Primality Testing

1
Primality Testing

• By
• Ho, Ching Hei
• Cheung, Wai Kwok

2
Introduction

• The primality test provides the probability of
  whether or not a large number is prime.
• Several theorems including Fermats theorem
  provide idea of primality test.
• Cryptography schemes such as RSA algorithm
  heavily based on primality test.

3
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.

4
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)

5
Algorithms (Cont.)

• Fermats 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 Fermats 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
• AP A(mod P) or AP mod P A
• For instances, if P 341, will P be PRIME?
• -gt from previous equalities, we would be able to
  obtain that
• 2(341-1)mod 341 1, if A 2

6
Algorithms (Cont.)

• It seems that 341 is a prime number under
  Fermats Theorem. However, if A is now equal to
  3
• 3(341-1)mod 341 56 !!!!!!!!!
• That means Fermats Theorem is not true in this
  case!
• ? Time complexity is O(log n) ?

7
Algorithms (Cont.)

• Rabin-Millers Probabilistic Primality Algorithm
  
• The Rabin-Millers Probabilistic Primality test
  was by Rabin, based on Millers 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 (2q)m, where m is an odd
  integer.
• Bm 1(mod n)
• ?i?0, q-1 such that b(m2)i -1(mod n)

8
Algorithm (Cont.)

• If the testing number satisfies either cases, it
  will be said as inconclusive. That means it
  could be a prime number.
• From Fermats Theorem, it concludes 341 is a
  prime but it is 11 31!
• Now try to use Rabin-Millers Algorithm.
• Let n be 341, b be 2. then assume
• q 2 and m 85 (since, n -1 2qm)
• 285 mod 341 32
• Since it is not equal to 1, 341 is composite!
• Time complexity is O(log N)

9
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.

10
RSA in action

• 1. Pick two large prime numbers namely p and q
  and compute their product and set it as n.
• n pq

11
RSA in action (cont.)

• 2. Set public key to send the message.
• public key (e, n)
• such that gcd(?(n), e) 1 1ltelt ?(n)
• sender uses public key to encrypt the
• message before sending it to the recipient.

12
RSA in action (cont.)

• 3. Retrieve message using private key.
• at the recipients side, private key(d, n), such
  that ed 1mod ?(n), need to be obtained in order
  to get the original message through decryption.

13
Demonstration for RSA

• Pick 2 primes p7, q17
• n pq
• n 119
• Compute
• ?(n) ?(119)
• ?(717)
• ?(7) ?(17)
• 6 16
• 96
• Find e such that gcd(?(n), e) 1 1ltelt ?(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) ?

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Demonstration for Encryption

• ! Base on RSA and the result we got !
• Encryption . . . (message 19)
• C Me mod n
• 195 mod 119
• 2476099 mod 119
• 66 ltthe original message will be encrypted
  with the value of 66gt

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Demonstration for Decryption

• ! Base on RSA and the result we got !
• Decryption . . .
• M Cd mod n
• M 6677 mod 119
• M 1.27 10140 mod 119
• M 19 ltthe original will now be recoveredgt

16
Reference

• An Online Calculator by Ulf Wostner from CCSF
  http//wiz.ccsf.edu/uwostner/calculator/number_th
  eory.php
• Definition of Rabin-Millers Probabilistic
  Primality Testing http//www.ma.iup.edu/MAA/procee
  dings/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