------------------------------------------------------------------
    -- | For doctest purposes only
    hex8 :: (SymVal a, Show a, Integral a) => SBV a -> String
    hex8 :: SBV a -> [Char]
    hex8 v :: SBV a
    v = Int -> Char -> [Char]
    forall a. Int -> a -> [a]
    replicate (8 Int -> Int -> Int
    forall a. Num a => a -> a -> a
    - [Char] -> Int
    forall (t :: * -> *) a. Foldable t => t a -> Int
    length [Char]
    s) '0' [Char] -> [Char] -> [Char]
    forall a. [a] -> [a] -> [a]
    ++ [Char]
    s
      where s :: [Char]
    s = (a -> [Char] -> [Char]) -> [Char] -> a -> [Char]
    forall a b c. (a -> b -> c) -> b -> a -> c
    flip a -> [Char] -> [Char]
    forall a. (Integral a, Show a) => a -> [Char] -> [Char]
    showHex "" (a -> [Char]) -> (SBV a -> a) -> SBV a -> [Char]
    forall b c a. (b -> c) -> (a -> b) -> a -> c
    . Maybe a -> a
    forall a. HasCallStack => Maybe a -> a
    fromJust (Maybe a -> a) -> (SBV a -> Maybe a) -> SBV a -> a
    forall b c a. (b -> c) -> (a -> b) -> a -> c
    . SBV a -> Maybe a
    forall a. SymVal a => SBV a -> Maybe a
    unliteral (SBV a -> [Char]) -> SBV a -> [Char]
    forall a b. (a -> b) -> a -> b
    $ SBV a
    v

    {-# ANN aesRound    ("HLint: ignore Use head" :: String) #-}
    {-# ANN aesInvRound ("HLint: ignore Use head" :: String) #-}