```-- |
-- Module:      Math.NumberTheory.Canon.AurifCyclo
-- Copyright:   (c) 2015-2019 Frederick Schneider
-- Licence:     MIT
-- Maintainer:  Frederick Schneider <fws.nyc@gmail.com>
-- Stability:   Provisional
--
-- <https://en.wikipedia.org/wiki/Aurifeuillean_factorization Aurifeuillian> and
-- <https://en.wikipedia.org/wiki/Cyclotomic_polynomial Cyclotomic>
-- <https://en.wikipedia.org/wiki/Factorization factorization> method functions.

{-# LANGUAGE PatternSynonyms, ViewPatterns #-}

module Math.NumberTheory.Canon.AurifCyclo (
aurCandDec,
aurDec,
applyCycloPair, applyCycloPairWithMap, CycloPair, Poly,
cyclo,          cycloWithMap,
cycloDivSet,    cycloDivSetWithMap,
chineseAurif,   chineseAurifWithMap,

crCycloAurifApply, applyCrCycloPair, divvy,
CycloMap, getIntegerBasedCycloMap, showCyclo, crCycloInitMap,
CR_, CanonRep_, CanonElement_
)
where

import Math.NumberTheory.Canon.Internals
import Math.NumberTheory.Moduli.Jacobi (JacobiSymbol(..), jacobi)
import Data.Array (array, (!), Array(), elems) -- to do: convert to unboxed? https://wiki.haskell.org/Arrays
import GHC.Real (numerator, denominator)
import Data.List (sort, sortBy, (\\))
import qualified Data.Map as M

-- CR_ Rep of 2
cr2 :: CR_
cr2 = fst \$ crFromI 2

-- | This function checks if the inputs along with operator flag have a cyclotomic or Aurifeuillian form to greatly simplify factoring.
--   If they do not, potentially much more expesive simple factorization is used via crSimpleApply.
--   Note: The cyclotomic map is threaded into the functions
crCycloAurifApply :: Bool -> CR_ -> CR_ -> CR_ -> Integer -> CycloMap -> (CR_, CycloMap)
crCycloAurifApply b x y g gi m
-- Optimization for prime g: If g is a prime (and exp not of from x^2 + y^2) but not Aurifeuillian (verify
| (crPrime g) && not (g == cr2 && b)
= eA ([term1, termNM1], m) -- split into  (x +/- y) and (x^(n-1) ... -/+ y^(n-1))

-- Factorize: grtx^g - grty^g via cyclotomic polynomials
| not b            = eA (cycA grtx grty g)

-- Factorize x^n + y^n using cyclotomic polynomials (if n = 2^x*m where x >= 0 and m > 2)
| b && not gpwr2   = eA (cycA (oddRoot x) (-1 * oddRoot y) odd')

| otherwise        = (fst \$ crSimpleApply op x y, m)
where op            = if b then (+) else (-)
((gp, _):gs)  = g
gpwr2         = gp == 2 && gs == []
gth_root v    = crToI \$ crRoot v gi
grtx          = gth_root x
grty          = gth_root y

-- used when factoring x^p +/- 1 where p is prime
term1         = integerApply op (crRoot x gi) (crRoot y gi) -- a +/- b
termNM1       = div (integerApply op x y) term1  -- divide a^g +/- b^g by the term above

cycA x' y' n  = (sort ia, m') -- sort the integers returned from low to high, should help if there are larger terms
where (ia, m') = applyCrCycloPair x' y' n m

-- eA stands for "enriched apply", v unlike to exceed factor cutoff
eA (a,mp)     = (foldr1 crMult \$ map (\v' -> fst \$ crFromI v') v, m')
where (v, m')   = case aurCandDecI x y gi g b of
Nothing       -> auL a mp  -- can't do anything Brent Aurif-wise, try Chinese method
Just (a1, a2) -> auL (divvy a a1 a2) mp -- meld in the 2 Aurif factors with input array
auL al ma = case c of -- aL stands for "augmented list)
Nothing       -> (al, mp')             -- just return what was passed in
Just (a3, a4) -> (divvy al a3 a4, mp') -- additional "Chinese" factors
where (c, mp') = chineseAurifCr x y b ma

odd'          | gp == 2   = tail g  -- grabs number sans any power of 2
| otherwise = g
oddRoot v     = crToI \$ crRoot v (crToI odd')

-- | This function checks if the input is a candidate for Aurifeuillian decomposition.  Its name is an abbreviation.
--   If it is a candidate, split it into two and evaluate it.  Otherwise, return nothing.
--   The code will "prep" the input params for the internal function so they will be relatively prime.
--   Possible solutions:  Non-zero multiples of xi and yi (where xi and yi are relatively prime and of the form
--                        xi = a^(a*b), yi = 1 where a and b are positive integers and b is odd. (xi and y1 may be interchanged)
--                        If a is even, we will find factors of a^(a*b) + 1.  The bool flag must be True
--                        If a is odd,  we will find factors of a^(a*b) - 1.  The bool flag must be False
--   This function and those it calls internally implement Richard Brent's algorithm for computing Aurifeuillian factors.
--   His logic is used in conjuction with cyclotomic polynomial decomposition.  They are explained in these two papers:
--   <http://maths-people.anu.edu.au/~brent/pd/rpb127.pdf rpb127> and <http://maths-people.anu.edu.au/~brent/pd/rpb135.pdf rpb135>.

aurCandDec :: Integer -> Integer -> Bool -> Maybe (Integer, Integer)
aurCandDec xi yi b = f (fst \$ crFromI xi) (fst \$ crFromI yi)
where f xp yp = aurCandDecI x y n (fst \$ crFromI n) b
where n      = gcd (crMaxRoot \$ crAbs x) (crMaxRoot \$ crAbs y)
gxy    = crGCD xp yp
(x, y) = (crDivStrict xp gxy, crDivStrict yp gxy) -- this will fix the input to be relatively prime
-- toDo: incorporate the factorization status when determining the correct gcd.

-- Don't call this I(nternal) function directly.  The function assumes that x and y are relatively prime.  Currently uses Brent logic only.
aurCandDecI :: CR_ -> CR_ -> Integer -> CR_ -> Bool -> Maybe (Integer, Integer)
aurCandDecI x y n cr b| (nm4 == 1 && b) || (nm4 /= 1 && not b) ||
(xdg == x && ydg == y)  || (m /= 0)
= Nothing --
| otherwise       = case aurDecI n' cr' of
Nothing             -> Nothing
Just (gamma, delta) -> apply gamma delta
where
-- override of n, to attempt decomp for g = gcd when number of form: g^gd +/-1,
-- this will only work when either x or y = 1 and not for any other divisor of g.
-- If both terms are not 1, we just attempt an Aurif. decomp for n

-- TODO: Need to integrate chineseAurif as it does something different
(n', cr') | x /= cr1 && y /= cr1 = (n, cr)
| otherwise            = (gcd1i, gcd1)
where x1    = if y /= cr1 then y else x
gcd1  = crRadical \$ crGCD x1 cr
gcd1i = crToI gcd1
nm4       = mod n' 4
divTry a  = case crDiv a (crExp cr' n' False) of -- check to divide by n^n, if not return original
Left _         -> a
Right quotient -> quotient
xdg       = divTry x
ydg       = divTry y
mrGCD     = gcd (crMaxRoot \$ crAbs xdg) (crMaxRoot \$ crAbs ydg)
m         = mod mrGCD (2*n')

-- need to consider cyclotomic translations, order the terms
(x', ml)  | ydg /= y  = ( (crDivRational ydg x), if (not b) then (-1) else 1)
| otherwise = ( (crDivRational xdg y), 1);

{-  The more familar form of the below is (C(x)^2 - nxD(x)^2):
gm(x)^2 -nx*dt(x)^2 =>
gamma +/- sqrt(nx) * delta     -}
xrtn      = crMult cr' (crRoot x' n')
xrtnr     = crToRational xrtn
sqrtnxr   = crToRational \$ crRoot (crMult cr' xrtn) 2

apply gm dt = Just (ml * numerator f1, numerator f2)
where f1                  = c - sqrtnxr * d
f2                  = c + sqrtnxr * d
c                   = aA gm xrtnr
d                   = aA dt xrtnr
-- aA means applyArray. array/lists are treated like polynomials (zero-base assumed)
aA a   x''  = f (elems a) 1 0
where f []     _  a' = a'
f (c:cs) m' a' = f cs (m'*x'') (a' + (toRational c)*m')

-- Example Aurif. decomp: C5(x) = x^2 + 3x + 1, D5(x) = x + 1 => Cyclotomic5(x) = C5(x)^2 − 5x*D5(x)^2

-- | This function returns a pair of polynomials (in array form) or Nothing (if it's squareful).
--   An illogical n (n <= 1) will generate an error.
aurDec :: Integer -> Maybe (Array Integer Integer, Array Integer Integer)
aurDec n | n <= 1    = error "aurifDecomp: n must be greater than 1"
| otherwise = aurDecI n (fst \$ crFromI n) -- no concern about factorization cutoff here

-- | Internal Aurifeuillian Decomposition Workhorse Function
aurDecI :: Integer -> CR_ -> Maybe (Array Integer Integer, Array Integer Integer)
aurDecI n cr | crHasSquare cr || n < 2 || n' < 2
= Nothing
| otherwise = Just (gamma, delta)
where nm4 = mod n 4
n'  = if (nm4 == 1) then n else (2*n)
d   = div (totient n') 2
dm2 = mod d 2

-- max gamma and delta subscripts to explicitly compute (add'l terms come from symmetry)
mg | dm2 == 1  = div (d-1) 2
| otherwise = div d     2
md | dm2 == 1  = div (d-1) 2
| otherwise = div (d-2) 2

-- create q array of size td2: q(2k+1) = jacobi n (2k+1),  q(2k+2  = mn * totient (2k+2)
q   = array (1, d) ([(i, f i) | i <- [1..d]])
where f i  | mod i 2 == 1 = convJacobi \$ jacobi n i
| otherwise    = eQ
where eQ       | ff == True = moeb cr' * (totient g) * (cos' \$ (n-1)*i)
| otherwise  = error "Moebius fcn can't be called if number not totally factored"
(cr', ff)= crFromI \$ div n' g
g        = gcd n' i

-- moebius fcn: 0 if has square, otherwise based on number of distinct prime factors
moeb crm | crHasSquare crm         = 0
| mod (length crm) 2 == 1 = -1
| otherwise               = 1

cos' c   | m8 == 2 || m8 == 6 = 0   -- "cosine" function
| m8 == 4            = -1
| m8 == 0            = 1
| otherwise          = error "Logic error: bad/odd value passed to cos'"
where m8 = mod c 8

-- These two arrarys have mutually recursive definitions
gamma = array (0, d) ([(0,1)] ++ [(i, gf i) | i <- [1..d]])
where gf k | k > mg    = gamma!(d-k)
| otherwise = div gTerm (2*k)
where gTerm = sum \$ map f [0..k-1]
where f j = n * q!(2*k-2*j-1) * delta!j - q!(2*k-2*j) * gamma!j

delta = array (0, d-1) ([(0,1)] ++ [(i, df i) | i <- [1..d-1]])
where df k | k > md    = delta!(d-k-1)
| otherwise = div dTerm (2*k+1)
where dTerm = gamma!k + sum (map f [0..k-1])
where f j = q!(2*k-2*j+1) * gamma!j - q!(2*k-2*j) * delta!j

{- Pseudocode for computing gammas G and deltas D
G(0) = 1
D(0) = 1

Evaluate G(k) for 1 .. floor(d/2)
G(k) = (1/2k) * sum(n*q(2k-2j-1)*D(j) - q(2k-2j)G(j)) [for j= 0 to k-1)

Evaluate D(k) for 1 .. floor(d-1/2)
D(k) = (1/2k+1) * ( G(k) + sum(q(2k+1-2j)*D(j) - q(2k-2j)D(j)) )

Evaluate G(k) for (floor(d/2)+1) to d => G(k) = G(d-k)
Evaluate D(k) for (floor(d+1/2)) to d-1 => D(k) = D(d-k)

Cyc(n) = C(x)^2 -nxD(x)^2 where gamma and delta are the coeffs for C(x) and D(x) respectively
-}

-- | Internal function requires two integers (computed via Aurif. methods) along with a list of Integers.  The product of
--   the Integers must be a divisor of the list's product otherwise an error will be generated.
--   It's called divvy because it splits the 2 integers across the array using the gcd.
--   This will help factoring because the larger term(s) will be broken up into smaller pieces.
divvy :: [Integer] -> Integer -> Integer -> [Integer]
divvy a x y = d (sortBy rev a) (abs x) (abs y)
where rev a' b'       = if (a' > b') then LT else GT
d []     x' y'  | x' == 1 && y' == 1             = []
| abs x' == 1 && abs y' == 1 = [x' * y']
| otherwise                    = error "Empty list passed as first param but x' and y' weren't both 1"
d (c:cs) x' y'  | x' == 1 && y' == 1 = c:cs
| otherwise          = v ++ d cs (div x' gnx) (div y' gny)
where v   = filter (>1) \$ [div q gny, gnx, gny]
gnx = gcd c x'
q   = div c gnx
gny = gcd q y'
{-  Test: Run divvy on this:
let a = [4,7,8401,62324358089100907319521,682969,61,374857981681,547]
let x = 50031545486420197
let y = 50031544711579219     -}

{-  Cyclotomic factorizations for numbers of the form x^n +/- y^n
Example:
x^15-y^15 = (x-y) (x^2+x y+y^2) (x^4+x^3 y+x^2 y^2+x y^3+y^4) (x^8-x^7*y +x^5*y^3 - x^4*y^4 +x^3*y^5 -x*y^7+y^8)

C(15) is a form of the last term where y = 1
It's possible in some cases to do an additional Aurifeuillian factorization (of the last term).   -}

-- | Crude polynomial type.  Look into optimizing at a later date
type Poly = [Integer]

-- | CycloPair: Pair of an Integer and its corresponding cyclotomic polynomial
type CycloPair         = (Integer, Poly)

-- | CycloMapInternal: Map internal to CycloMap newtype
type CycloMapInternal  = M.Map CR_ CycloPair

-- | CycloMap is a newtype hiding the details of a map of CR_ to pairs of integers and corresponding cyclotomic polynomials.
newtype CycloMap       = MakeCM CycloMapInternal deriving (Eq, Show)

-- | Unwrap the CycloMap newtype.
fromCM :: CycloMap -> CycloMapInternal
fromCM (MakeCM cm) = cm

-- | Unwrap the CycloMap and convert the internal canon rep keys to Integers, returning a "raw" map
getIntegerBasedCycloMap :: CycloMap -> M.Map Integer CycloPair
getIntegerBasedCycloMap cm = M.mapKeys crToI (fromCM cm)

-- | This is an initial map with the cyclotomic polynomials for 1.
crCycloInitMap :: CycloMap
crCycloInitMap = MakeCM \$ M.insert cr1 (1, gen_xNm1 1) M.empty

-- | Wrapper function to query map internals
cmLookup :: CR_ -> CycloMap -> Maybe CycloPair
cmLookup c m = M.lookup c (fromCM m)

-- Internal function for updating map internals
cmInsert :: CR_ -> CycloPair -> CycloMap -> CycloMap
cmInsert c p m = MakeCM \$ M.insert c p (fromCM m)

-- Computing the cyclotomic polynomials for the divisor set of a number:
--   Begin with with a map of 2 elements to the cylotomic polynomial for 1 and 2
--   If the cr doesn't equal the radical,  then open it up to the other factors and the square-free factors must be in the map
--   Identity: x^n -1 is the product of cyclotomic polynomial for each  d where d | n.

-- | Integer wrapper for crCyclo with default CycloMap parameter
cyclo :: Integer -> (CycloPair, CycloMap)
cyclo n = crCyclo (fst \$ crFromI n) crCycloInitMap -- no concern over factorization cutoff here

-- | Integer wrapper for crCyclo
cycloWithMap :: Integer -> CycloMap -> (CycloPair, CycloMap)
cycloWithMap n m = crCyclo (fst \$ crFromI n) m -- no concern over factorization cutoff here

-- | Integer wrapper for crCycloDivSet with default CycloMap parameter
cycloDivSet :: Integer -> CycloMap
cycloDivSet n = fst \$ crCycloDivSet (fst \$ crFromI n) crCycloInitMap -- no concern over factorization cutoff here

-- | Integer wrapper for crCycloDivSet
cycloDivSetWithMap :: Integer -> CycloMap -> (CycloMap, CycloMap)
cycloDivSetWithMap n m  = crCycloDivSet (fst \$ crFromI n) m -- no concern over factorization cutoff here

-- polynomial along with working cyclotomic map.
crCyclo :: CR_ -> CycloMap -> (CycloPair, CycloMap)
crCyclo cr m | crPositive cr   = ((crToI \$ crDivStrict cr r, p), m')
| otherwise       = error "crCyclo: Positive integer needed"
((_,p), m') = crCycloRad r m

-- | Return a pair of cyclo maps just for the divisors and then a master map.
crCycloDivSet :: CR_ -> CycloMap -> (CycloMap, CycloMap)
crCycloDivSet cr m | crPositive cr = m2
| otherwise     = error "crCycloDivSet: Positive integer needed"
where (_,m2) = c
crd = crDivisors cr
c   = case cmLookup cr m of
Nothing -> c' -- need to compute it
Just p  -> (p, (mf, m))       -- found it.  add a filtered version.  ToDo: optimize this
where mf = MakeCM \$ M.fromList \$ filter (\(n,_) -> elem n crd) \$ M.toList \$ fromCM m

{- Performance note: The filter version above tended to be somewhat faster than the lookup
version below so I used that
where mf = MakeCM \$ M.fromList \$ map l crd     -- "lookup version"
where l d = case cmLookup d m of
Nothing -> error \$ e d
Just p -> (d, p)
e d = error "crCycloDivSet: Logic error: Divisor = '" ++ (show d) ++ "' not found!"
-}

c'  | r == cr   = (pr, (pm, pm))
| otherwise = (cm, (mm, mm))
where (cm, mm)      = mfn sqFulDivs pm
(pr, pm)      = crCycloRad r m
sqFulDivs     = crd \\ crDivisors r  -- "squareful" divisors
mfn []     _  = error "Logic Error in mfn: Empty list is forbidden"
mfn (n:ns) mp | ns == []  = cp
| otherwise = mfn ns mp'
where cp@(_, mp') = crCycloAll n mp

-- Compute the "radical" divisors first and then the non-square free entries.
crCycloRad :: CR_ -> CycloMap -> (CycloPair, CycloMap)
crCycloRad cr m = case cmLookup cr m of
Nothing -> c' -- need to compute it
Just p  -> (p, m)           -- found it
where c' | cs == []  = (cycpr, cmInsert cr cycpr m)
| otherwise = (cyc_n, cmInsert cr cyc_n mp)
where (_ : cs)   = cr
-- for primes, because the tail of the cr is [] meaning only one prime factor
r          = fromInteger \$ crToI \$ crRadical cr
-- ToDo: Optimize cycpr to be quotient of (r^n -1)/(r-1) when r is a big prime
cycpr      = genPrimePoly r
-- for composites
-- Create polynomial of the form : x^n -1
xNm1       = gen_xNm1 r
(cPrd, mp) = mf (init \$ crDivisors cr)
cyc_n      = (1, divPoly xNm1 cPrd)

-- mf (Memo Fold) takes a list of divisors and returns the pair: (cyclotomic product, memoized map)
mf (n:ns) = f ns p m'
where ((_,p), m')      = crCycloRad n m
f (n':ns') p' mp = f ns' (multPoly p' p'') m'' -- cycloMap-threaded mult. fold
where ((_,p''), m'') = crCycloRad n' mp
f _        p' mp = (p', mp)
mf []     = error "Logic error: Blank list can't be passed to mf aka crCycloMemoFold"

-- | Return a pair of Integer and its cyclotomic polynomial while efficiently building up a map
crCycloAll :: CR_ -> CycloMap -> (CycloPair, CycloMap)
crCycloAll cr m | p == cr1     = case cmLookup cr m of
Just cb -> (cb, m)         -- found it
| otherwise    = (crp, cmInsert cr crp md)
where (p, d)      = crPullSq
((i,y), md) = case cmLookup d m of
Nothing -> crCycloAll d m -- need to compute it
Just c  -> (c, m)         -- found it
-- Optimization: (1, x + 1) => (4, x + 1). Note: Cyc2(x) = x + 1
-- The first value is the exponential multiplier Cyc8(x) = C2(x^4) = (x^4) + 1
crp         = ((fst \$ head p) * i, y)
crPullSq    = f [] cr
where f h []             = (cr1, h)
f h (c@(cp,ce):cs) | ce > 1    = ([(cp, 1)], h ++ (cp, ce-1):cs)
| otherwise = f (h ++ [c]) cs

-- | These "apply cyclo" functions will use cyclotomic polynomial methods to factor x^e - b^e.
applyCrCycloPair :: Integer -> Integer -> CR_ -> CycloMap -> ([Integer], CycloMap)
applyCrCycloPair l r cr m = (applyCrCycloPairI l r cr (M.elems \$ fromCM md), mn)
where (md, mn) = crCycloDivSet cr m

applyCrCycloPairI :: Integer -> Integer -> CR_ -> [CycloPair] -> [Integer]
applyCrCycloPairI l r cr cds           = map applyPoly cds
where nd              = crTotient cr
pA v            = a where a = array (0,nd) ([(0,1)] ++ [(i, v*a!(i-1)) | i <- [1..nd]]) -- array of powers
lpa             = pA l
rpa             = pA r
applyPoly (m,p) = foldr1 (+) (map f \$ zip p [0..])
where f (a, b) | a == 0    = 0
| otherwise = a * lpa!(m*b) * rpa!(m*(maxExp - b))
maxExp   = toInteger \$ length p - 1

-- | Wraps applyCycloPairWithMap with default CycloMap argument.
applyCycloPair :: Integer -> Integer -> Integer -> [Integer]
applyCycloPair x y e = fst \$ applyCycloPairWithMap x y e crCycloInitMap

-- | This will use cyclotomic polynomial methods to factor x^e - b^e.
applyCycloPairWithMap :: Integer -> Integer -> Integer -> CycloMap -> ([Integer], CycloMap)
applyCycloPairWithMap  x y e m = applyCrCycloPair x y (fst \$ crFromI e) m -- no concern over factorization cutoff here

-- | This will display the cyclotomic polynomials for a CR.
showCyclo :: CR_ -> CycloMap -> [Char]
showCyclo n m = p \$ snd \$ fst \$ crCyclo n m
where p  (c:cs)   = show c ++ (p' cs (1 :: Int)) -- "LE" endianness is assumed here
p  _        = []
p' (c:cs) s | c == 0    = r
| otherwise = (if c > 0 then " + " else  " - ") ++ (if ac == 1 then "" else show ac) ++
"x" ++ (if s == 1 then "" else "^" ++ show s) ++ r
where r  = p' cs (s+1)
ac = abs c
p' _      _ = []

-- | All the exponents must be even to return True.
crSquareFlag :: CR_ -> Bool
crSquareFlag = all (\(_, ce) -> mod ce 2 == 0)

-- | Wrapper for chineseAurifWithMap with default CycloMap parameter
--
--   Integral examples to try.  These both will return answers of the form: Just (bigFactor1, bigFactor2).  The product of the
--   two big factors will divide the equivalent value in each comment.
--
--   >>> (q,m,p,k,r,n)=(7,1,19,13,1,1)
--   >>> chineseAurif ((q^(2*m)*p)^(p*k)) ((r^(2*n))^(p*k)) True -- Equivalent to 931^247+1
--
--   >>> (q,m,p,k,r,n)=(5,8,29,13,11,1)
--   >>> chineseAurif ((q^(2*m)*p)^(p*k)) ((r^(2*n))^(p*k)) False -- Equivalent to (5^16*29)^377 - 121^377
--
chineseAurif :: Integer -> Integer -> Bool -> Maybe (Integer, Integer)
chineseAurif x y b = fst \$ chineseAurifWithMap x y b crCycloInitMap

-- Integer wrapper for chineseAurifCr
-- | The source for this algorithm is this <http://www.jams.or.jp/scm/contents/Vol-2-3/2-3-16.pdf paper>
--   The formula at 2.7 there is implemented in the internal chineseAurifCr function. This will handle a
--   subset of the cases that the main Aurif. routines handle.
chineseAurifWithMap :: Integer -> Integer -> Bool -> CycloMap -> (Maybe (Integer, Integer), CycloMap)
chineseAurifWithMap x y b m = chineseAurifCr (fst \$ crFromI x) (fst \$ crFromI y) b m
-- practically speaking, factor cutoff is a non-issue

-- This function reduces the two CR parameters by gcd before calling an internal function to find a "Chinese" Aurif. factorization.
-- Solutions will be found for any non-zero multiple of xp yp b (where xp and yp are relatively prime)
-- where xp is of the form: (q^2m * p) ^ (p * k)
--       yp is of the form: (r^2n) ^ (p * k)
--       (That said, the order of xp and yp can be switched and the same result or non-result would be obtained.)
--       op = + if p == 3 mod 4 and op
--       op = - if p == 1 mod 4
--       For xp and yp, all of variables (q, m, p, k, r, n) are (CRs equivalent to) positive integers.
--       p and k are also odd. p must be square-free. q must be relatively prime to k.
chineseAurifCr :: CR_ -> CR_ -> Bool -> CycloMap -> (Maybe (Integer, Integer), CycloMap)
chineseAurifCr xp yp b m = case c of
Nothing -> chineseAurifI mbyx n myx (crToI myx) b m' -- if first try fails, try the reverse
r       -> (r, m')
where (c, m') = chineseAurifI mbxy n mxy (crToI mxy) b m
gcdxy   = crGCD xp yp
(x, y)  = (crDivStrict xp gcdxy, crDivStrict yp gcdxy) -- strip out any commonality
n       = gcd (crMaxRoot \$ crAbs x) (crMaxRoot \$ crAbs y)
ncr     = fst \$ crFromI n -- factorization cutoff not an issue here
mbxy    = crRoot (crDivRational x y) n
mxy     = crGCD (crNumer mbxy)  ncr
mbyx    = crRecip mbxy
myx     = crGCD (crNumer mbyx) ncr

-- Internal function to find factor of mb^n +/- 1 (mb would be M from paper, mb meaning m "big".
-- Solution forms: Any non-zero multiple of (q^2m * p) ^ (p * k) op (r^2n)^(p * k)  where k is an odd, postive number, m, n > 0.
-- This will work if the op is "+" when mod p 4 = 3 OR when the op is "-" for when mod p 4 = 1.
chineseAurifI :: CR_ -> Integer -> CR_ -> Integer -> Bool -> CycloMap -> (Maybe (Integer, Integer), CycloMap)
chineseAurifI mbcr n mcr m b mp | mod n 2 == 0        || mod m 2 == 0 ||          -- n and m must both be odd
m < 3               || km /= 0      ||          -- m must be odd and > 1 and m | n
(mm4 == 1 && b)     || -- sign and modulus
(mm4 == 3 && not b) || -- must be in synch
mbdm == cr0         || not (crSquareFlag mbdm)  -- mb/m must be a square and integral
= (Nothing, mp)
| otherwise = case cv - (gd1 * gd2) of
0 -> (Just (gd1, gd2), mp')
_ -> (Nothing, mp) -- addl check since paper doesn't indicate rationals are supported
where mm4      = mod m 4
e        = toRational \$ if (mm4 == 3) then (-1) else mm4
(k, km)  = quotRem n m
mbdm     = case crDiv mbcr mcr of
Left  _ -> cr0 -- error condition if not a multiple
Right q -> q
r        = crToRational \$ crRoot (crMult mbcr mcr) 2 -- sqrt (m*M) from paper
mb       = crToRational mbcr
jR c     = toRational \$ convJacobi \$ jacobi c m
eM       = e * mb
v1       = (toRational m) * mb^(div (k * (m + 1)) 2)
v2       = t * s
where t = (jR 2) * r * (mb ^ (div (k-1) 2))
s = sum \$ map (\c -> (jR c) * eM^(k*c))
\$ filter (\c -> gcd c m == 1) [1..m] -- rel. prime
ncr      = fst \$ crFromI n -- factor cutoff not an issue here
-- get cyclotomic value
cv        = head \$ applyCrCycloPairI (numerator eM) (denominator eM) ncr [cp]
(cp, mp') = crCyclo ncr mp
gd1       = gcd cv (numerator \$ v1 - v2) -- delta1
gd2       = gcd cv (numerator \$ v1 + v2) -- delta2

-- workaround after arithmoi changes
convJacobi :: JacobiSymbol -> Integer
convJacobi j = case j of
MinusOne -> -1
Zero     -> 0
One      -> 1

-- Simple Polynomial functions just used internally
-- Generate a polynomial of the form: x^n - 1
gen_xNm1 :: Int -> Poly
gen_xNm1 r = -1 : (replicate (r-1) 0) ++ [1]

genPrimePoly :: Int -> (Integer, Poly)
genPrimePoly r = (1, replicate r 1)

-- | Multiply two polynomials
multPoly :: Num a => [a] -> [a] -> [a]
multPoly [] _ = []
multPoly (p:p1) p2 = let pTimesP2        = multiplyBy p p2
xTimesP1Timesp2 = multiplyByX \$ multPoly p1 p2
where multiplyBy  a p' = map (a*) p'
multiplyByX p'   = 0:p'

-- | Div function for polynomials.  The assumption is that p1 is a multiple of p2
divPoly :: Integral a => [a] -> [a] -> [a]
divPoly p1 p2 = go [] p1 (length p1 - length p2)
where go q u n
| n < 0     = q
| otherwise = go (q0:q) u' (n-1)