Wednesday, May 9, 2012

Miller Rabin primality test

I had exercise to implement Miller Rabin primality test

According to my text book, the test algorithms was as following

image

Test (n)

1. Find integers k,q,d with K>0, q odd, so that (n-1 = 2^k * q);

2. Select a random integer a,1<a<n-1;

3. if a^q mod n = 1 then return probably prime;

4. for j = 0 to k-1 do;

5. if a^2jq mod n = n-1 then return probably prime;

6. return not prime;

I also found another way of the algorithm in http://rosettacode.org/wiki/Miller-Rabin_primality_test as following

Input: n > 2, an odd integer to be tested for primality;
k, a parameter that determines the accuracy of the test
Output: composite if n is composite, otherwise probably prime
write n − 1 as 2s·d with d odd by factoring powers of 2 from n − 1
LOOP: repeat k times:
pick a randomly in the range [2, n − 1]
xad mod n
if x = 1 or x = n − 1 then do next LOOP
for r = 1 .. s − 1
xx2 mod n
if x = 1 then return composite
if x = n − 1 then do next LOOP
return composite
return probably prime


When I tried both in  C#, both worked as expected



//textbook
public static bool IsProbabilyPrime(BigInteger n, int k)
{
bool result = false;
if (n < 2)
return false;
if (n == 2)
return true;
// return false if n is even -> divisbla by 2
if (n % 2 == 0)
return false;
//writing n-1 as 2^s.d
BigInteger d = n - 1;
BigInteger s = 0;
while (d % 2 == 0)
{
d >>= 1;
s = s + 1;
}
for (int i = 0; i < k; i++)
{
BigInteger a;
do
{
a = RandomIntegerBelow(n - 2);
}
while (a < 2 || a >= n - 2);

if ( BigInteger.ModPow(a,d,n) == 1) return true;
for (int j = 0; j < s - 1; j++)
{
if (BigInteger.ModPow(a,2*j*d,n) == n - 1)
return true;
}
result = false;
}
return result;
}

//rosettacode.org
public static bool IsProbablePrime(this BigInteger source, int certainty)
{
if (source == 2 || source == 3)
return true;
if (source < 2 || source % 2 == 0)
return false;

BigInteger d = source - 1;
int s = 0;

while (d % 2 == 0)
{
d /= 2;
s += 1;
}

RandomNumberGenerator rng = RandomNumberGenerator.Create();
byte[] bytes = new byte[source.ToByteArray().LongLength];
BigInteger a;

for (int i = 0; i < certainty; i++)
{
do
{
rng.GetBytes(bytes);
a = new BigInteger(bytes);
}
while (a < 2 || a >= source - 2);

BigInteger x = BigInteger.ModPow(a, d, source);
if (x == 1 || x == source - 1)
continue;

for (int r = 1; r < s; r++)
{
x = BigInteger.ModPow(x, 2, source);
if (x == 1)
return false;
if (x == source - 1)
break;
}

if (x != source - 1)
return false;
}

return true;
}