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Primality Testing
By
Ho, Ching Hei
Cheung, Wai Kwok
Introduction
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The primality test provides the probability
of whether or not a large number is
prime.
Several theorems including Fermat’s
theorem provide idea of primality test.
Cryptography schemes such as RSA
algorithm heavily based on primality test.
Definitions
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A Prime number is an integer that has no
integer factors other than 1 and itself. On the
other hand, it is called composite number.
A primality testing is a test to determine
whether or not a given number is prime, as
opposed to actually decomposing the number
into its constituent prime factors (which is
known as prime factorization) Use multiple
points if necessary.
Algorithms
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A Naïve Algorithm
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Pick any integer P that is greater than 2.
Try to divide P by all odd integers starting from 3
to square root of P.
If P is divisible by any one of these odd integers,
we can conclude that P is composite.
The worst case is that we have to go through all
odd number testing cases up to square root of P.
Time complexity is O(square root of N)
Algorithms (Cont.)
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Fermat’s Theorem
 Given that P is an integer that we would like to test that it is either
a PRIME or not.
 And A is another integer that is greater than zero and less than P.
 From Fermat’s Theorem, if P is a PRIME, it will satisfy this two
equalities:
 A^(p-1) = 1(mod P) or A^(p-1)mod P = 1
 A^P = A(mod P) or A^P mod P = A
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For instances, if P = 341, will P be PRIME?
-> from previous equalities, we would be able to obtain
that:
2^(341-1)mod 341 = 1, if A = 2
Algorithms (Cont.)
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It seems that 341 is a prime number under
Fermat’s Theorem. However, if A is now
equal to 3:
3^(341-1)mod 341 = 56 !!!!!!!!!
That means Fermat’s Theorem is not true in
this case!
 Time complexity is O(log n) 
Algorithms (Cont.)
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Rabin-Miller’s Probabilistic Primality Algorithm
 The Rabin-Miller’s Probabilistic Primality test was
by Rabin, based on Miller’s idea. This algorithm
provides a fast method of determining of
primality of a number with a controllably small
probability of error.
 Given (b, n), where n is the number to be tested
for primality, and b is randomly chosen in [1, n1]. Let n-1 = (2^q)*m, where m is an odd
integer.
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B^m = 1(mod n)
i[0, q-1] such that b^(m2)^i = -1(mod n)
Algorithm (Cont.)
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If the testing number satisfies either cases, it will be
said as “inconclusive”. That means it could be a prime
number.
From Fermat’s Theorem, it concludes 341 is a prime
but it is 11 * 31!
Now try to use Rabin-Miller’s Algorithm.
Let n be 341, b be 2. then assume:
q = 2 and m = 85 (since, n -1 = 2^q*m)
2^85 mod 341 = 32
Since it is not equal to 1, 341 is composite!
Time complexity is O(log N)
RSA Algorithm
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The scheme was developed by Rivest,
Shamir, and Adleman.
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The scheme was used to encrypt
plaintext into blocks in order to prevent
third party to gain access to private
message.
RSA in action:
1. Pick two large prime numbers namely p
and q and compute their product and
set it as n.
n = p*q
RSA in action (cont.):
2. Set public key to send the message.
 public key (e, n)
such that: gcd((n), e) = 1; [1<e< (n)]
 sender uses public key to encrypt the
message before sending it to the
recipient.
RSA in action (cont.):
3. Retrieve message using private key.
 at the recipient’s side, private key(d, n),
such that ed = 1mod (n), need to be
obtained in order to get the original
message through decryption.
Demonstration for RSA:
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Pick 2 primes: p=7, q=17
n = p*q
n = 119
Compute:
(n) = (119)
= (7*17)
= (7) * (17)
= 6 * 16
= 96
Find e such that gcd((n), e) = 1; [1<e< (n)]
gcd(e, 96) = 1
e=5
 public key(e, n) 
Find d such that ed = 1mod (n)
5d = 1mod96
5d = 96 * k +1, where k is some constant
5d = 96 * 4 + 1, assume k = 4
5d = 385
d = 77
 private key(d, n) 
Demonstration for Encryption:
! Base on RSA and the result we got !
 Encryption . . . (message = 19)
C = M^e mod n
= 19^5 mod 119
= 2476099 mod 119
= 66
<the original message will be encrypted with the value of 66>
Demonstration for Decryption
! Base on RSA and the result we got !
 Decryption . . .
M = C^d mod n
M = 66^77 mod 119
M = 1.27 * 10^140 mod 119
M = 19 <the original will now be recovered>
Reference:
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An Online Calculator by Ulf Wostner from CCSF
http://wiz.ccsf.edu/~uwostner/calculator/number_theory.
php
Definition of Rabin-Miller’s Probabilistic Primality Testing
http://www.ma.iup.edu/MAA/proceedings/vol1/higgins.pdf
Definition of Primality Testing
http://mathworld.wolfram.com/AKSPrimalityTest.html
Primality Test for Applications
http://www-math.mit.edu/phase2/UJM/vol1/DORSEYF.PDF