Primality Testing - Data Structures - Lecture Slides, Slides of Data Structures and Algorithms

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Primality Testing

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.

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)

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

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)

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)

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(φ(n), e) = 1; [1<e< φ(n)]

• sender uses public key to encrypt the

message before sending it to the recipient.

Demonstration for RSA:

• Pick 2 primes: p=7, q= 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 = 1mod 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>