Unit 3.ppt

Unit 3.ppt

• 1.
  Cryptography and Network Security
• 2.
  Prime Numbers  primenumbers only have divisors of 1 and self  they cannot be written as a product of other numbers  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  tofactor 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.
  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  conversely can determine the greatest common divisor by comparing their prime factorizations and using least powers  Eg.18=21x32
• 5.
  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 ap = p (mod p)  useful in public key and primality testing
• 6.
  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)
• 7.
  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(pq) =(p-1)x(q-1)  eg. ø(37) = 36 ø(21) = (3–1)x(7–1) = 2x6 = 12
• 8.
  Euler's Theorem  ageneralisation of Fermat's Theorem  aø(n) = 1 (mod n)  for any a,n where gcd(a,n)=1  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
• 9.
  Primality Testing  oftenneed 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
• 10.
  Prime Distribution  primenumber theorem states that primes occur roughly every (ln n) integers  but can immediately ignore evens  so in practice need only test 0.5 ln(n) numbers of size n to locate a prime  note this is only the “average”  sometimes primes are close together  other times are quite far apart
• 11.
  Chinese Remainder Theorem used to speed up modulo computations  if working modulo a product of numbers  eg. mod M = m1m2..mk  Chinese Remainder theorem lets us work in each moduli mi separately  since computational cost is proportional to size, this is faster than working in the full modulus M
• 12.
  Chinese Remainder Theorem can implement CRT in several ways  to compute A(mod M)  first compute all ai = A mod mi separately  determine constants ci below, where Mi = M/mi  then combine results to get answer using:
• 13.
  Primitive Roots  fromEuler’s theorem have aø(n)mod n=1  consider am=1 (mod n), GCD(a,n)=1  must exist for m = ø(n) but may be smaller  once powers reach m, cycle will repeat  if smallest is m = ø(n) then a is called a primitive root  if p is prime, then successive powers of a "generate" the group mod p  these are useful but relatively hard to find
• 14.
  Discrete Logarithms  theinverse problem to exponentiation is to find the discrete logarithm of a number modulo p  that is to find x such that y = gx (mod p)  this is written as x = logg y (mod p)  if g is a primitive root then it always exists, otherwise it may not, eg. x = log3 4 mod 13 has no answer x = log2 3 mod 13 = 4 by trying successive powers  whilst exponentiation is relatively easy, finding discrete logarithms is generally a hard problem
• 15.
  Summary  have considered: prime numbers  Fermat’s and Euler’s Theorems & ø(n)  Primality Testing  Chinese Remainder Theorem  Discrete Logarithms