Finding a prime with Miller Rabin - lua

Question

I have what I believe is a proper implementation of the miller-rabin algorithm using Lua, and I am trying to get a consistent return for prime numbers. It seems my implementation only works half of the time. Although if I try implementing similar code within python, that code works 100% of the time. Could someone point me in the right direction?
--decompose n-1 as (2^s)*d
local function decompose(negOne)
exponent, remainder = 0, negOne
while (remainder%2) == 0 do
exponent = exponent+1
remainder = remainder/2
end
assert((2^exponent)*remainder == negOne and ((remainder%2) == 1), "Error setting up s and d value")
return exponent, remainder
end
local function isNotWitness(n, possibleWitness, exponent, remainder)
witness = (possibleWitness^remainder)%n
if (witness == 1) or (witness == n-1) then
return false
end
for _=0, exponent do
witness = (witness^2)%n
if witness == (n-1) then
return false
end
end
return true
end
--using miller-rabin primality testing
--n the integer to be tested, k the accuracy of the test
local function isProbablyPrime(n, accuracy)
if n <= 3 then
return n == 2 or n == 3
end
if (n%2) == 0 then
return false
end
exponent, remainder = decompose(n-1)
--checks if it is composite
for i=0, accuracy do
math.randomseed(os.time())
witness = math.random(2, n - 2)
if isNotWitness(n, witness, exponent, remainder) then
return false
end
end
--probably prime
return true
end
if isProbablyPrime(31, 30) then
print("prime")
else
print("nope")
end

Answer 1

Python has arbitrary length integers. Lua doesn't.
The problem is in witness = (possibleWitness^remainder)%n.
Lua is unable to calculate exact result of 29^15 % 31 directly.
There is a workaround working for numbers n < sqrt(2^53):
witness = mulmod(possibleWitness, remainder, n)
where
local function mulmod(a, e, m)
local result = 1
while e > 0 do
if e % 2 == 1 then
result = result * a % m
e = e - 1
end
e = e / 2
a = a * a % m
end
return result
end