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Ingy döt Net
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## Nim currently doesn't have a BigInt standard library
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## so we translate the version from Go which uses a
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## deterministic approach, which is correct for all
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## possible values in uint32.
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proc isPrime*(n: uint32): bool =
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# bases of 2, 7, 61 are sufficient to cover 2^32
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case n
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of 0, 1: return false
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of 2, 7, 61: return true
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else: discard
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var
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nm1 = n-1
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d = nm1.int
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s = 0
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n = n.uint64
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while d mod 2 == 0:
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d = d shr 1
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s += 1
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for a in [2, 7, 61]:
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var
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x = 1.uint64
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p = a.uint64
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dr = d
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while dr > 0:
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if dr mod 2 == 1:
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x = x * p mod n
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p = p * p mod n
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dr = dr shr 1
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if x == 1 or x.uint32 == nm1:
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continue
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var r = 1
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while true:
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if r >= s:
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return false
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x = x * x mod n
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if x == 1:
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return false
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if x.uint32 == nm1:
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break
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r += 1
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return true
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proc isPrime*(n: int32): bool =
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## Overload for int32
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n >= 0 and n.uint32.isPrime
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when isMainModule:
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const primeNumber1000 = 7919 # source: https://en.wikipedia.org/wiki/List_of_prime_numbers
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var
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i = 0u32
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numberPrimes = 0
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while true:
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if isPrime(i):
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if numberPrimes == 999:
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break
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numberPrimes += 1
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i += 1
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assert i == primeNumber1000
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assert isPrime(2u32)
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assert isPrime(31u32)
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assert isPrime(37u32)
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assert isPrime(1123u32)
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assert isPrime(492366587u32)
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assert isPrime(1645333507u32)
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@ -0,0 +1,140 @@
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# Compile as: $ nim c -d:release mrtest.nim
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# Run using: $ ./mrtest
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import math # for gcd and mod
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import bitops # for countTrailingZeroBits
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import strutils, typetraits # for number input
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import times, os # for timing code execution
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proc addmod*[T: SomeInteger](a, b, modulus: T): T =
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## Modular addition
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let a_m = if a < modulus: a else: a mod modulus
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if b == 0.T: return a_m
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let b_m = if b < modulus: b else: b mod modulus
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# Avoid doing a + b that could overflow here
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let b_from_m = modulus - b_m
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if a_m >= b_from_m: return a_m - b_from_m
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return a_m + b_m # safe to add here; a + b < modulus
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proc mulmod*[T: SomeInteger](a, b, modulus: T): T =
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## Modular multiplication
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var a_m = if a < modulus: a else: a mod modulus
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var b_m = if b < modulus: b else: b mod modulus
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if b_m > a_m: swap(a_m, b_m)
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while b_m > 0.T:
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if (b_m and 1) == 1: result = addmod(result, a_m, modulus)
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a_m = (a_m shl 1) - (if a_m >= (modulus - a_m): modulus else: 0)
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b_m = b_m shr 1
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proc expmod*[T: SomeInteger](base, exponent, modulus: T): T =
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## Modular exponentiation
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result = 1 # (exp 0 = 1)
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var (e, b) = (exponent, base)
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while e > 0.T:
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if (e and 1) == 1: result = mulmod(result, b, modulus)
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e = e shr 1
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b = mulmod(b, b, modulus)
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# Returns true if +self+ passes Miller-Rabin Test on witnesses +b+
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proc miller_rabin_test[T: SomeInteger](num: T, witnesses: seq[uint64]): bool =
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var d = num - 1
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let (neg_one_mod, n) = (d, d)
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d = d shr countTrailingZeroBits(d) # suck out factors of 2 from d
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for b in witnesses: # do M-R test with each witness base
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if b.T mod num == 0: continue # **skip base if a multiple of input**
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var s = d
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var y = expmod(b.T, d, num)
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while s != n and y != 1 and y != neg_one_mod:
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y = mulmod(y, y, num)
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s = s shl 1
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if y != neg_one_mod and (s and 1) != 1: return false
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true
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proc selectWitnesses[T: SomeInteger](num: T): seq[uint64] =
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## Best known deterministic witnesses for given range and number of bases
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## https://miller-rabin.appspot.com/
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## https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test
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if num < 341_531u:
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result = @[9345883071009581737u64]
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elif num < 1_050_535_501u:
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result = @[336781006125u64, 9639812373923155u64]
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elif num < 350_269_456_337u:
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result = @[4230279247111683200u64, 14694767155120705706u64, 16641139526367750375u64]
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elif num < 55_245_642_489_451u:
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result = @[2u64, 141889084524735u64, 1199124725622454117u64, 11096072698276303650u64]
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elif num < 7_999_252_175_582_851u:
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result = @[2u64, 4130806001517u64, 149795463772692060u64, 186635894390467037u64, 3967304179347715805u64]
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elif num < 585_226_005_592_931_977u:
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result = @[2u64, 123635709730000u64, 9233062284813009u64, 43835965440333360u64, 761179012939631437u64, 1263739024124850375u64]
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elif num.uint64 < 18_446_744_073_709_551_615u64:
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result = @[2u64, 325, 9375, 28178, 450775, 9780504, 1795265022]
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else:
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result = @[2u64, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
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proc primemr*[T: SomeInteger](n: T): bool =
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let primes = @[2u64, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
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if n <= primes[^1].T: return (n in primes) # for n <= primes.last
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let modp47 = 614889782588491410u # => primes.product, largest < 2^64
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if gcd(n, modp47) != 1: return false # eliminates 86.2% of all integers
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let witnesses = selectWitnesses(n)
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miller_rabin_test(n, witnesses)
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echo "\nprimemr?"
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echo("n = ", 1645333507u)
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var te = epochTime()
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echo primemr 1645333507u
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echo (epochTime()-te).formatFloat(ffDecimal, 6)
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echo "\nprimemr?"
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echo("n = ", 2147483647u)
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te = epochTime()
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echo primemr 2147483647u
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echo (epochTime()-te).formatFloat(ffDecimal, 6)
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echo "\nprimemr?"
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echo("n = ", 844674407370955389u)
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te = epochTime()
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echo primemr 844674407370955389u
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echo (epochTime()-te).formatFloat(ffDecimal, 6)
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echo "\nprimemr?"
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echo("n = ", 1844674407370954349u)
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te = epochTime()
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echo primemr 1844674407370954349u
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echo (epochTime()-te).formatFloat(ffDecimal, 6)
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echo "\nprimemr?"
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echo("n = ", 1844674407370954351u)
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te = epochTime()
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echo primemr 1844674407370954351u
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echo (epochTime()-te).formatFloat(ffDecimal, 6)
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echo "\nprimemr?"
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echo("n = ", 9223372036854775783u)
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te = epochTime()
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echo primemr 9223372036854775783u
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echo (epochTime()-te).formatFloat(ffDecimal, 6)
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echo "\nprimemr?"
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echo("n = ", 9241386435364257883u64)
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te = epochTime()
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echo primemr 9241386435364257883u64
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echo (epochTime()-te).formatFloat(ffDecimal, 6)
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echo "\nprimemr?"
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echo("n = ", 18446744073709551533u64, ", is largest prime < 2^64")
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te = epochTime()
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echo 18446744073709551533u64.primemr
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echo (epochTime()-te).formatFloat(ffDecimal, 6)
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echo "\nprimemr?"
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let num = 5_000_000u # => 348_513 primes
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var primes: seq[uint] = @[]
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echo("find primes < ", num)
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te = epochTime()
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for n in 0u..num:
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if n.primemr: primes.add(n)
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stdout.write("\r",((float64(n) / float64(num))*100).formatFloat(ffDecimal, 1), "%")
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echo("\nnumber of primes < ",num, " are ", primes.len)
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echo (epochTime()-te).formatFloat(ffDecimal, 6)
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@ -0,0 +1,41 @@
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(* Translated from the wikipedia pseudo-code *)
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let miller_rabin n ~iter:k =
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(* return r and d where n = 2^r*d (from scheme implementation) *)
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let get_rd n =
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let rec loop r d =
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(* not even *)
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if Z.(equal (logand d one) one) then
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(r,d)
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else
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loop Z.(r + one) Z.(div d ~$2)
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in
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loop Z.zero n
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in
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let single_miller n r d =
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(* (random (n - 4)) + 2 *)
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let a = Bigint.to_zarith_bigint
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Bigint.((random ((of_zarith_bigint n) - (of_int 4))) + (of_int 2))
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in
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let x = Z.(powm a d n) in
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if Z.(equal x ~$1) || Z.(equal x (n - ~$1)) then true
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else
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let rec loop i x =
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if Z.(equal ~$i (r - ~$1)) then false
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else
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let x = Z.(powm x ~$2 n) in
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if Z.(equal x (n - ~$1)) then true
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else loop (i + 1) x
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in
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loop 0 x
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in
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let n = Z.abs n in
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if Z.(equal n one) then false
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else if Z.(equal (logand n one) zero) then false
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else if Z.(equal (n mod ~$3) zero) then false
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else
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let r, d = get_rd Z.(n - one) in
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let rec loop i bool =
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if i = k then bool
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else loop (i + 1) (bool && single_miller n r d)
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in
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loop 0 true
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