Information and network security 34 primality

Information and network security 34 primality

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  Information and NetworkSecurity:34 Primality Prof Neeraj Bhargava Vaibhav Khanna Department of Computer Science School of Engineering and Systems Sciences Maharshi Dayanand Saraswati University Ajmer
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  Prime Numbers prime numbersonly 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
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  Prime Factorisation to factora 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
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  • The ideaof "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.
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  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
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  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.
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  Euler Totient Functionø(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)
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  Euler Totient Functionø(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
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  Euler's Theorem • ageneralisation 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)
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  Primality Testing often needto 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
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  • For manycryptographic 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.
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  Assignment • Explain indetail the working of Primality Algorithm