The document discusses primality testing, which determines whether a number is prime, highlighting its importance in cryptography, particularly the RSA algorithm. Various techniques such as naive algorithms, Fermat's theorem, and the Rabin-Miller probabilistic primality test are presented, along with their time complexities and examples. It also details the RSA encryption and decryption process using prime numbers and their derived keys.

1. Bharathiar University
Department of Computer Applications
N.Mohanasundaram
II - MCA

2. Primality Testing

3. Introduction
 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.

4. Definitions
 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.

5. Algorithms
 A Naïve Algorithm
 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)

6. Algorithms (Cont.)
 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
 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

7. Algorithms (Cont.)
 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) ⇐

8. Algorithms (Cont.)
 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, n-1].
Let n-1 = (2^q)*m, where m is an odd integer.
 B^m = 1(mod n)
 ∃i∈[0, q-1] such that b^(m2)^i = -1(mod n)

9. Algorithm (Cont.)
 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)

10. RSA Algorithm
 The scheme was developed by Rivest,
Shamir, and Adleman.
 The scheme was used to encrypt
plaintext into blocks in order to prevent
third party to gain access to private
message.

11. 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

12. RSA in action (cont.):
2. Set public key to send the message.
 public key (e, n)
such that: gcd(φ(n), e) = 1; [1<e< φ(n)]
 sender uses public key to encrypt the
message before sending it to the
recipient.

13. RSA in action (cont.):
3. Retrieve message using private key.
 at the recipient’s side, private key(d, n),
such that ed = 1mod φ(n), need to be
obtained in order to get the original
message through decryption.

14. Demonstration for RSA:
 Pick 2 primes: p=7, q=17
n = p*q
n = 119
 Compute:
φ(n) = φ(119)
= φ(7*17)
= φ(7) * φ(17)
= 6 * 16
= 96
 Find e such that gcd(φ(n), e) = 1; [1<e< φ(n)]
gcd(e, 96) = 1
e = 5 ⇒ public key(e, n) ⇐
 Find d such that ed = 1mod φ(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) ⇐

15. 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>

16. 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>

17. Reference:
 An Online Calculator by Ulf Wostner from CCSF
http://wiz.ccsf.edu/~uwostner/calculator/number_theory.p
hp
 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/DORSEY-
F.PDF

18. Thank you