GNFS implementation + GF2 gaussian elimination fix

src/GNFS/PolySelect.hs

@@ -0,0 +1,47 @@
+module GNFS.PolySelect
+  ( NFSPoly(..)
+  , selectPolynomial
+  ) where
+
+import Data.Maybe (listToMaybe)
+import GNFS.FiniteField (irreducibleModP)
+import GNFS.Polynomial
+import GNFS.Util (iroot)
+
+-- | A monic polynomial f of degree d and an integer m with f(m) = n.
+data NFSPoly = NFSPoly
+  { npDegree :: Int
+  , npCoeffs :: Poly      -- ^ little-endian, monic
+  , npM      :: Integer
+  } deriving Show
+
+-- | Base-m polynomial selection. Returns a monic polynomial of the requested degree, irreducible over Q (certified by irreducibility modulo some small prime), with f(m) = n.
+-- Balanced digits are tried first (smaller coefficients => smaller norms).
+selectPolynomial :: Int -> Integer -> Maybe NFSPoly
+selectPolynomial d n = listToMaybe
+  [ NFSPoly d f m
+  | m <- takeWhile (\m' -> m' > 1 && 2 * m' ^ d > n) [m0, m0 - 1 ..]
+  , digits <- [balanced m (plainDigits m n), plainDigits m n]
+  , let f = trimP digits
+  , length f == d + 1
+  , last f == 1
+  , irreducibleSomewhere f
+  ]
+  where
+    m0 = iroot d n
+    smallPrimes = [3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73]
+    irreducibleSomewhere f = any (\p -> irreducibleModP p f) smallPrimes
+
+plainDigits :: Integer -> Integer -> [Integer]
+plainDigits _ 0 = []
+plainDigits m x = let (q, r) = x `divMod` m in r : plainDigits m q
+
+-- | Convert digits in [0, m) to balanced digits in (-m/2, m/2], propagating carries.
+balanced :: Integer -> [Integer] -> [Integer]
+balanced m = go 0
+  where
+    go 0 []     = []
+    go c []     = [c]
+    go c (x:xs) =
+      let v = x + c
+      in if 2 * v > m then (v - m) : go 1 xs else v : go 0 xs