finite-fields-0.2: Arithmetic in finite fields

Math.FiniteField.Primes

Description

Prime numbers and related number theoretical stuff.

Synopsis

# Integer logarithm

Largest integer k such that 2^k is smaller or equal to n

Smallest integer k such that 2^k is larger or equal to n

# Integer square root

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.

Smallest integer whose square is larger or equal to the input

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)

Newton's method without an initial guess. For very small numbers (<10^10) it is somewhat faster than the above version.

# Elementary number theory

d divides n

Divisors of n (note: the result is not ordered!)

List of square-free divisors together with their Mobius mu value (note: the result is not ordered!)

List of square-free divisors (note: the result is not ordered!)

Sum ofthe of the divisors

Sum of k-th powers of the divisors

moebiusMu :: (Integral a, Num b) => a -> b Source #

The Moebius mu function (naive implementation)

Euler's totient function

liouvilleLambda :: (Integral a, Num b) => a -> b Source #

The Liouville lambda function (naive implementation)

# List of prime numbers

primes :: [Integer] Source #

Infinite list of primes, using the TMWE algorithm.

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

List of primes, using tree merge with wheel. Code by Will Ness.

# Prime factorization

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

Efficient powers modulo m.

powerMod a k m == (a^k) mod m

# Prime testing

Prime testing using trial division

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.

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...

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