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Information and Network Security:34
Primality
Prof Neeraj Bhargava
Vaibhav Khanna
Department of Computer Science
School of Engineering and Systems Sciences
Maharshi Dayanand Saraswati University Ajmer
Prime Numbers
prime numbers only have divisors of 1 and self
they cannot be written as a product of other numbers
note: 1 is prime, but is generally not of interest
eg. 2,3,5,7 are prime, 4,6,8,9,10 are not
prime numbers are central to number theory
list of prime number less than 200 is:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79
83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163
167 173 179 181 191 193 197 199
Prime Factorisation
to factor a number n is to write it as a product of other numbers:
n=a x b x c
note that factoring a number is relatively hard compared to
multiplying the factors together to generate the number
the prime factorisation of a number n is when its written as a
product of primes
eg. 91=7x13 ; 3600=24x32x52
• The idea of "factoring" a number is important - finding numbers
which divide into it.
• Taking this as far as can go, by factorising all the factors, we can
eventually write the number as a product of (powers of) primes - its
prime factorisation.
• Note also that factoring a number is relatively hard compared to
multiplying the factors together to generate the number.
Relatively Prime Numbers & GCD
• two numbers a, b are relatively prime if have no common divisors apart
from 1
• eg. 8 & 15 are relatively prime since factors of 8 are 1,2,4,8 and of 15 are 1,3,5,15
and 1 is the only common factor
• Have the concept of “relatively prime” if two number share no common
factors other than 1.
• Another common problem is to determine the "greatest common divisor”
GCD(a,b) which is the largest number that divides into both a & b.
• conversely can determine the greatest common divisor by comparing their
prime factorizations and using least powers
• eg. 300=21x31x52 18=21x32 hence GCD(18,300)=21x31x50=6
Fermat's Theorem
• ap-1 = 1 (mod p)
• where p is prime and gcd(a,p)=1
• also known as Fermat’s Little Theorem
• also have: ap = a (mod p)
• useful in public key and primality testing
• Two theorems that play important roles in public-key cryptography
are Fermat’s theorem and Euler’s theorem.
• Fermat’s theorem (also known as Fermat’s Little Theorem) as listed
above, states an important property of prime numbers.
Euler Totient Function ø(n)
• when doing arithmetic modulo n
• complete set of residues is: 0..n-1
• reduced set of residues is those numbers (residues) which are
relatively prime to n
• eg for n=10,
• complete set of residues is {0,1,2,3,4,5,6,7,8,9}
• reduced set of residues is {1,3,7,9}
• number of elements in reduced set of residues is called the Euler
Totient Function ø(n)
Euler Totient Function ø(n)
• to compute ø(n) need to count number of residues to be excluded
• in general need prime factorization, but
• for p (p prime) ø(p)=p-1
• for p.q (p,q prime) ø(p.q)=(p-1)x(q-1)
• compute ø(n) need to count the number of residues to be excluded.
In general you need use a complex formula on the prime factorization
of n, but have a couple of special cases as shown.
• eg.
ø(37) = 36
ø(21) = (3–1)x(7–1) = 2x6 = 12
Euler's Theorem
• a generalisation of Fermat's Theorem
• aø(n) = 1 (mod n)
• for any a,n where gcd(a,n)=1
• Euler's Theorem is a generalization of Fermat's Theorem for any number n.
• As is the case for Fermat's theorem, an alternative form of the theorem is also
useful. Again, similar to the case with Fermat's theorem, the first form of Euler's
theorem requires that a be relatively prime to n.
• eg.
a=3;n=10; ø(10)=4;
hence 34 = 81 = 1 mod 10
a=2;n=11; ø(11)=10;
hence 210 = 1024 = 1 mod 11
• also have: aø(n)+1 = a (mod n)
Primality Testing
often need to find large prime numbers
traditionally sieve using trial division
ie. divide by all numbers (primes) in turn less than the square root of the
number
only works for small numbers
alternatively can use statistical primality tests based on properties of
primes
for which all primes numbers satisfy property
but some composite numbers, called pseudo-primes, also satisfy the property
can use a slower deterministic primality test
• For many cryptographic functions it is necessary to select one or
more very large prime numbers at random. Thus we are faced with
the task of determining whether a given large number is prime.
• There is no simple yet efficient means of accomplishing this task.
• Traditionally sieve for primes using trial division of all possible prime
factors of some number, but this only works for small numbers.
• Alternatively can use repeated statistical primality tests based on
properties of primes, and then for certainty, use a slower
deterministic primality test, such as the AKS test.
Assignment
• Explain in detail the working of Primality Algorithm

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Information and network security 34 primality

  • 1. Information and Network Security:34 Primality Prof Neeraj Bhargava Vaibhav Khanna Department of Computer Science School of Engineering and Systems Sciences Maharshi Dayanand Saraswati University Ajmer
  • 2. Prime Numbers prime numbers only have divisors of 1 and self they cannot be written as a product of other numbers note: 1 is prime, but is generally not of interest eg. 2,3,5,7 are prime, 4,6,8,9,10 are not prime numbers are central to number theory list of prime number less than 200 is: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199
  • 3. Prime Factorisation to factor a number n is to write it as a product of other numbers: n=a x b x c note that factoring a number is relatively hard compared to multiplying the factors together to generate the number the prime factorisation of a number n is when its written as a product of primes eg. 91=7x13 ; 3600=24x32x52
  • 4. • The idea of "factoring" a number is important - finding numbers which divide into it. • Taking this as far as can go, by factorising all the factors, we can eventually write the number as a product of (powers of) primes - its prime factorisation. • Note also that factoring a number is relatively hard compared to multiplying the factors together to generate the number.
  • 5. Relatively Prime Numbers & GCD • two numbers a, b are relatively prime if have no common divisors apart from 1 • eg. 8 & 15 are relatively prime since factors of 8 are 1,2,4,8 and of 15 are 1,3,5,15 and 1 is the only common factor • Have the concept of “relatively prime” if two number share no common factors other than 1. • Another common problem is to determine the "greatest common divisor” GCD(a,b) which is the largest number that divides into both a & b. • conversely can determine the greatest common divisor by comparing their prime factorizations and using least powers • eg. 300=21x31x52 18=21x32 hence GCD(18,300)=21x31x50=6
  • 6. Fermat's Theorem • ap-1 = 1 (mod p) • where p is prime and gcd(a,p)=1 • also known as Fermat’s Little Theorem • also have: ap = a (mod p) • useful in public key and primality testing • Two theorems that play important roles in public-key cryptography are Fermat’s theorem and Euler’s theorem. • Fermat’s theorem (also known as Fermat’s Little Theorem) as listed above, states an important property of prime numbers.
  • 7. Euler Totient Function ø(n) • when doing arithmetic modulo n • complete set of residues is: 0..n-1 • reduced set of residues is those numbers (residues) which are relatively prime to n • eg for n=10, • complete set of residues is {0,1,2,3,4,5,6,7,8,9} • reduced set of residues is {1,3,7,9} • number of elements in reduced set of residues is called the Euler Totient Function ø(n)
  • 8. Euler Totient Function ø(n) • to compute ø(n) need to count number of residues to be excluded • in general need prime factorization, but • for p (p prime) ø(p)=p-1 • for p.q (p,q prime) ø(p.q)=(p-1)x(q-1) • compute ø(n) need to count the number of residues to be excluded. In general you need use a complex formula on the prime factorization of n, but have a couple of special cases as shown. • eg. ø(37) = 36 ø(21) = (3–1)x(7–1) = 2x6 = 12
  • 9. Euler's Theorem • a generalisation of Fermat's Theorem • aø(n) = 1 (mod n) • for any a,n where gcd(a,n)=1 • Euler's Theorem is a generalization of Fermat's Theorem for any number n. • As is the case for Fermat's theorem, an alternative form of the theorem is also useful. Again, similar to the case with Fermat's theorem, the first form of Euler's theorem requires that a be relatively prime to n. • eg. a=3;n=10; ø(10)=4; hence 34 = 81 = 1 mod 10 a=2;n=11; ø(11)=10; hence 210 = 1024 = 1 mod 11 • also have: aø(n)+1 = a (mod n)
  • 10. Primality Testing often need to find large prime numbers traditionally sieve using trial division ie. divide by all numbers (primes) in turn less than the square root of the number only works for small numbers alternatively can use statistical primality tests based on properties of primes for which all primes numbers satisfy property but some composite numbers, called pseudo-primes, also satisfy the property can use a slower deterministic primality test
  • 11. • For many cryptographic functions it is necessary to select one or more very large prime numbers at random. Thus we are faced with the task of determining whether a given large number is prime. • There is no simple yet efficient means of accomplishing this task. • Traditionally sieve for primes using trial division of all possible prime factors of some number, but this only works for small numbers. • Alternatively can use repeated statistical primality tests based on properties of primes, and then for certainty, use a slower deterministic primality test, such as the AKS test.
  • 12. Assignment • Explain in detail the working of Primality Algorithm