| Safe Haskell | Safe-Inferred |
|---|---|
| Language | Haskell2010 |
Math.FiniteField.Primes
Description
Prime numbers and related number theoretical stuff.
Synopsis
- integerLog2 :: Integer -> Integer
- ceilingLog2 :: Integer -> Integer
- isSquare :: Integer -> Bool
- integerSquareRoot :: Integer -> Integer
- ceilingSquareRoot :: Integer -> Integer
- integerSquareRoot' :: Integer -> (Integer, Integer)
- integerSquareRootNewton' :: Integer -> (Integer, Integer)
- divides :: Integer -> Integer -> Bool
- divisors :: Integer -> [Integer]
- squareFreeDivisors :: Integer -> [(Integer, Sign)]
- squareFreeDivisors_ :: Integer -> [Integer]
- divisorSum :: Integer -> Integer
- divisorSum' :: Int -> Integer -> Integer
- moebiusMu :: (Integral a, Num b) => a -> b
- eulerTotient :: Integer -> Integer
- liouvilleLambda :: (Integral a, Num b) => a -> b
- primes :: [Integer]
- primesSimple :: [Integer]
- primesTMWE :: [Integer]
- factorize :: Integer -> [(Integer, Int)]
- factorizeNaive :: Integer -> [(Integer, Int)]
- productOfFactors :: [(Integer, Int)] -> Integer
- integerFactorsTrialDivision :: Integer -> [Integer]
- groupIntegerFactors :: [Integer] -> [(Integer, Int)]
- powerMod :: Integer -> Integer -> Integer -> Integer
- isPrimeTrialDivision :: Integer -> Bool
- millerRabinPrimalityTest :: Integer -> Integer -> Bool
- isProbablyPrime :: Integer -> Bool
- isVeryProbablyPrime :: Integer -> Bool
Integer logarithm
integerLog2 :: Integer -> Integer Source #
Largest integer k such that 2^k is smaller or equal to n
ceilingLog2 :: Integer -> Integer Source #
Smallest integer k such that 2^k is larger or equal to n
Integer square root
integerSquareRoot :: Integer -> Integer Source #
Integer square root (largest integer whose square is smaller or equal to the input) using Newton's method, with a faster (for large numbers) inital guess based on bit shifts.
ceilingSquareRoot :: Integer -> Integer Source #
Smallest integer whose square is larger or equal to the input
integerSquareRoot' :: Integer -> (Integer, Integer) Source #
We also return the excess residue; that is
(a,r) = integerSquareRoot' n
means that
a*a + r = n a*a <= n < (a+1)*(a+1)
integerSquareRootNewton' :: Integer -> (Integer, Integer) Source #
Newton's method without an initial guess. For very small numbers (<10^10) it is somewhat faster than the above version.
Elementary number theory
squareFreeDivisors :: Integer -> [(Integer, Sign)] Source #
List of square-free divisors together with their Mobius mu value (note: the result is not ordered!)
squareFreeDivisors_ :: Integer -> [Integer] Source #
List of square-free divisors (note: the result is not ordered!)
divisorSum :: Integer -> Integer Source #
Sum ofthe of the divisors
eulerTotient :: Integer -> Integer Source #
Euler's totient function
liouvilleLambda :: (Integral a, Num b) => a -> b Source #
The Liouville lambda function (naive implementation)
List of prime numbers
primesSimple :: [Integer] Source #
A relatively simple but still quite fast implementation of list of primes. By Will Ness http://www.haskell.org/pipermail/haskell-cafe/2009-November/068441.html
primesTMWE :: [Integer] Source #
List of primes, using tree merge with wheel. Code by Will Ness.
Prime factorization
integerFactorsTrialDivision :: Integer -> [Integer] Source #
The naive trial division algorithm.
groupIntegerFactors :: [Integer] -> [(Integer, Int)] Source #
Groups integer factors. Example: from [2,2,2,3,3,5] we produce [(2,3),(3,2),(5,1)]
Modulo m arithmetic
powerMod :: Integer -> Integer -> Integer -> Integer Source #
Efficient powers modulo m.
powerMod a k m == (a^k) `mod` m
Prime testing
isPrimeTrialDivision :: Integer -> Bool Source #
Prime testing using trial division
millerRabinPrimalityTest :: Integer -> Integer -> Bool Source #
Miller-Rabin Primality Test (taken from Haskell wiki).
We test the primality of the first argument n by using the second argument a as a candidate witness.
If it returs False, then n is composite. If it returns True, then n is either prime or composite.
A random choice between 2 and (n-2) is a good choice for a.
isProbablyPrime :: Integer -> Bool Source #
For very small numbers, we use trial division, for larger numbers, we apply the
Miller-Rabin primality test log4(n) times, with candidate witnesses derived
deterministically from n using a pseudo-random sequence
(which should be based on a cryptographic hash function, but isn't, yet).
Thus the candidate witnesses should behave essentially like random, but the resulting function is still a deterministic, pure function.
TODO: implement the hash sequence, at the moment we use Random instead...
isVeryProbablyPrime :: Integer -> Bool Source #
A more exhaustive version of isProbablyPrime, this one tests candidate
witnesses both the first log4(n) prime numbers and then log4(n) pseudo-random
numbers