arithmoi-0.4.0.2: Efficient basic number-theoretic functions. Primes, powers, integer logarithms.

Portability Non-portable (GHC extensions) Provisional Daniel Fischer Safe-Infered

Math.NumberTheory.Primes.Factorisation

Description

Various functions related to prime factorisation. Many of these functions use the prime factorisation of an `Integer`. If several of them are used on the same `Integer`, it would be inefficient to recalculate the factorisation, hence there are also functions working on the canonical factorisation, these require that the number be positive and in the case of the Carmichael function that the list of prime factors with their multiplicities is ascending.

Synopsis

# Factorisation functions

Factorisation of `Integer`s by the elliptic curve algorithm after Montgomery. The algorithm is explained at http://programmingpraxis.com/2010/04/23/modern-elliptic-curve-factorization-part-1/ and http://programmingpraxis.com/2010/04/27/modern-elliptic-curve-factorization-part-2/

The implementation is not very optimised, so it is not suitable for factorising numbers with several huge prime divisors. However, factors of 20-25 digits are normally found in acceptable time. The time taken depends, however, strongly on how lucky the curve-picking is. With luck, even large factors can be found in seconds; on the other hand, finding small factors (about 12-15 digits) can take minutes when the curve-picking is bad.

Given enough time, the algorithm should be able to factor numbers of 100-120 digits, but it is best suited for numbers of up to 50-60 digits.

factorise :: Integer -> [(Integer, Int)]Source

`factorise n` produces the prime factorisation of `n`, including a factor of `(-1)` if `n < 0`. `factorise 0` is an error and the factorisation of `1` is empty. Uses a `StdGen` produced in an arbitrary manner from the bit-pattern of `n`.

`defaultStdGenFactorisation` first strips off all small prime factors and then, if the factorisation is not complete, proceeds to curve factorisation. For negative numbers, a factor of `-1` is included, the factorisation of `1` is empty. Since `0` has no prime factorisation, a zero argument causes an error.

stepFactorisation :: Integer -> [(Integer, Int)]Source

`stepFactorisation` is like `factorise'`, except that it doesn't use a pseudo random generator but steps through the curves in order. This strategy turns out to be surprisingly fast, on average it doesn't seem to be slower than the `StdGen` based variant.

factorise' :: Integer -> [(Integer, Int)]Source

Like `factorise`, but without input checking, hence `n > 1` is required.

Like `defaultStdGenFactorisation`, but without input checking, so `n` must be larger than `1`.

### Factor sieves

A compact store of smallest prime factors.

`factorSieve n` creates a store of smallest prime factors of the numbers not exceeding `n`. If you need to factorise many smallish numbers, this can give a big speedup since it avoids many superfluous divisions. However, a too large sieve leads to a slowdown due to cache misses. The prime factors are stored as `Word16` for compactness, so `n` must be smaller than `2^32`.

sieveFactor :: FactorSieve -> Integer -> [(Integer, Int)]Source

`sieveFactor fs n` finds the prime factorisation of `n` using the `FactorSieve` `fs`. For negative `n`, a factor of `-1` is included with multiplicity `1`. After stripping any present factors `2`, the remaining cofactor `c` (if larger than `1`) is factorised with `fs`. This is most efficient of course if `c` does not exceed the bound with which `fs` was constructed. If it does, trial division is performed until either the cofactor falls below the bound or the sieve is exhausted. In the latter case, the elliptic curve method is used to finish the factorisation.

### Trial division

trialDivisionTo :: Integer -> Integer -> [(Integer, Int)]Source

`trialDivisionTo bound n` produces a factorisation of `n` using the primes `= bound@. If @n@ has prime divisors @ bound`, the last entry in the list is the product of all these. If `n <= bound^2`, this is a full factorisation, but very slow if `n` has large prime divisors.

## Partial factorisation

smallFactors :: Integer -> Integer -> ([(Integer, Int)], Maybe Integer)Source

`smallFactors bound n` finds all prime divisors of `n > 1` up to `bound` by trial division and returns the list of these together with their multiplicities, and a possible remaining factor which may be composite.

Arguments

 :: Maybe Integer Lower bound for composite divisors -> StdGen Standard PRNG -> Maybe Int Estimated number of digits of smallest prime factor -> Integer The number to factorise -> [(Integer, Int)] List of prime factors and exponents

A wrapper around `curveFactorisation` providing a few default arguments. The primality test is `bailliePSW`, the `prng` function - naturally - `randomR`. This function also requires small prime factors to have been stripped before.

Arguments

 :: Maybe Integer Lower bound for composite divisors -> (Integer -> Bool) A primality test -> (Integer -> g -> (Integer, g)) A PRNG -> g Initial PRNG state -> Maybe Int Estimated number of digits of the smallest prime factor -> Integer The number to factorise -> [(Integer, Int)] List of prime factors and exponents

`curveFactorisation` is the driver for the factorisation. Its performance (and success) can be influenced by passing appropriate arguments. If you know that `n` has no prime divisors below `b`, any divisor found less than `b*b` must be prime, thus giving `Just (b*b)` as the first argument allows skipping the comparatively expensive primality test for those. If `n` is such that all prime divisors must have a specific easy to test for structure, a custom primality test can improve the performance (normally, it will make very little difference, since `n` has not many divisors, and many curves have to be tried to find one). More influence has the pseudo random generator (a function `prng` with `6 <= fst (prng k s) <= k-2` and an initial state for the PRNG) used to generate the curves to try. A lucky choice here can make a huge difference. So, if the default takes too long, try another one; or you can improve your chances for a quick result by running several instances in parallel.

`curveFactorisation` requires that small prime factors have been stripped before. Also, it is unlikely to succeed if `n` has more than one (really) large prime factor.

### Single curve worker

`montgomeryFactorisation n b1 b2 s` tries to find a factor of `n` using the curve and point determined by the seed `s` (`6 <= s < n-1`), multiplying the point by the least common multiple of all numbers `<= b1` and all primes between `b1` and `b2`. The idea is that there's a good chance that the order of the point in the curve over one prime factor divides the multiplier, but the order over another factor doesn't, if `b1` and `b2` are appropriately chosen. If they are too small, none of the orders will probably divide the multiplier, if they are too large, all probably will, so they should be chosen to fit the expected size of the smallest factor.

It is assumed that `n` has no small prime factors.

The result is maybe a nontrivial divisor of `n`.

# Totients

Calculates the totient of a positive number `n`, i.e. the number of `k` with `1 <= k <= n` and `gcd n k == 1`, in other words, the order of the group of units in `ℤ/(n)`.

Alias of `totient` for people who prefer Greek letters.

A compact store of totients.

`totientSieve n` creates a store of the totients of the numbers not exceeding `n`. A `TotientSieve` only stores values for numbers coprime to `30` to reduce space usage. The maximal admissible value for `n` is `fromIntegral (maxBound :: Word)`.

`sieveTotient ts n` finds the totient `π(n)`, i.e. the number of integers `k` with `1 <= k <= n` and `gcd n k == 1`, in other words, the order of the group of units in `ℤ/(n)`, using the `TotientSieve` `ts`. First, factors of `2, 3` and `5` are handled individually, if the remaining cofactor of `n` is within the sieve range, its totient is looked up, otherwise the calculation falls back on factorisation, first by trial division and if necessary, elliptic curves.

totientFromCanonical :: [(Integer, Int)] -> IntegerSource

Calculate the totient from the canonical factorisation.

# Carmichael function

Calculates the Carmichael function for a positive integer, that is, the (smallest) exponent of the group of units in `&8484;/(n)`.

Alias of `carmichael` for people who prefer Greek letters.

A compact store of values of the Carmichael function.

`carmichaelSieve n` creates a store of values of the Carmichael function for numbers not exceeding `n`. Like a `TotientSieve`, a `CarmichaelSieve` only stores values for numbers coprime to `30` to reduce space usage. The maximal admissible value for `n` is `fromIntegral (maxBound :: Word)`.

`sieveCarmichael cs n` finds the value of `λ(n)` (or `ψ(n)`), the smallest positive integer `e` such that for all `a` with `gcd a n == 1` the congruence `a^e ≡ 1 (mod n)` holds, in other words, the (smallest) exponent of the group of units in `ℤ/(n)`. The strategy is analogous to `sieveTotient`.

Calculate the Carmichael function from the factorisation. Requires that the list of prime factors is strictly ascending.

# Divisors

`divisors n` is the set of all (positive) divisors of `n`. `divisors 0` is an error because we can't create the set of all `Integer`s.

`tau n` is the number of (positive) divisors of `n`. `tau 0` is an error because `0` has infinitely many divisors.

Alias for `tau` for people preferring Greek letters.

Alias for `tau`.

The sum of all (positive) divisors of a positive number `n`, calculated from its prime factorisation.

`sigma k n` is the sum of the `k`-th powers of the (positive) divisors of `n`. `k` must be non-negative and `n` positive. For `k == 0`, it is the divisor count (`d^0 = 1`).

σ :: Int -> Integer -> IntegerSource

Alias for `sigma` for people preferring Greek letters.

Alias for `sigma`.

The set of divisors, efficiently calculated from the canonical factorisation.

tauFromCanonical :: [(a, Int)] -> IntegerSource

The number of divisors, efficiently calculated from the canonical factorisation.

The sum of all divisors, efficiently calculated from the canonical factorisation.

sigmaFromCanonical :: Int -> [(Integer, Int)] -> IntegerSource

The sum of the powers (with fixed exponent) of all divisors, efficiently calculated from the canonical factorisation.