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make isprime(::BigInt) use deterministic algo when possible #54

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make isprime(::BigInt) use deterministic algo when possible #54

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51 changes: 26 additions & 25 deletions src/Primes.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -135,12 +135,23 @@ const PRIMES = primes(2^16)

"""
isprime(n::Integer) -> Bool
isprime(n::Integer, reps) -> Bool

Returns `true` if `n` is prime, and `false` otherwise.

If `n ≤ 2^64` and `reps` is not specified, this is a deterministic test.
Otherwise, the test is probabilistic, with a false positive rate less than
`1/4^reps`.

A value of `reps = 25` is considered safe for cryptographic applications
(Knuth, Seminumerical Algorithms), and is the default when `n > 2^64`.

```julia
julia> isprime(3)
true

juli> isprime(big(2)^130-1, 10)
false
```
"""
function isprime(n::Integer)
Expand All @@ -166,20 +177,8 @@ function isprime(n::Integer)
return true
end

"""
isprime(x::BigInt, [reps = 25]) -> Bool

Probabilistic primality test. Returns `true` if `x` is prime with high probability (pseudoprime);
and `false` if `x` is composite (not prime). The false positive rate is about `0.25^reps`.
`reps = 25` is considered safe for cryptographic applications (Knuth, Seminumerical Algorithms).

```julia
julia> isprime(big(3))
true
```
"""
isprime(x::BigInt, reps=25) = ccall((:__gmpz_probab_prime_p,:libgmp), Cint, (Ptr{BigInt}, Cint), &x, reps) > 0

isprime(x::Integer, reps::Integer) =
ccall((:__gmpz_probab_prime_p,:libgmp), Cint, (Ptr{BigInt}, Cint), &(big(x)), reps) > 0

# Miller-Rabin witness choices based on:
# http://mathoverflow.net/questions/101922/smallest-collection-of-bases-for-prime-testing-of-64-bit-numbers
Expand Down Expand Up @@ -217,19 +216,21 @@ function _witnesses(n::UInt64)
i = xor((n >> 16), n) * 0x45d9f3b
i = xor((i >> 16), i) * 0x45d9f3b
i = xor((i >> 16), i) & 255 + 1
@inbounds return (Int(bases[i]),)
@inbounds return (bases[i] % Int,)
end
witnesses(n::Integer) =
n < 4294967296 ? _witnesses(UInt64(n)) :
n < 2152302898747 ? (2, 3, 5, 7, 11) :
n < 3474749660383 ? (2, 3, 5, 7, 11, 13) :
(2, 325, 9375, 28178, 450775, 9780504, 1795265022)

isprime(n::UInt128) =
n ≤ typemax(UInt64) ? isprime(UInt64(n)) : isprime(BigInt(n))
isprime(n::Int128) = n < 2 ? false :
n ≤ typemax(Int64) ? isprime(Int64(n)) : isprime(BigInt(n))

n < 4_294_967_296 ? _witnesses(n % UInt64) :
n < 2_152_302_898_747 ? (2, 3, 5, 7, 11) :
n < 3_474_749_660_383 ? (2, 3, 5, 7, 11, 13) :
(2, 325, 9375, 28178, 450775, 9780504, 1795265022)

# the isprime implementation works faster with Unsigned types
# than with their signed counterparts
isprime(n::Base.BitSigned) = n < 2 ? false : isprime(unsigned(n))
isprime(n::Union{UInt128,BigInt}) =
n < 2 ? false :
n ≤ typemax(UInt64) ? isprime(n % UInt64) :
isprime(n, 25)

# Trial division of small (< 2^16) precomputed primes +
# Pollard rho's algorithm with Richard P. Brent optimizations
Expand Down