Base implementation of signed numbers and EGCD, with tests.

This commit is contained in:
2018-10-15 19:16:25 -07:00
parent d43f0bcd42
commit 89e2dfc6ef
164 changed files with 560116 additions and 67 deletions

View File

@@ -1,8 +1,10 @@
{-# LANGUAGE RecordWildCards #-}
import Control.Exception(assert)
import Control.Monad(foldM_,forM_,when)
import Data.Bits(shiftL,shiftR)
import Data.List(sort)
import qualified Data.Map.Strict as Map
import GHC.Integer.GMP.Internals(powModInteger)
import GHC.Integer.GMP.Internals(powModInteger,recipModInteger)
import Numeric(showHex)
import Prelude hiding (log)
import System.Directory(createDirectoryIfMissing)
@@ -11,6 +13,8 @@ import System.FilePath((</>))
import System.IO(Handle,IOMode(WriteMode),hPutStrLn,withFile,hFlush,hPutStr,stderr)
import System.Random(StdGen,newStdGen,random,split)
import Debug.Trace
data Operation = Add
| BaseOps
| Barretts
@@ -23,6 +27,14 @@ data Operation = Add
| Square
| Sub
| Convert Int
| SignedAdd
| SignedBase
| SignedCmp
| SignedShift
| SignedSub
| SigConvert Int
| EGCD
| ModInv
deriving (Eq, Ord, Show)
data Requirement = Req Int Operation
@@ -31,44 +43,64 @@ data Requirement = Req Int Operation
data Need = Need Operation (Int -> [Requirement])
needs :: [Need]
needs = [ Need ModExp (\ size -> [Req size ModMul
,Req size ModSq
,Req size Barretts])
, Need ModSq (\ size -> [Req (size * 2) Div
,Req size Barretts
,Req size Square])
, Need ModMul (\ size -> [Req size Mul
,Req size Barretts
,Req size (Convert (size * 2))
,Req (size * 2) Div])
, Need Barretts (\ size -> [Req (size + 64) BaseOps
,Req size (Convert (size + 64))
,Req (size + 64) (Convert ((size * 2) + 64))
,Req size (Convert ((size * 2) + 64))
,Req ((size * 2) + 64) Add
,Req ((size * 2) + 64) Sub
,Req (size + 64) Mul
,Req (size * 2) (Convert ((size * 2) + 64))
,Req ((size * 2) + 64) Shifts
,Req ((size * 2) + 128) Shifts
,Req ((size * 2) + 64) Div
,Req (size + 64) (Convert (size * 2))
,Req (size + 64) (Convert ((size * 2) + 128))
,Req ((size * 2) + 64)
(Convert ((size * 2) + 128))
])
, Need Div (\ size -> [Req size (Convert (size * 2))
,Req 192 BaseOps
,Req 384 BaseOps
,Req 192 Mul
,Req size Mul
,Req size Shifts
,Req (size * 2) Sub
])
, Need Mul (\ size -> [Req (size * 2) BaseOps])
, Need Sub (\ size -> [Req size Add])
, Need Add (\ size -> [Req (size + 64) BaseOps
,Req size (Convert (size + 64))])
needs = [ Need ModExp (\ size -> [Req size ModMul
,Req size ModSq
,Req size Barretts])
, Need ModSq (\ size -> [Req (size * 2) Div
,Req size Barretts
,Req size Square])
, Need ModMul (\ size -> [Req size Mul
,Req size Barretts
,Req size (Convert (size * 2))
,Req (size * 2) Div])
, Need Barretts (\ size -> [Req (size + 64) BaseOps
,Req size (Convert (size + 64))
,Req (size + 64) (Convert ((size * 2) + 64))
,Req size (Convert ((size * 2) + 64))
,Req ((size * 2) + 64) Add
,Req ((size * 2) + 64) Sub
,Req (size + 64) Mul
,Req (size * 2) (Convert ((size * 2) + 64))
,Req ((size * 2) + 64) Shifts
,Req ((size * 2) + 128) Shifts
,Req ((size * 2) + 64) Div
,Req (size + 64) (Convert (size * 2))
,Req (size + 64) (Convert ((size * 2) + 128))
,Req ((size * 2) + 64)
(Convert ((size * 2) + 128))
])
, Need Div (\ size -> [Req size (Convert (size * 2))
,Req 192 BaseOps
,Req 384 BaseOps
,Req 192 Mul
,Req size Mul
,Req size Shifts
,Req (size * 2) Sub
])
, Need Mul (\ size -> [Req (size * 2) BaseOps])
, Need Sub (\ size -> [Req size Add])
, Need Add (\ size -> [Req (size + 64) BaseOps
,Req size (Convert (size + 64))])
, Need ModInv (\ size -> [Req size SignedBase,
Req size EGCD])
, Need EGCD (\ size -> [Req size BaseOps,
Req (size + 64) SignedBase,
Req size (SigConvert (size + 64)),
Req (size + 64) SignedShift,
Req (size + 64) SignedAdd,
Req (size + 64) SignedSub,
Req (size + 64) SignedCmp
])
, Need SignedShift (\ size -> [Req size Shifts, Req size Add])
, Need SignedAdd (\ size -> [Req size Sub,
Req (size + 64) Add,
Req (size + 64) SignedBase,
Req size (SigConvert (size + 64))
])
, Need SignedSub (\ size -> [Req (size + 64) SignedBase,
Req size (SigConvert (size + 64)),
Req size Sub
])
]
newRequirements :: Requirement -> [Requirement]
@@ -81,7 +113,7 @@ bitSizes :: [Int]
bitSizes = [192,256,384,512,576,1024,2048,3072,4096,7680,8192,15360]
baseRequirements :: [Requirement]
baseRequirements = map (\ x -> Req x ModExp) bitSizes
baseRequirements = concatMap (\ x -> [Req x ModExp, Req x ModInv]) bitSizes
requirements :: [Requirement]
requirements = go baseRequirements
@@ -123,8 +155,31 @@ generateTestBlock hndl name level useRT ignoreAt addOns =
");")
hPutStrLn hndl " }"
generateSigTestBlock :: Handle ->
String -> Operation -> Bool -> Int ->
[Int -> Int] -> [Int -> Int] ->
IO ()
generateSigTestBlock hndl name level useRT ignoreAt addOns uaddOns =
do hPutStrLn hndl (" mod " ++ name ++ " {")
when useRT $
do hPutStrLn hndl (" use super::super::*;")
hPutStrLn hndl (" use testing::run_test;")
hPutStrLn hndl ""
forM_ requirements $ \ (Req size kind) ->
when (kind == level) $
hPutStrLn hndl (" generate_" ++ name ++
"_tests!(" ++
(if size >= ignoreAt then "ignore " else "") ++
"I" ++ show size ++ ", " ++
"U" ++ show size ++ ", " ++
"i" ++ show size ++
concatMap (\ f -> ", I" ++ show (f size)) addOns ++
concatMap (\ f -> ", U" ++ show (f size)) uaddOns ++
");")
hPutStrLn hndl " }"
generateInvocs :: IO ()
generateInvocs =
generateInvocs = do
withFile "src/unsigned/invoc.rs" WriteMode $ \ hndl ->
do forM_ requirements $ \ (Req size oper) ->
case oper of
@@ -141,6 +196,7 @@ generateInvocs =
Square -> hPutStrLn hndl ("square_impls!(U" ++ show size ++ ", U" ++ show (size * 2) ++ ", " ++ show size ++ ");")
Sub -> hPutStrLn hndl ("subtraction_impls!(U" ++ show size ++ ", " ++ show (size `div` 64) ++ ");")
Convert to -> hPutStrLn hndl ("conversion_impls!(U" ++ show size ++ ", U" ++ show to ++ ");")
_ -> return ()
hPutStrLn hndl ""
hPutStrLn hndl "\n#[cfg(test)]"
hPutStrLn hndl "mod tests {"
@@ -164,18 +220,43 @@ generateInvocs =
generateTestBlock hndl "barrett_modmul" Barretts True 4000 [(+ 64)]
generateTestBlock hndl "barrett_modexp" Barretts True 1024 [(+ 64)]
hPutStrLn hndl "}"
withFile "src/signed/invoc.rs" WriteMode $ \ hndl ->
do forM_ requirements $ \ (Req size oper) ->
case oper of
SignedAdd -> hPutStrLn hndl ("add_impls!(I" ++ show size ++ ", I" ++ show (size + 64) ++ ", U" ++ show (size + 64) ++ ");")
SignedBase -> hPutStrLn hndl ("signed_impls!(I" ++ show size ++ ", U" ++ show size ++ ");")
SignedCmp -> hPutStrLn hndl ("cmp_impls!(I" ++ show size ++ ");")
SignedShift -> hPutStrLn hndl ("shift_impls!(I" ++ show size ++ ", U" ++ show size ++ ");")
SignedSub -> hPutStrLn hndl ("subtraction_impls!(I" ++ show size ++ ", I" ++ show (size + 64) ++ ", U" ++ show (size + 64) ++ ");")
EGCD -> hPutStrLn hndl ("egcd_impls!(I" ++ show (size + 64) ++ ", U" ++ show size ++ ", I" ++ show size ++ ");")
ModInv -> hPutStrLn hndl ("modinv_impls!(I" ++ show size ++ ");")
SigConvert v -> hPutStrLn hndl ("conversion_impls!(I" ++ show size ++ ", U" ++ show size ++ ", I" ++ show v ++ ", U" ++ show v ++ ");")
_ -> return ()
hPutStrLn hndl ""
hPutStrLn hndl "\n#[cfg(test)]"
hPutStrLn hndl "mod tests {"
generateSigTestBlock hndl "sigadd" SignedAdd True 16384 [(+ 64)] [(+ 64)]
generateSigTestBlock hndl "sigsub" SignedSub True 16384 [(+ 64)] [(+ 64)]
generateSigTestBlock hndl "signed" SignedBase True 90000 [] []
generateSigTestBlock hndl "sigconversion" SignedBase False 90000 [] []
generateSigTestBlock hndl "sigcmp" SignedCmp True 90000 [] []
generateSigTestBlock hndl "sigshiftl" SignedShift True 16384 [] []
generateSigTestBlock hndl "sigshiftr" SignedShift True 16384 [] []
generateSigTestBlock hndl "egcd" EGCD True 1024 [(+ 64)] [(+ 64)]
generateSigTestBlock hndl "modinv" ModInv True 9000 [] []
hPutStrLn hndl "}"
log :: String -> IO ()
log str = hPutStr stderr str >> hFlush stderr
generateTests :: Operation -> String -> Database ->
generateTests :: String -> Operation -> String -> Database ->
(Int -> Database -> (Map.Map String String, Integer, Database)) ->
IO ()
generateTests op directory init runner = do
generateTests prefix op directory init runner = do
forM_ (getSizes op requirements) $ \ size ->
do createDirectoryIfMissing True ("testdata" </> directory)
log $ "Generating " ++ show size ++ "-bit " ++ directory ++ " tests: 000%"
let dest = "testdata" </> directory </> ("U" ++ show size ++ ".tests")
let dest = "testdata" </> directory </> (prefix ++ show size ++ ".tests")
withFile dest WriteMode $ \ hndl ->
foldM_ (writer hndl size runner) init [0..numberOfTests]
log "\n"
@@ -216,11 +297,20 @@ generateNum (db, rng0) varname size =
then (x', (Map.insert varname (x':before) db, rng1))
else generateNum (db, rng1) varname size
genSign :: (Integer, Database) -> (Integer, Database)
genSign (x, (db, rng0)) =
let (n, rng1) = random rng0
in if n then (0 - x, (db, rng1)) else (x, (db, rng1))
modulate :: Integer -> Int -> Integer
modulate x size = x `mod` (2 ^ size)
modulate' :: Integer -> Int -> Integer
modulate' x size = signum x * (abs x `mod` (2 ^ size))
showX :: (Integral a, Show a) => a -> String
showX x = showHex x ""
showX x | x < 0 = "-" ++ showX (abs x)
| otherwise = showHex x ""
showB :: Bool -> String
showB False = "0"
@@ -244,7 +334,7 @@ generateAllTheTests :: IO ()
generateAllTheTests =
do gen0 <- newStdGen
let (db1, gen1) = emptyDatabase gen0
generateTests Add "add" db1 $ \ size memory0 ->
generateTests "U" Add "add" db1 $ \ size memory0 ->
let (a, memory1) = generateNum memory0 "a" size
(b, memory2) = generateNum memory1 "b" size
c = a + b
@@ -252,7 +342,7 @@ generateAllTheTests =
("c", showX c)]
in (res, c, memory2)
let (db2, gen2) = emptyDatabase gen1
generateTests Barretts "barrett_gen" db2 $ \ size memory0 ->
generateTests "U" Barretts "barrett_gen" db2 $ \ size memory0 ->
let (m, memory1) = generateNum memory0 "m" size
k = computeK m
u = barrett m
@@ -260,7 +350,7 @@ generateAllTheTests =
("u", showX u)]
in (res, u, memory1)
let (db3, gen3) = emptyDatabase gen2
generateTests Barretts "barrett_reduce" db3 $ \ size memory0 ->
generateTests "U" Barretts "barrett_reduce" db3 $ \ size memory0 ->
let (m, memory1) = generateNum memory0 "m" size
(x, memory2) = generateNum memory1 "x" (min (2 * size) (2 * k * 64))
k = computeK m
@@ -271,7 +361,7 @@ generateAllTheTests =
("r", showX r)]
in (res, r, memory2)
let (db4, gen4) = emptyDatabase gen3
generateTests BaseOps "base" db4 $ \ size memory0 ->
generateTests "U" BaseOps "base" db4 $ \ size memory0 ->
let (x, memory1) = generateNum memory0 "x" size
(m, memory2) = generateNum memory1 "m" size
m' = m `mod` (fromIntegral size `div` 64)
@@ -281,7 +371,7 @@ generateAllTheTests =
("m", showX m'), ("r", showX r)]
in (res, x, memory2)
let (db5, gen5) = emptyDatabase gen4
generateTests BaseOps "cmp" db5 $ \ size memory0 ->
generateTests "U" BaseOps "cmp" db5 $ \ size memory0 ->
let (a, memory1) = generateNum memory0 "a" size
(b, memory2) = generateNum memory1 "b" size
res = Map.fromList [("a", showX a), ("b", showX b),
@@ -289,7 +379,7 @@ generateAllTheTests =
("e", showB (a == b))]
in (res, a, memory2)
let (db6, gen6) = emptyDatabase gen5
generateTests Div "div" db6 $ \ size memory0 ->
generateTests "U" Div "div" db6 $ \ size memory0 ->
let (a, memory1) = generateNum memory0 "a" size
(b, memory2) = generateNum memory1 "b" size
q = a `div` b
@@ -298,7 +388,7 @@ generateAllTheTests =
("q", showX q), ("r", showX r)]
in (res, q, memory2)
let (db7, gen7) = emptyDatabase gen6
generateTests Mul "mul" db7 $ \ size memory0 ->
generateTests "U" Mul "mul" db7 $ \ size memory0 ->
let (a, memory1) = generateNum memory0 "a" size
(b, memory2) = generateNum memory1 "b" size
c = a * b
@@ -306,7 +396,7 @@ generateAllTheTests =
("c", showX c)]
in (res, c, memory2)
let (db8, gen8) = emptyDatabase gen7
generateTests Shifts "shiftl" db8 $ \ size memory0 ->
generateTests "U" Shifts "shiftl" db8 $ \ size memory0 ->
let (a, memory1) = generateNum memory0 "a" size
(l, memory2) = generateNum memory1 "l" size
l' = l `mod` fromIntegral (computeK a * 64)
@@ -314,7 +404,7 @@ generateAllTheTests =
res = Map.fromList [("a", showX a), ("l", showX l'), ("r", showX r)]
in (res, r, memory2)
let (db9, gen9) = emptyDatabase gen8
generateTests Shifts "shiftr" db9 $ \ size memory0 ->
generateTests "U" Shifts "shiftr" db9 $ \ size memory0 ->
let (a, memory1) = generateNum memory0 "a" size
(l, memory2) = generateNum memory1 "l" size
l' = l `mod` fromIntegral (computeK a * 64)
@@ -322,7 +412,7 @@ generateAllTheTests =
res = Map.fromList [("a", showX a), ("l", showX l'), ("r", showX r)]
in (res, l, memory2)
let (dbA, genA) = emptyDatabase gen9
generateTests Sub "sub" dbA $ \ size memory0 ->
generateTests "U" Sub "sub" dbA $ \ size memory0 ->
let (a, memory1) = generateNum memory0 "a" size
(b, memory2) = generateNum memory1 "b" size
c = modulate (a - b) size
@@ -330,7 +420,7 @@ generateAllTheTests =
("c", showX c)]
in (res, c, memory2)
let (dbB, genB) = emptyDatabase genA
generateTests ModSq "modsq" dbB $ \ size memory0 ->
generateTests "U" ModSq "modsq" dbB $ \ size memory0 ->
let (a, memory1) = generateNum memory0 "a" size
(m, memory2) = generateNum memory1 "m" size
k = computeK m
@@ -341,7 +431,7 @@ generateAllTheTests =
("k", showX k)]
in (res, c, memory2)
let (dbC, genC) = emptyDatabase genB
generateTests ModMul "modmul" dbC $ \ size memory0 ->
generateTests "U" ModMul "modmul" dbC $ \ size memory0 ->
let (a, memory1) = generateNum memory0 "a" size
(b, memory2) = generateNum memory1 "b" size
(m, memory3) = generateNum memory2 "m" size
@@ -353,7 +443,7 @@ generateAllTheTests =
("u", showX u), ("k", showX k)]
in (res, c, memory3)
let (dbD, genD) = emptyDatabase genC
generateTests ModExp "modexp" dbD $ \ size memory0 ->
generateTests "U" ModExp "modexp" dbD $ \ size memory0 ->
let (b, memory1) = generateNum memory0 "b" size
(e, memory2) = generateNum memory1 "e" size
(m, memory3) = generateNum memory2 "m" size
@@ -365,11 +455,140 @@ generateAllTheTests =
("u", showX u), ("k", showX k)]
in (res, r, memory3)
let (dbE, genE) = emptyDatabase genD
generateTests Square "square" dbE $ \ size memory0 ->
generateTests "U" Square "square" dbE $ \ size memory0 ->
let (a, memory1) = generateNum memory0 "a" size
r = modulate (a * a) (2 * size)
res = Map.fromList [("a", showX a), ("r", showX r)]
in (res, r, memory1)
--
let (dbF, genF) = emptyDatabase genE
generateTests "I" SignedAdd "sigadd" dbF $ \ size memory0 ->
let (a, memory1) = genSign (generateNum memory0 "a" size)
(b, memory2) = genSign (generateNum memory1 "b" size)
c = a + b
res = Map.fromList [("a", showX a), ("b", showX b),
("c", showX c)]
in (res, c, memory2)
let (dbG, genG) = emptyDatabase genF
generateTests "I" SignedSub "sigsub" dbG $ \ size memory0 ->
let (a, memory1) = genSign (generateNum memory0 "a" size)
(b, memory2) = genSign (generateNum memory1 "b" size)
c = a - b
res = Map.fromList [("a", showX a), ("b", showX b),
("c", showX c)]
in (res, c, memory2)
let (dbH, genH) = emptyDatabase genG
generateTests "I" SignedBase "signed" dbH $ \ size memory0 ->
let (x, memory1) = genSign (generateNum memory0 "x" size)
res = Map.fromList [("x", showX x), ("z", showB (x == 0)),
("e", showB (even x)), ("o", showB (odd x))]
in (res, x, memory1)
let (dbI, genI) = emptyDatabase genH
generateTests "I" SignedShift "sigshiftl" dbI $ \ size memory0 ->
let (a, memory1) = genSign (generateNum memory0 "a" size)
(l, memory2) = generateNum memory1 "l" size
l' = l `mod` fromIntegral (computeK (abs a) * 64)
r = modulate' (a `shiftL` fromIntegral l') size
res = Map.fromList [("a", showX a), ("l", showX l'), ("r", showX r)]
in (res, r, memory2)
let (dbJ, genJ) = emptyDatabase genI
generateTests "I" SignedShift "sigshiftr" dbJ $ \ size memory0 ->
let (a, memory1) = genSign (generateNum memory0 "a" size)
(l, memory2) = generateNum memory1 "l" size
l' = l `mod` fromIntegral (computeK (abs a) * 64)
r = modulate' (a `shiftR` fromIntegral l') size
res = Map.fromList [("a", showX a), ("l", showX l'), ("r", showX r)]
in (res, l, memory2)
let (dbK, genK) = emptyDatabase genJ
generateTests "I" SignedCmp "sigcmp" dbK $ \ size memory0 ->
let (a, memory1) = genSign (generateNum memory0 "a" size)
(b, memory2) = genSign (generateNum memory1 "b" size)
res = Map.fromList [("a", showX a), ("b", showX b),
("g", showB (a > b)), ("l", showB (a < b)),
("e", showB (a == b))]
in (res, a, memory2)
let (dbL, genL) = emptyDatabase genK
generateTests "I" EGCD "egcd" dbL $ \ size memory0 ->
let (x, memory1) = generateNum memory0 "x" size
(y, memory2) = generateNum memory1 "y" size
ans@(a, b, v) = extendedGCD x y
res = Map.fromList [("x", showX x), ("y", showX y),
("a", showX a), ("b", showX b),
("v", showX v)]
in assert (v == gcd x y) (res, v, memory2)
let (dbM, genM) = emptyDatabase genL
generateTests "I" ModInv "modinv" dbM $ \ size memory0 ->
let (a, memory1) = genSign (generateNum memory0 "a" size)
(b, memory2) = genSign (generateNum memory1 "b" size)
c = recipModInteger a b
res = Map.fromList [("a", showX a), ("b", showX b),
("c", showX c)]
in (res, c, memory2)
data AlgState = AlgState {
u :: Integer,
v :: Integer,
bigA :: Integer,
bigB :: Integer,
bigC :: Integer,
bigD :: Integer
}
printState :: AlgState -> IO ()
printState a =
do putStrLn ("u: " ++ showX (u a))
putStrLn ("v: " ++ showX (v a))
putStrLn ("A: " ++ showX (bigA a))
putStrLn ("B: " ++ showX (bigB a))
putStrLn ("C: " ++ showX (bigC a))
putStrLn ("D: " ++ showX (bigD a))
extendedGCD :: Integer -> Integer -> (Integer, Integer, Integer)
extendedGCD x y = (a, b, g * (v finalState))
where
(x', y', g, initState) = initialState x y 1
finalState = runAlgorithm x' y' initState
a = bigC finalState
b = bigD finalState
initialState :: Integer -> Integer -> Integer -> (Integer, Integer, Integer, AlgState)
initialState x y g | even x && even y = initialState (x `div` 2) (y `div` 2) (g * 2)
| otherwise = (x, y, g, AlgState x y 1 0 0 1)
runAlgorithm :: Integer -> Integer -> AlgState -> AlgState
runAlgorithm x y state | u state == 0 = state
| otherwise = runAlgorithm x y state6
where
state4 = step4 x y state
state5 = step5 x y state4
state6 = step6 state5
step4 :: Integer -> Integer -> AlgState -> AlgState
step4 x y input@AlgState{..} | even u = step4 x y input'
| otherwise = input
where
input' = AlgState u' v bigA' bigB' bigC bigD
u' = u `div` 2
bigA' | even bigA && even bigB = bigA `div` 2
| otherwise = (bigA + y) `div` 2
bigB' | even bigA && even bigB = bigB `div` 2
| otherwise = (bigB - x) `div` 2
step5 :: Integer -> Integer -> AlgState -> AlgState
step5 x y input@AlgState{..} | even v = step5 x y input'
| otherwise = input
where
input' = AlgState u v' bigA bigB bigC' bigD'
v' = v `div` 2
bigC' | even bigC && even bigD = bigC `div` 2
| otherwise = (bigC + y) `div` 2
bigD' | even bigC && even bigD = bigD `div` 2
| otherwise = (bigD - x) `div` 2
step6 :: AlgState -> AlgState
step6 input@AlgState{..}
| u >= v = AlgState (u - v) v (bigA - bigC) (bigB - bigD) bigC bigD
| otherwise = AlgState u (v - u) bigA bigB (bigC - bigA) (bigD - bigB)
main :: IO ()
main =
@@ -377,3 +596,31 @@ main =
let args' = if null args then ["invocs", "tests"] else args
when ("invocs" `elem` args') generateInvocs
when ("tests" `elem` args') generateAllTheTests
---
run :: Integer -> Integer -> IO ()
run inputx inputy =
do let (x, y, g, initState) = initialState inputx inputy 1
finalState <- go x y initState
putStrLn ("-- FINAL STATE -----------------------")
printState finalState
putStrLn ("Final value: " ++ showX (g * v finalState))
putStrLn ("-- RUN ------")
printState (runAlgorithm x y initState)
putStrLn ("-- NORMAL ------")
let (a, b, v) = extendedGCD inputx inputy
putStrLn ("a: " ++ showX a)
putStrLn ("b: " ++ showX b)
putStrLn ("v: " ++ showX v)
where
go x y state =
do putStrLn "-- STATE -----------------------------"
printState state
if u state == 0
then return state
else do let state' = step4 x y state
state'' = step5 x y state'
state''' = step6 state''
go x y state'''